MATHEMATICAL MODELLING AND ANALYSIS MODELLING/vo… · During the reign of Kestutis Naujieji Trakai...

ISSN 1392-6292

Institute of Mathematics, Latvian Academy of Sciences·University of Latvia

Institute of Mathematics and Informatics, Vilnius

University of Tartu

Vilnius Gediminas Technical University

MATHEMATICAL

MODELLING

AND ANALYSIS

THE BALTIC JOURNAL

ON MATHEMATICAL

APPLICATIONS,

NUMERICAL ANALYSIS

AND DIFFERENTIAL EQUATIONS

Volume 9 Number 1 2004

Electronical edition available at

http://www.vtu.lt/rc/mma/

Vilnius Technika 2004

Editor-in-Chief R. ČiegisVilnius Gediminas Technical University

Saulėtekio al. 11, LT-10223 Vilnius, Lithuania

[email protected]

Executive Editor A. ŠtikonasInstitute of Mathematics and Informatics

Akademijos 4, LT-08663 Vilnius, Lithuania

[email protected]

International Editorial Board

V. Abrashin (Belarus) Institute of Mathematics, Minsk [email protected]

A. Buikis (Latvia) Institute of Mathematics of LAS and UL, Riga [email protected]

A. Fitt (UK) University of Southampton [email protected]

V. Gromak (Belarus) Belarussian State Univ., Minsk [email protected]

O. Iliev (Germany) Institute for Industrial Mathematics, Kaiserslautern

[email protected]

F. Ivanauskas (Lithuania) Vilnius University [email protected]

J. Janno (Estonia) Tallinn Technical University [email protected]

H. Kalis (Latvia) Institute of Mathematics of LAS and UL, Riga [email protected]

V. Korzyuk (Belarus) Belarussian State Univ., Minsk [email protected]

A. Krylovas (Lithuania) Vilnius Gediminas Technical University [email protected]

Ü. Lepik (Estonia) University of Tartu [email protected]

P. Matus (Belarus) Institute of Mathematics, Minsk [email protected]

M. Meilūnas (Lithuania) Vilnius Gediminas Technical University [email protected]

H. Ockendon (UK) Oxford Centre for Industrial and Applied Mathematics

[email protected]

W. Okrasinski (Poland) University of Zielona Gora [email protected]

A. Pedas (Estonia) University of Tartu [email protected]

K. Pileckas (Lithuania) Institute of Mathematics and Informatics [email protected]

U. Raitums (Latvia) Institute of Mathematics of LAS and UL, Riga

[email protected]

A. Reinfelds (Latvia) Institute of Mathematics of LAS and UL, Riga [email protected]

J. Rokicki (Poland) Institute of Aeronautics, Warsaw [email protected]

S. Rutkauskas (Lithuania) Institute of Mathematics and Informatics, Vilnius

[email protected]

M. Sapagovas (Lithuania) Institute of Mathematics and Informatics, Vilnius

[email protected]

E. Tamme (Estonia) University of Tartu [email protected]

P. Vabishchevich (Russia) Institute for Mathematical Modelling, Moscow

[email protected]

G. Vainikko (Estonia) Estonian Academy of Sciences [email protected]

M. Weber (Germany) Technische Universität Dresden

[email protected]

G. Yelenin (Russia) M.V. Lomonosov Moscow State University [email protected]

A. Zemitis (Latvija) Ventspils University College [email protected]

A. Zlotnik (Russia) Power Engineering Institute, Moscow

[email protected]

©Vilnius Gediminas

Technical University, 2004

Mathematical Modelling and Analysis, Volume 9 Number 1, 2004 i

CONTENTS

K. Batrakov, S. Sytova

Modelling of quasi-Cherenkov electron beam instability in periodical struc-

tures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

J. Dhar

A prey-predator model with diffusion and a supplementary resource for the

prey in a two-patch environment . . . . . . . . . . . . . . . . . . . . . . . . 9

O. Dumbrajs, H. Kalis, A. Reinfelds

Numerical solution of single mode gyrotron equation . . . . . . . . . . . . . 25

R. Garška, I.Krūminienė

Spatial analysis and prediction of Curonian lagoon data with Gstat . . . . . . 39

J. Sieber, M. Radžiūnas,K. R. SchneiderDynamics of multisection semiconductor lasers . . . . . . . . . . . . . . . 51

J. Socolowsky

On the existence and uniqueness of two-fluid channel flows . . . . . . . . . . 67

M. Tarang

Stability of the spline collocation method for second order Volterra integro-

differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

ISSN 1392-6292

MATHEMATICALMODELLINGAND ANALYSISTHE BALTIC JOURNALON MATHEMATICALAPPLICATIONS,NUMERICAL ANALYSISAND DIFFERENTIAL EQUATIONS


Electronical edition available at << : ;<1= : ?< 6 647

Abstracted/Indexed in: 7 < 67 < 7? ; <7?#?17 <1< 9 7 < 67 < 9 7<; = 7? !"#"%$"& 9' 67 <' =#? 7 <(' *)


ISSN 1392-6292



University of TartuVilnius Gediminas Technical University


Electronical edition available at << >:;<1= : ?<( 6 647


Editor-in-Chief R. CiegisVilnius Gediminas Technical UniversitySauletekio al. 11, LT-10223 Vilnius, Lithuania

Executive Editor A. ŠtikonasInstitute of Mathematics and InformaticsAkademijos 4, LT-08663 Vilnius, Lithuania

International Editorial BoardV. Abrashin (Belarus) Institute of Mathematics, Minsk !" # $A. Buikis (Latvia) Institute of Mathematics of LAS and UL, Riga &% " #'A. Fitt (UK) University of Southampton ( )*+ +) %V. Gromak (Belarus) Belarussian State Univ., Minsk , +&&-. - -"/0$O. Iliev (Germany) Institute for Industrial Mathematics, Kaiserslautern" 1 !%-)("F. Ivanauskas (Lithuania) Vilnius University 2 "%3*4%5#' J. Janno (Estonia) Tallinn Technical University 6 ++3""H. Kalis (Latvia) Institute of Mathematics of LAS and UL, Riga % " .V. Korzyuk (Belarus) Belarussian State Univ., Minsk 7 +8$%+ , 9 "/ $A. Krylovas (Lithuania) Vilnius Gediminas Technical University % Ü. Lepik (Estonia) University of Tartu $":-% #""P. Matus (Belarus) Institute of Mathematics, Minsk -0!" . $M. Meilunas (Lithuania) Vilnius Gediminas Technical University ; ' H. Ockendon (UK) Oxford Centre for Industrial and Applied Mathematics+&%"(+ )'+<#) %W. Okrasinski (Poland) University of Zielona Gora = '>%-%? 8.;8 , +/ :-A. Pedas (Estonia) University of Tartu " . :"( #""K. Pileckas (Lithuania) Institute of Mathematics and Informatics :"%% /@A)' U. Raitums (Latvia) Institute of Mathematics of LAS and UL, Riga-() A53''A. Reinfelds (Latvia) Institute of Mathematics of LAS and UL, Riga " " #'J. Rokicki (Poland) Institute of Aeronautics, Warsaw 6 %&"/0:19"( :-S. Rutkauskas (Lithuania) Institute of Mathematics and Informatics, Vilnius $% /B5)' M. Sapagovas (Lithuania) Institute of Mathematics and Informatics, VilniusC D: , +% /@A)' E. Tamme (Estonia) University of Tartu "-E .""P. Vabishchevich (Russia) Institute for Mathematical Modelling, Moscow"#3G. Vainikko (Estonia) Estonian Academy of Sciences , "();&%%-+ .M. Weber (Germany) Technische Universität Dresden1""- 5!("(";("G. Yelenin (Russia) M.V. Lomonosov Moscow State University $""-+)A. Zemitis (Latvija) Ventspils University College 8"5 " #'A. Zlotnik (Russia) Power Engineering Institute, Moscow8+ %:5&@:")3

c©Vilnius GediminasTechnical University, 2004

Mathematical Modelling and Analysis, Volume 9 Number 2, 2004 a

10th International ConferenceMathematical Modelling and Analysis

and2nd International Conference

Computational Methods in Applied MathematicsJune 1 – 5, 2005, Trakai, Lithuania

"!#%$'&%($*))%+,.--/

Conference organizers: The European Consortium for Mathematics in Industry(ECMI), Vilnius Gediminas Technical University, Institute of Mathematics and In-formatics, Vilnius University and Journal Computational Methods in Applied Math-ematics.

Aims: The Conference focuses on various aspects of mathematical modellingand usage of finite difference and finite element methods for numerical solution ofmodern problems of science and engineering. It aims, in particular, at fostering co-operation among practitioners and theoreticians in this field. Another very importantgoal of the MMA and CMAM meetings is to assist the creation and maintenanceof contacts between scientists from the West and the East. Working language of theConference is English.

The basic topics:Analysis of numerical methods for solving problems of mathematical physics;Parallel algorithms and parallel computing;Application of numerical methods to engineering problems;Analysis of ODE and PDE problems and applications;Navier - Stokes equations and Computational Fluid Dynamics;Image processing;Financial mathematics and mathematics in economics;Scientific computation.

The scientific program includes invited plenary talks (40 min) and contributedtalks (20 min). We also invite participants to organize minisymposiums. The formatfor a minisymposium might be a principal lecture and a number of communicationsof 15-20 minutes given by the other speakers. The minisymposia will be scheduledin parallel sessions.

International Organizing CommitteeR. Ciegis (Lithuania - Chairman), P. Matus (Belarus - Vice-chairman), A. Štikonas

(Lithuania - Scientific secretary), R. Belevicius (Lithuania), A. Buikis (Latvia),I. Gaishun (Belarus), P. Hemker (Netherlands), O. Iliev (Germany), A. Iakubenia(Belarus), F. Ivanauskas (Lithuania), A. Krylovas (Lithuania), R. Lazarov (USA),M. Meil unas (Lithuania), H. Neunzert (Germany), A. Pedas (Estonia), U. Raitums(Latvia), A. Reinfelds (Latvia), J. Rokicki (Poland), M. Sapagovas (Lithuania),P. Vabishchevich (Russia), G. Vainikko (Estonia), A. Zemitis (Latvia)

b Editor-in-Chief R. Ciegis

Abstracts and ProceedingsAuthors are requested to send an abstract (1 page) before March 30, 2005. In-

structions and a style file for the preparation of the abstracts are available at << : ;<1= :@? < 66#7Conference materialsThe selected papers of the Conference will be published in

Vol.10 of "Mathematical Modelling and Analysis"(The Baltic Journal on Mathematical Applications,

Numerical Analysis and Differential Equations), << : ;<1= : ?< 6 647and inVol.5 of Journal "Computational Methods in Applied Mathematics" << : 6476 : 9'All papers will be peer-reviewed.

Correspondence addressDr. A. Štikonas (MMA2005 & CMAM2),Institute of Mathematics and Informatics,Akademijos 4, LT-08663,Vilnius, LithuaniaPh.: (+370) 5 210 97 34, (+370) 5 269 89 28, Fax: (+370) 5 272 92 09

Questions regarding MMA2005 & CMAM2 should be addressed toe-mail: [email protected]

To receive the Second Announcement you must register at: << :;< = :@?< 61647or to fill the registration form and send it by e-mail.

Trakai is a popular center of tourism. It is situated 30 km from Vilnius. The townas well as its surroundings started developing in the XIII century as a state center.According to annals, Grand Duke Gediminas after a successful hunt found a beau-tiful place not far from the then capital Kernave and decided to build a castle here.That is how a new castle was built in Senieji Trakai which at that time was calledTrakai. The town of Trakai was first mentioned in German annals in 1337, whichis regarded to be the official date of its foundation. When Grand Duke Gediminasfinally settled in Vilnius, Senieji Trakai was inherited by his son the Duke Kestutis.During the reign of Kestutis Naujieji Trakai was a place of intensive construction:one castle was built in the strait between lakes Galve and Lukos, another one - onan island in lake Galve. A village grew around the castle. The distinctive featureof Trakai is that the town was built and preserved by people of different nationali-ties. Here lived communities of Karaites, Tartars, Lithuanians, Russians and Polish.Both Christian and Karaites communities were granted separate self-government -Magdeburg - rights. The island castle was rebuilt in the second half of the XX cen-tury. In summer different festivals and concerts take place in the island castle. Moreinformation at http: :<77 : ?<

Mathematical Modelling and Analysis, Volume 9 Number 2, 2004 c

INFORMATION FOR AUTHORS

Aims and Scope

$ $" & "# # # " the Baltic Journal on MathematicalApplications, Numerical Analysis and Differential Equations publishes carefully se-lected papers of the high quality presenting new and important developments in allareas of mathematical modelling and analysis. The scope of the journal includes thefollowing:

• All fields of Numerical analysis,• Mathematical aspects of Scientific Computing,• Parallel algorithms,• Mathematical modelling,• Analysis of ODE and PDE,• Approximation theory,• Optimization.

Instructions for Authors

Manuscript submitted to this journal will be considered for the publication with theunderstanding that the same work has not been published and is not under consid-eration for publication elswere. All the papers will be reviewed. Any manuscriptwhich does not conform to the below instructions may be returned for the necessaryrevision.

Papers submitted for publication should be written in English. The length of apaper is up to 20 pages.

Each paper should have the following structure: the title, the name(s) and insti-tutional affiliation(s) of the author(s), the abstract (50 - 80 words), the text, the list ofreferences.

All illustrations must be supplied on separate sheets and must be marked on theback with figure number, title of paper and name of author. We welcome illustrationsgiven in format.

The list of references should always be in alphabetical order. We ask to submityour references writen with TEX. The complete reference should be listed asfollows.

d Editor-in-Chief R. Ciegis

References

[1] A. Author. Article in proceedings. In: H. Ammann and V.A. Solonnikov(Eds.), Proc. ofthe 6th Intern. Conference NSEC-6, Palanga, Lithuania, 1997, Navier-Stokes Equationsand Related Nonlinear Problems, VSP/TEV, Utrect/Vilnius, 255 – 264, 1998.

[2] A. Author. Difference methods for initial value problems. Interscience Publishers, NewYork, 1998. (in Russian)

[3] F. Author, S. Author and T. Author. Article in journal. Journal, 1(2), 3 – 40, 1998.

We ask to submit your papers written in LATEX 2ε . You may obtain the $ $" & "# # # " style files from our ftp server( 91< 7%:;<1=>: ?< '= 7? ). We supply a style file61647%: ? and a LATEX 2ε template file < 6#?17 < :< and #?17 6647+: < file.

Papers for publications (2 copies) should be send to:

Editorial OfficeMATHEMATICAL MODELLING AND ANALYSISThe Baltic Journal on Mathematical Applications,Numerical Analysis and Differential EquationsAttn.: Dr. A. ŠtikonasInstitute of Mathematics and Informatics,Akademijos 4, LT-08663 Vilnius, LithuaniaPhone: (+370 5) 2109734, 2109346Fax:(+370 5) 2729209

You can send your papers by E-mail:61647819 6 :;<1= : ?<

Internet Address

Information about journal $ $" & "# # # " the Baltic

Journal on Mathematical Applications, Numerical Analysis and Differential Equa-tions and online edition of all papers are presented on Internet: << : ;<1= : ?< 6 647

Mathematical Modelling and Analysis

The Baltic Journal on Mathematical Applications,Numerical Analysis and Differential Equations

Editor-in-Chief R. Ciegis

Volume 9, Number 2, 2004

<< : ;<1= :@? < 66#7

Matematinis modeliavimas ir analize

Vyriausiasis redaktorius R. Ciegis

Tomas 9, Numeris 2, 2004

SL 136. 2004 05 15. 6,0 apsk. leid. l. Tiražas 150 egz.Leido Vilniaus Gedimino technikos universitetoleidykla „Technika“, Sauletekio al. 11, LT-10223 Vilnius-40Spausdino UAB „Sapnu sala“, S. Moniuškos g. 21-10, LT-08121 Vilnius

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ISSN 1392-6292

Institute of Mathematics,Latvian Academy of Sciences·University of Latvia

Institute of Mathematics and Informatics,Vilnius


MATHEMATICALMODELLINGAND ANALYSISTHE BALTIC JOURNALON MATHEMATICALAPPLICATIONS,NUMERICAL ANALYSISAND DIFFERENTIAL EQUATIONSVolume 9 Number 3 2004

Electronical edition available athttp://www.vtu.lt/r /mma/Vilnius Technika 2004

Editor-in-Chief R. CiegisVilnius Gediminas Technical UniversitySauletekio al. 11, LT-10223 Vilnius, Lithuaniar fm.vtu.lt Executive Editor A. Štikonas

Institute of Mathematics and InformaticsAkademijos 4, LT-08663 Vilnius, Lithuaniaashfm.vtu.lt

International Editorial BoardV. Abrashin (Belarus) Institute of Mathematics, Minskabrashinim.bas-net.byA. Buikis (Latvia) Institute of Mathematics of LAS and UL, Rigabuikislatnet.lvG. Erbacci (Italy) Supercomputing Group - InterUniversity ComputingCentererba i ine a.itA. Fitt (UK) University of Southampton adfmaths.soton.a .ukV. Gromak (Belarus) Belarussian State Univ., Minskgrombsu.byO. Iliev (Germany) Institute for Industrial Mathematics, Kaiserslauternilievitwm.uni-kl.deF. Ivanauskas(Lithuania) Vilnius University Feliksas.Ivanauskasmif.vu.ltJ. Janno (Estonia) Tallinn Technical Universityjannoio .eeH. Kalis (Latvia) Institute of Mathematics of LAS and UL, Rigakalislatnet.lvV. Korzyuk (Belarus) Belarussian State Univ., MinskKorzyukorg.bsu.unibel.byA. Krylovas (Lithuania) Vilnius Gediminas Technical Universityakrfm.vtu.ltÜ. Lepik (Estonia) University of Tartu ylepikmath.ut.eeP. Matus (Belarus) Institute of Mathematics, Minskmatusim.bas-net.byM. Meil unas(Lithuania) Vilnius Gediminas Technical Universitymmfm.vtu.ltH. Ockendon (UK) Oxford Centre for Industrial and Applied Mathematicso kendonmaths.ox.a .ukW. Okrasinski (Poland) University of Zielona GoraW.Okrasinskiim.uz.zgora.plA. Pedas(Estonia) University of Tartu arvet.pedasut.eeK. Pileckas(Lithuania) Institute of Mathematics and Informaticspile kasktl.mii.ltU. Raitums (Latvia) Institute of Mathematics of LAS and UL, Rigauldis.raitumsmii.lu.lvA. Reinfelds(Latvia) Institute of Mathematics of LAS and UL, Rigareinflatnet.lvJ. Rokicki (Poland) Institute of Aeronautics, Warsawja kmeil.pw.edu.plS. Rutkauskas(Lithuania) Institute of Mathematics and Informatics, Vilniusstasysrktl.mii.ltM. Sapagovas(Lithuania) Institute of Mathematics and Informatics, VilniusM.Sapagovasktl.mii.ltE. Tamme (Estonia) University of Tartu enn_tut.eeP. Vabishchevich(Russia) Institute for Mathematical Modelling, Moscowvabibrae.a .ruG. Vainikko (Estonia) Estonian Academy of Sciencesgennadi.vainikkohut.fiM. Weber (Germany) Technische Universität Dresdenwebermath.tu-dresden.deG. Yelenin (Russia) M.V. Lomonosov Moscow State Universityyeleninor .ruA. Zemitis (Latvija) Ventspils University College zemitisventa.lvA. Zlotnik (Russia) Power Engineering Institute, Moscowzlotnikapmsun.mpei.a .ru


Mathematical Modelling and Analysis, Volume 9 Number 3, 2004 aFirst Announ ement10th International Conference

Mathematical Modelling and Analysisand

2nd International ConferenceComputational Methods in Applied Mathematics

June 1 – 5, 2005, Trakai, Lithuaniawww.vtu.lt/r /mma2005Conference organizers:The International Association for Mathematics and

Computers in Simulation (IMACS), The European Consortium for Mathematics inIndustry (ECMI), Vilnius Gediminas Technical University,Institute of Mathematicsand Informatics, Vilnius University and Journal Computational Methods in AppliedMathematics.

Aims: The Conference focuses on various aspects of mathematical modellingand usage of finite difference and finite element methods for numerical solution ofmodern problems of science and engineering. It aims, in particular, at fostering co-operation among practitioners and theoreticians in this field. Another very importantgoal of the MMA and CMAM meetings is to assist the creation andmaintenanceof contacts between scientists from the West and the East. Working language of theConference is English.

The basic topics:Analysis of numerical methods for solving problems of mathematical physics;Parallel algorithms and parallel computing;Application of numerical methods to engineering problems;Analysis of ODE and PDE problems and applications;Navier - Stokes equations and Computational Fluid Dynamics;Image processing;Financial mathematics and mathematics in economics;Scientific computation.

The scientific program includes invited plenary talks (40 min) and contributedtalks (20 min). We also invite participants to organize minisymposiums. The formatfor a minisymposium might be a principal lecture and a numberof communicationsof 15-20 minutes given by the other speakers. The minisymposia will be scheduledin parallel sessions.

International Organizing CommitteeR.Ciegis (Lithuania - Chairman), P. Matus (Belarus - Vice-chairman), A. Štikonas

(Lithuania - Scientific secretary), R. Belevicius (Lithuania), A. Buikis (Latvia),I. Gaishun (Belarus), P. Hemker (Netherlands), O. Iliev (Germany), A. Iakubenia(Belarus), F. Ivanauskas (Lithuania), A. Krylovas (Lithuania), R. Lazarov (USA),M. Meil unas (Lithuania), H. Neunzert (Germany), A. Pedas (Estonia), U. Raitums(Latvia), A. Reinfelds (Latvia), J. Rokicki (Poland), M. Sapagovas (Lithuania),

b Editor-in-Chief R.Ciegis

I. Sloan (Australia), P. Vabishchevich (Russia), G. Vainikko (Estonia), A. Zemitis(Latvia)

Abstracts and ProceedingsAuthors are requested to send an abstract (1 page) before March 30, 2005. In-

structions and a style file for the preparation of the abstracts are available athttp://www.vtu.lt/r /mma2005/Conference materialsThe selected papers of the Conference will be published in

Vol. 10 of "Mathematical Modelling and Analysis"(The Baltic Journal on Mathematical Applications,

Numerical Analysis and Differential Equations),http://www.vtu.lt/r /mma/and inVol. 5 of Journal "Computational Methods in Applied Mathematics"http://www. mam.info/All papers will be peer-reviewed.

Correspondence addressDr. A. Štikonas (MMA2005 & CMAM2),Institute of Mathematics and Informatics,Akademijos 4, LT-08663,Vilnius, LithuaniaPh.: (+370) 5 210 97 34, (+370) 5 269 89 28, Fax: (+370) 5 272 92 09

Questions regarding MMA2005 & CMAM2 should be addressed toe-mail: [email protected]

To receive the Second Announcement you must register at:http://www.vtu.lt/r /mma2005/or to fill the registration form and send it by e-mail.

Trakai is a popular center of tourism. It is situated 30 km from Vilnius. The townas well as its surroundings started developing in the XIII century as a state center.According to annals, Grand Duke Gediminas after a successful hunt found a beau-tiful place not far from the then capital Kernave and decidedto build a castle here.That is how a new castle was built in Senieji Trakai which at that time was calledTrakai. The town of Trakai was first mentioned in German annals in 1337, whichis regarded to be the official date of its foundation. When Grand Duke Gediminasfinally settled in Vilnius, Senieji Trakai was inherited by his son the Duke Kestutis.During the reign of Kestutis Naujieji Trakai was a place of intensive construction:one castle was built in the strait between lakes Galve and Lukos, another one - onan island in lake Galve. A village grew around the castle. Thedistinctive featureof Trakai is that the town was built and preserved by people ofdifferent nationali-ties. Here lived communities of Karaites, Tartars, Lithuanians, Russians and Polish.Both Christian and Karaites communities were granted separate self-government -Magdeburg - rights. The island castle was rebuilt in the second half of the XX cen-tury. In summer different festivals and concerts take placein the island castle. Moreinformation at http://www.trakai.lt



Aims and ScopeMATHEMATICAL MODELLING AND ANALYSIS the Baltic Journal on MathematicalApplications, Numerical Analysis and Differential Equations publishes carefully se-lected papers of the high quality presenting new and important developments in allareas ofmathematical modelling and analysis. The scope of the journal includes thefollowing:



Manuscript submitted to this journal will be considered forthe publication with theunderstanding that the same work has not been published and is not under consid-eration for publication elswere. All the papers will be reviewed. Any manuscriptwhich does not conform to the below instructions may be returned for the necessaryrevision.

Papers submitted for publication should be written inEnglish. The length of apaper is up to 20 pages.

Each paper should have the following structure: the title, the name(s) and insti-tutional affiliation(s) of the author(s), the abstract (50 -80 words), the text, the list ofreferences.

All illustrations must be supplied on separate sheets and must be marked on theback with figure number, title of paper and name of author. We welcome illustrationsgiven inEPS format.

The list of references should always be in alphabetical order. We ask to submityour references writen withBibTEX. The complete reference should be listed asfollows.

d Editor-in-Chief R.Ciegis

References

[1] A. Author. Article in proceedings. In: H. Ammann and V.A.Solonnikov(Eds.),Proc. ofthe 6th Intern. Conference NSEC-6, Palanga, Lithuania, 1997, Navier-Stokes Equationsand Related Nonlinear Problems, VSP/TEV, Utrect/Vilnius,255 – 264, 1998.


[3] F. Author, S. Author and T. Author. Article in journal.Journal, 1(2), 3 – 40, 1998.

We ask to submit your papers written in LATEX 2ε . You may obtain theMATHEMATICAL MODELLING AND ANALYSIS style files from our ftp server(ftp://inga.vtu.lt/MMA/journal/). We supply a style filemma. ls and a LATEX 2ε template filetemplate.tex andplain_mma.bst file.



You can send your papers by E-mail:mmafm.vtu.ltInternet Address

Information about journalMATHEMATICAL MODELLING AND ANALYSIS the BalticJournal on Mathematical Applications, Numerical Analysisand Differential Equa-tions and online edition of all papers are presented on Internet:http://www.vtu.lt/r /mma/



Editor-in-Chief R.Ciegis

Volume 9, Number 3, 2004http://www.vtu.lt/r /mma/Matematinis modeliavimas ir analize

Vyriausiasis redaktorius R.Ciegis



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ISSN 1392-6292



Electronical edition available athttp://www.vtu.lt/rc/mma/

Abstracted/Indexed in:Mathematical Reviews

Zentralblatt für Mathematik

Referativnyi Zhurnal(VINITI information publication)


ISSN 1392-6292






Electronical edition available athttp://www.vtu.lt/rc/mma/


Editor-in-Chief R. CiegisVilnius Gediminas Technical University

Sauletekio al. 11, LT-10223 Vilnius, [email protected]

Executive Editor A. ŠtikonasInstitute of Mathematics and Informatics

Akademijos 4, LT-08663 Vilnius, [email protected]

International Editorial Board

V. Abrashin (Belarus) Institute of Mathematics, Minsk [email protected]

A. Buikis (Latvia) Institute of Mathematics of LAS and UL, Riga [email protected]

G. Erbacci (Italy) Supercomputing Group - InterUniversity Computing [email protected]

A. Fitt (UK) University of Southampton [email protected]

V. Gromak (Belarus) Belarussian State Univ., Minsk [email protected]

O. Iliev (Germany) Institute for Industrial Mathematics, [email protected]

F. Ivanauskas (Lithuania) Vilnius University [email protected]

J. Janno (Estonia) Tallinn Technical University [email protected]

H. Kalis (Latvia) Institute of Mathematics of LAS and UL, Riga [email protected]

V. Korzyuk (Belarus) Belarussian State Univ., Minsk [email protected]

A. Krylovas (Lithuania) Vilnius Gediminas Technical University [email protected]

Ü. Lepik (Estonia) University of Tartu [email protected]

P. Matus (Belarus) Institute of Mathematics, Minsk [email protected]

M. Meilunas (Lithuania) Vilnius Gediminas Technical University [email protected]

H. Ockendon (UK) Oxford Centre for Industrial and Applied [email protected]

W. Okrasinski (Poland) University of Zielona Gora [email protected]

A. Pedas (Estonia) University of Tartu [email protected]

K. Pileckas (Lithuania) Institute of Mathematics and Informatics [email protected]

U. Raitums (Latvia) Institute of Mathematics of LAS and UL, [email protected]

A. Reinfelds (Latvia) Institute of Mathematics of LAS and UL, Riga [email protected]

J. Rokicki (Poland) Institute of Aeronautics, Warsaw [email protected]

S. Rutkauskas (Lithuania) Institute of Mathematics and Informatics, [email protected]

M. Sapagovas (Lithuania) Institute of Mathematics and Informatics, [email protected]

E. Tamme (Estonia) University of Tartu [email protected]

P. Vabishchevich (Russia) Institute for Mathematical Modelling, [email protected]

G. Vainikko (Estonia) Estonian Academy of Sciences [email protected]

M. Weber (Germany) Technische Universität [email protected]

G. Yelenin (Russia) M.V. Lomonosov Moscow State University [email protected]

A. Zemitis (Latvija) Ventspils University College [email protected]

A. Zlotnik (Russia) Power Engineering Institute, [email protected]


Authors Index

Mathematical Modelling and Analysis, Volume 9, 2004

Avenhaus R.: Applications of In-spection Games. . . . . .179

Batrakov K.: Modelling of Quasi-Cherenkov Electron BeamInstability in PeriodicalStructures . . . . . . . . . . . . . 1

Bolotin I.B.: The First Basic Boun-dary Value Problem of Rie-mann’s Type for Biana-lytical Functions in a Planewith Slots . . . . . . . . . . . . 91

Bolotin I.B.: The Second Bounda-ry Value Problem of Rie-mann’s Type for Biana-lytical Functions with Dis-continuous Coefficients193

Cebers A.: see Cırulis T.. . . . . .287Chakravarty S.: Numerical Simu-

lation of Unsteady Two-Layered Pulsatile BloodFlow in a Stenosed Flex-ible Artery: Effect of Pe-ripheral Layer Viscosity99

Chtcheritsa O.V.: Implicit Numer-ical Algorithm for the So-lution of Phase TransitionProblems in Multi-Com-ponent Alloys . . . . . . . 253

Čiegis R.: see Krylovas A. . . . . 209Čiegis R.: Parallel Numerical Model-

ling of Short Laser PulseCompression. . . . . . . . .115

Čiegis R.: Mathematical Modellingof Water Flow in PaperPress Machines . . . . . . 267

Cırulis T.: Comparative Analysisby Means of Finite Differ-ences and DM Methods for

Linearized Problem of Gy-rotrons . . . . . . . . . . . . . . 127

Cırulis T.: Dynamics of small Bub-ble Interface Perturbationsin Vertical Hele-Shaw Cellwith Magnetic Liquid un-der the Action of NormalMagnetic Field . . . . . . 287

Dement’ev A.: see Čiegis R. . . 115Denisovas V.: see Švitra D. . . . 327Dhar J.: Diffusion of Population

under the Influence Indus-trialization in a Twin-CityEnvironment . . . . . . . . 201

Dubatovskaya M.: On an Exact De-scription of the SchottkyGroups of Symmetries137

Dumbrajs O.: Numerical Solutionof Single Mode GyrotronEquation . . . . . . . . . . . . . 25

Essoufi El-H.: see Sofonea M. . 229

Garška R.: Spatial Analysis andPrediction of Curonian La-goon Data with Gstat 39

Gromyko G.: On the Cooling ofa Free Thin Film at thePresence of the van derWaals Forces . . . . . . . . 299

J. Dhar: A Prey-predator Modelwith Diffusion and a Sup-plementary Resource forthe Prey in a Two-PatchEnvironment. . . . . . . . . . .9

Juščenko N.: see Švitra D. . . . . 327

Kalis H.: see Cırulis T. . . . . . . . 127Kalis H.: see Dumbrajs O. . . . . . 25Kruminiene I.: see Garška R. . . 39

ii Editor-in-Chief R. Čiegis

Krylovas A.: A Review of Numeri-cal Asymptotic Averagingfor Weakly Nonlinear Hy-perbolic Waves . . . . . . 209

Lietuvietis O.: see Cırulis T. . 127,287

Mandal A.: see Chakravarty S..99Mandal P.K.: see Chakravarty S.

99Mazhorova O.S.: see Chtcheritsa

O.V. . . . . . . . . . . . . . . . . 253Meilunas M.: see Čiegis R. . . . .267Misevičius A.: An Improved Hy-

brid Optimization Algo-rithm for the QuadraticAssignment Problem.149

Omrani K.: On Fully Discrete Ga-lerkin Approximations forthe Cahn-Hilliard Equation313

Popov Yu.P.: see Chtcheritsa O.V.253

Popova L.: see Gromyko G. . . .299

Radžiunas M.: see Sieber J. . . . . 51Rasulov K.M.: About the Solution

in Closed Form of Genera-lized Markushevich Boun-dary Value Problem in theClass of Analytical Func-tions . . . . . . . . . . . . . . . . 223

Rasulov K.M.: see Bolotin I.B. .91Reinfelds A.: see Dumbrajs O. . 25Rogosin S.: see Dubatovskaya M.

137Rybak I.V.: Monotone and Con-

servative Difference Schemesfor Elliptic Equations withMixed Derivatives . . . 169

Schneider K.R.: see Sieber J. . . 51Sieber J.: Dynamics of Multisec-

tion Semiconductor Lasers51

Šilko G.: see Čiegis R. . . . . . . . . 115

Singh H.:see Dhar J. . . . . . . . . . . 201Socolowsky J.: On the Existence

and Uniqueness of Two-Fluid Channel Flows. .67

Sofonea M.: A Piezoelectric Con-tact Problem with Slip De-pendent Coefficient of Fric-tion . . . . . . . . . . . . . . . . . 229

Štikonas A.: see Čiegis R.. . . . .267Švitra D.: Computer Modelling of

Density Dynamics of Single-Species Laboratory Insects’Population . . . . . . . . . . 327

Sytova S.: see Batrakov K. . . . . . . 1

Tabakova S.: see Gromyko G. .299Tammeraid I.: Generalized Riesz

Method and ConvergenceAcceleration . . . . . . . . . 341

Tarang I.: Stability of the SplineCollocation Method for Se-cond Order Volterra Inte-gro-Differential Equations79

Zyuzina E.L.: Stability of Three-Level Difference Schemeswith Respect to the Right-Hand Side . . . . . . . . . . . 243

Mathematical Modelling and Analysis, Volume 9 Number 4, 2004 a

Second Announcement

10th International ConferenceMathematical Modelling and Analysis

and2nd International Conference

Computational Methods in Applied MathematicsJune 1 – 5, 2005, Trakai, Lithuania

www.vtu.lt/rc/mma2005

Conference organizers: The International Association for Mathematics andComputers in Simulation (IMACS), the European Consortium for Mathematics inIndustry (ECMI), Vilnius Gediminas Technical University, Institute of Mathematicsand Informatics, Vilnius University and Computational Methods in Applied Mathe-matics (CMAM).

International Organizing Committee. R. Ciegis (Lithuania - Chairman), P. Ma-tus (Belarus - Vice-chairman), A. Štikonas (Lithuania - Scientific secretary), R. Bele-vicius (Lithuania), A. Buikis (Latvia), I. Gaishun (Belarus), P. Hemker (Nether-lands), O. Iliev (Germany), A. Iakubenia (Belarus), F. Ivanauskas (Lithuania), A. Kry-lovas (Lithuania), R. Lazarov (USA), M. Meilunas (Lithuania), H. Neunzert (Ger-many), A. Pedas (Estonia), U. Raitums (Latvia), A. Reinfelds (Latvia), J. Rokicki(Poland), M. Sapagovas (Lithuania), I. Sloan (Australia), P. Vabishchevich (Russia),G. Vainikko (Estonia), A. Zemitis (Latvia)

Aims: The Conference focuses on various aspects of mathematical modellingand usage of finite difference and finite element methods for numerical solution ofmodern problems of science and engineering. It aims, in particular, at fostering co-operation among practitioners and theoreticians in this field. Another very importantgoal of the MMA and CMAM meetings is to assist the creation and maintenance ofcontacts between scientists from the West and East. Working language of the Con-ference is English.

The basic topics:Analysis of numerical methods for solving problems of mathematical physics;Parallel algorithms and parallel computing;Application of numerical methods;Analysis of ODE and PDE problems and applications;Navier - Stokes equations and Computational Fluid Dynamics;Image processing;Financial mathematics and mathematics in economics;Scientific computation.

The scientific program includes Invited Plenary Talks (40 min), Invited Semi-plenary lectures (30 min) and Contributed Talks (20 min). The program also includesPoster Sessions.

b Editor-in-Chief R. Ciegis

We also invite participants to organize mini-symposia. The format for a mini-symposium is a principal lecture (30 min) and a number of communications of 20minutes given by the other speakers. The mini-symposia will be scheduled in parallelsessions. The deadline for proposals of mini-symposia is January 30, 2005.

The following mini-symposia are approved:1. "Modelling and simulation of hydrogeological and geo-environmental

problems", co-organizers: H. Neunzert, O. Iliev (ITWM, Kaiserslautern, Germany),P.Vabishchevich (Moscow university, Russia), e-mail: neunzert,[email protected]

2. "Navier - Stokes equations and Computational Fluid Dynamics", co-organizers: W.Zajaczkowski, J. Rokicki (Poland), K. Pileckas (IMI, Vilnius, Lithua-nia) e-mail: [email protected], [email protected], [email protected]

3. "Computational Mechanics", organizers: R. Belevicius (VGTU, Vilnius,Lithuania), e-mail: [email protected]

4. "Parallel and grid computing: algorithms and applications", co-organizers:M. Sosonkina (Ames Laboratory and Iowa State University, USA), R. Ciegis (VGTU,Vilnius, Lithuania), e-mail: [email protected]

5. "Mathematical models including non-local boundary conditions", co-organi-zers: M. Sapagovas (IMI, Vilnius, Lithuania), V.L. Makarov (Kiev, Ukraine), e-mail:[email protected]

Abstracts and Proceeding. Authors are requested to send an abstract (1 page)before March 30, 2005. Instructions and a style file for the preparation of the abstractsare available at http://www.vtu.lt/rc/mma2005/

Conference materials. The selected papers of the Conference will be publishedin Vol. 10 of "Mathematical Modelling and Analysis"http://www.vtu.lt/rc/mma/

and in Vol. 5 of Journal "Computational Methods in Applied Mathematics"http://www.cmam.info/

All papers will be peer-reviewed.

Registration. For registration please fill in the registration form at our web-site.Please inform us about your arrival and departure dates.

The registration fee is 240 EUR and can be paid directly at the registration desk.It includes the abstract volume, all local expenses (accommodation in two bed roomsfor four nights and three meals per day), conference dinner and coffee breaks. Per-sons who wish accommodation in single rooms should pay the registration fee 300EUR. The fee for accompanying persons is 140 EUR. It icludes the reception, con-ference dinner, social events and accommodation in two bed rooms for four nightsand three meals per day.

Full traveling and registration information will be given on our web-site.

Deadlines. Abstracts: March 30, 2005; Notification of participation and reserva-tion of accommodation: May 10, 2005

Correspondence address. Dr. A. Štikonas (MMA2005 & CMAM2), Instituteof Mathematics and Informatics, Akademijos 4, LT-08663, Vilnius, Lithuania. Ph.:(+370) 5 210 97 34, (+370) 5 269 89 28, Fax: (+370) 5 272 92 09. Questions regard-ing MMA2005 & CMAM2 should be addressed to e-mail: [email protected]



Aims and Scope

MATHEMATICAL MODELLING AND ANALYSIS the Baltic Journal on MathematicalApplications, Numerical Analysis and Differential Equations publishes carefully se-lected papers of the high quality presenting new and important developments in allareas of mathematical modelling and analysis. The scope of the journal includes thefollowing:



Manuscript submitted to this journal will be considered for the publication with theunderstanding that the same work has not been published and is not under consid-eration for publication elswere. All the papers will be reviewed. Any manuscriptwhich does not conform to the below instructions may be returned for the necessaryrevision.

Papers submitted for publication should be written in English. The length of apaper is up to 20 pages.

Each paper should have the following structure: the title, the name(s) and insti-tutional affiliation(s) of the author(s), the abstract (50 - 80 words), the text, the list ofreferences.

All illustrations must be supplied on separate sheets and must be marked on theback with figure number, title of paper and name of author. We welcome illustrationsgiven in EPS format.

The list of references should always be in alphabetical order. We ask to submityour references writen with BibTEX. The complete reference should be listed asfollows.

d Editor-in-Chief R. Ciegis

References

[1] A. Author. Article in proceedings. In: H. Ammann and V.A. Solonnikov(Eds.), Proc. of

the 6th Intern. Conference NSEC-6, Palanga, Lithuania, 1997, Navier-Stokes Equationsand Related Nonlinear Problems, VSP/TEV, Utrect/Vilnius, 255 – 264, 1998.


[3] F. Author, S. Author and T. Author. Article in journal. Journal, 1(2), 3 – 40, 1998.

We ask to submit your papers written in LATEX 2ε . You may obtain theMATHEMATICAL MODELLING AND ANALYSIS style files from our ftp server(ftp://inga.vtu.lt/MMA/journal/). We supply a style filemma.cls and a LATEX 2ε template file template.tex and plain_mma.bst file.



You can send your papers by E-mail:[email protected]

Internet Address

Information about journal MATHEMATICAL MODELLING AND ANALYSIS the BalticJournal on Mathematical Applications, Numerical Analysis and Differential Equa-tions and online edition of all papers are presented on Internet:




Editor-in-Chief R. Ciegis

Volume 9, Number 4, 2004


Matematinis modeliavimas ir analize

Vyriausiasis redaktorius R. Ciegis



Mathematical Modelling and Analysis Volume 9, Number 4, 2004

CONTENTS

O.V. Chtcheritsa, O.S. Mazhorova, Yu.P. PopovImplicit Numerical Algorithm for the Solution of Phase Transition Prob-lems in Multi-Component Alloys . . . . . . . . . . . . . . . . . . . . . . 253

R. Čiegis, M. Meilûnas, A. ŠtikonasMathematical Modelling of Water Flow in Paper Press Machines . . . . 267

T. Cırulis, O. Lietuvietis, A.CebersDynamics of small Bubble Interface Perturbations in Vertical Hele-

Shaw Cell with Magnetic Liquid under the Action of Normal MagneticField . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

G. Gromyko, S. Tabakova, L. PopovaOn the Cooling of a Free Thin Film at the Presence of the van der WaalsForces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

K. OmraniOn Fully Discrete Galerkin Approximations for the Cahn-Hilliard Equa-tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

D. Švitra, V. Denisovas, N. JuščenkoComputer Modelling of Density Dynamics of Single-Species LaboratoryInsects’ Population . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

I. TammeraidGeneralized Riesz Method and Convergence Acceleration . . . . . . . . 341

!"#$%'&)($+*,.-0/213435/768

1–8c© 2004 Technika ISSN 1392-6292

MODELLING OF QUASI-CHERENKOVELECTRON BEAM INSTABILITY INPERIODICAL STRUCTURES 1

K. BATRAKOV and S. SYTOVA

Institute for Nuclear Problems, Belarussian State University

Bobruiskaya 11, 220050 Minsk, Belarus

E-mail: 9:0;<:=>0?@#ACBDFEHGIAJB7KL=%EM9N%OPKLN4;>0?:@AQBDFEHGRAQB7KL=%EM9N

Received October 2 2003; revised December 8 2003

Abstract. Nonlinear stage of quasi-Cherenkov instability of electron beam under condi-tions of two- and three-dimensional distributed feedback is simulated. The scheme of dis-tributed feedback with two strong coupled waves is considered. Mathematical model of quasi-Cherenkov electron beam instability is proposed. Numerical method to solve the nonlinearintegro-differential system, describing such instability, is worked out. Results of numericalexperiments are discussed.

Key words: quasi-Cherenkov instability, numerical modelling, nonlinear integro-differentialsystem

1. Introduction

This contribution is devoted to modelling of nonlinear stage of quasi–Cherenkovelectron beam instability under the conditions of two- and three-dimensional dis-tributed feedback. Quasi-Cherenkov instability takes place when one or more waverefraction index satisfy the Cherenkov condition [4]. In this case electrons radiate co-herently. Such instability mechanism can be considered as a technique for realizationof Free Electron Laser (FEL). FELs are devices which use the electron beam energyto generate coherent electromagnetic radiation. Such devices are very perspective forelectromagnetic radiation generation in wide spectral range. Nowadays FEL lasingis obtained in different wavelength ranges: from centimeter to ultra-violet [2, 9, 14].The high expensive International X-ray FEL project is on the preparation stage now[16]. Volume FEL (VFEL) based on the mechanism of multi wave volume distributedfeedback (VDFB) was proposed in [3, 5]. VFELs give possibility to reduce the start-ing currents of generation, to provide generation in large volume, to tune generation

1 Authors thank prof. V. G. Baryshevsky for his interest to this work

2 K. Batrakov, S. Sytova

frequency [3]. VFEL generation in large volume essentially increases the electricstrength of resonator and allows to produce electromagnetic pulses of great power(> 10 GWt) in mm–cm range. Besides the multiwave distributed feedback in VFELprovides the modes discrimination in the case when the linear sizes of resonator(waveguide) cross section exceed generated wavelength (the so-called oversized sys-tems).

First lasing of volume FEL (VFEL) in millimeter range was recently obtained bya group of scientists of Institute for Nuclear Problems [7].

A lot of papers are devoted to FEL simulation (for example [10, 11]). In ourworks [1, 6, 8, 18, 19, 20, 21, 22] we have considered mathematical models of dif-ferent types of VFEL in X-ray, optical and millimeter wave ranges. Earlier electronbeam was simulated as a hydrodynamical approximation [1, 18] or as distributionfunctions [8, 19]. It turned out, however, that hydrodynamical approximation is veryrough. And in millimeter range electron beam presentation as distribution functionsfor sufficiently large beam current density leads to appearance of non-physical insta-bility related to the computational error. Therefore in this work simulation of quasi-Cherenkov instability is performed by means of phase averaging method which isfrequently used in large number of works. The main distinction of this work is inapplying such method to VFEL simulation.

2. Mathematical Model

Let us consider quasi-Cherenkov stimulated radiation of wide electron beam passingthrough spatial periodic structure. In Fig. 1 four schemes of simple VFEL are pre-sented. A target of length L is a medium possessing spatially periodic permittivity.There are several different possibilities. Fig.1a corresponds to the case when there areno incident waves emerging on system. For distributed feedback forming the specificso-called diffraction (or the Bragg) conditions can be fulfilled. These conditions havethe form |k| = |kτ | for the two waves case. Here k is the radiation wave vector andτ is reciprocal vector of the periodical structure τ = 2πn1/d1, 2πn2/d2, 2πn3/d3,d1, d2, d3 are basic translation periods, n1, n2, n3 are integers. Here we consider so-called Bragg diffraction geometry when one wave propagates in forward directionand the other – in backward (see Fig.1a – Fig.1d).

The developed mathematical model allows to consider such geometry when inci-dent wave emerges from the side z = 0 (forward incident wave Fig.1b), or from theside z = L (backward incident wave Fig.1c), or from both sides simultaneously (see,Fig.1d). Moreover two mirrors can be placed on each side of the target to accumulateradiation.

Let us consider the system of equations describing quasi–Cherenkov instability.Equations for this process are written for stationary regime of nonlinear saturation.This system with appropriate boundary conditions is written as follows:

Modelling of quasi–Cherenkov electron beam instability 3

zL0

kkt

t

Electro

n beam

target

zL0

kkt

t

Electro

n beam

target

Incid

ent radia

tion

a b

zL0

kkt

t

Electro

n beam

target

Incident radiation

zL0

kkt

t

Electro

n beam

target

Incid

ent radia

tion

Incident radiation

c d

Figure 1. Scheme of quasi–Cherenkov VFEL in Bragg geometry.

dE

dz

+ a11E + a12Eτ

= Φ

2π∫

0

2π − p

8π2

(exp(−iΘ(z, p)) + exp(−iΘ(z,−p))) dp,

E(0) = E0, z ∈ [0, L], p ∈ [−2π, 2π] ,

dEτ

dz

+ a21E + a22Eτ = 0, Eτ (L) = E1,

d2Θ(z, p)

dz2

= Ψ

(

k −

dΘ(z, p)

dz

)3

Re (E(z) exp(iΘ(z, p))) ,

Θ(0, p) = p,

dΘ(0, p)

dz

= 0;

(2.1)

where i is the imaginary unit.There are two independent arguments in system (2.1): spatial coordinate z and

initial electron phase p. Amplitudes of electromagnetic fields E(z), Eτ (z) and co-efficients a are complex-valued. Function Θ(z, p) describes phase of electron beamrelative to the electromagnetic wave. Θ(z, p) and coefficients Φ and Ψ are real. k isa projection of wave vector k on z axis. We suppose that all functions are smooth,bounded and slowly changing.


Stationary solution under zero boundary conditions is due to the essentially non-linear contribution of electron beam. In linear regime the homogeneous system withzero boundary conditions has infinite number of solutions.

3. Numerical Algorithm

To solve the system of integro–differential equations with nonlinearity on right–handsides an iterative algorithm is proposed. We use notations from [17].

Introducing in domain Ω = 0 ≤ z ≤ L, −2π ≤ p ≤ 2π uniform grids on z

and p:

ωz = zi = ihz, i = 0, 1, . . . , M, Mhz = L,

ωp = pj = hpj, j = −N, . . . ,−1, 0, 1, . . . , N, hpN = 2π .

The discrete functions, defined on the grid, will be denoted by

Θji = Θ(zi, pj), E

ji = E(zi, pj) .

We approximate the differential problem with the following finite–difference scheme:

s

Θjzz = Ψ

(

k −

s

Θj

z

)3

Re

(

s−1

E exp(is

Θj

)

, j = 0,±1, . . . ,±N, (3.1)

s

Ez + a11

s

E + a12

s

Eτ = Φ

N∑

j=0

cj

(

exp(−i

s

Θj) + exp(−i

s

Θ−j)

)

, (3.2)

s

Eτz + a21

s

E + a22

s

Eτ = 0, (3.3)

where s ≥ 0 is a number of iteration. As an initial approximation we define:

0

Θj = hpj,

0

E = 0,

0

Eτ = 0 .

Here cj are coefficients of quadrature formula. We use the trapezoidal rule here.Let us write the difference equation (3.1) in the following form:

s

Θ

j

i+1 −2s

Θ

j

i ++s

Θ

j

i−1

h2z

= Ψ

k −

s

Θ

j

i+1 −

s

Θ

j

i−1

2hz

3

Re

(

s−1

E exp(is

Θji )

)

, (3.4)

i.e. it is an implicit difference equation with respect tos+1

Θ i+1. Solving this cubic

equation get three solutions, we choose the one which is close tos+1

Θ i. The tworest roots are meaningless. As it was shown in numerical experiments this approachworks very well.

But it is not a very efficient strategy to solve numerically cubic equations. It ispossible to solve it by using the Picard type iterative process:


l+1

Θ i+1 −2s

Θ

j

i +s

Θ

j

i−1

h2z

= Ψ

k −

l

Θi+1 −

s

Θ

j

i−1

2hz

3

Re

(

s−1

E exp(is+1

Θji )

)

,

0

Θi+1=s

Θ

j

i+1 ,

where l ≥ 0 is a number of inner iterations. As it was shown in numerical experi-ments, it is enough to make only two inner iterations to solve the cubic equation.

Inasmuch as our iterative process (3.1)–(3.3) is nonlinear, it seems to be impos-sible to investigate its convergence. If we consider a linearized case of this processthen some conclusions are evident. We restrict ourselves to Fig. 2, which demon-strates the convergence of the iterative process. It is stabilized after approximately40 iterations.

0 40 80 120 160

N iterations

0

100

200

300

400

500

| E |

Figure 2. Numerical solution depending on iterations number.

So, in accordance to numerical experiments, our schemes are stable and the dis-crete solution converges to the solution of initial differential system because numer-ical results coincided in full with analytical estimations.

4. Numerical Results

Let us discuss results of numerical experiments carried out. Among them we haveconsidered the case when there was no incident radiation, in other words E(0) = 0and Eτ (L) = 0, as well as E(0) 6= 0.

Starting currents of electron beam, radiation power and radiation frequency de-pend on the feedback geometry. Therefore changing this geometry we can changethese quantities and even to turn regime of generation to amplification regime andvice versa.

Threshold current density is very important value characterizing the system.There is no generation process if current is lower than some critical value. In thecase when current density is between this critical value and the value of genera-tion threshold current (jth) the system operates in amplification regime. This is theregime of regenerative amplification for the Bragg geometry. The regime of genera-tor is realized when the current density exceeds the threshold j > jth. In that case


radiation of electron beam should exceed the losses on boundaries of the resonatorand absorption losses. Such threshold is depicted in Fig.3 depending on the lengthof the target. It demonstrates threshold current density depending on thickness of thetarget with and without incident radiation and with and without absorption of thetarget. We can see that the larger is the target length the lower is the threshold. Ab-sorption Im(χ0) = 0.01 raises the threshold. Presence of incident wave with E 6= 0decreases it.

8 10 12 14 16 18 20L, cm

0

1000

2000

3000

4000

5000

6000

7000

8000

j , A/cm

c =0.1, E=0

c =(0.1, 0.01), E=0

0

0

th

2

l l l

Figure 3. Electron beam current threshold in Bragg geometry.

Two different geometries were studied: Bragg geometry (see Fig.1 and numeri-cal results above), in which two waves propagate to opposite sides of resonator andLaue geometry, in which waves propagate to one side of a resonator (see Fig.4). Tworegimes are possible in the Bragg case as stated above: regime of regenerative ampli-fication and regime of generation. Two regimes are possible in Laue case too. Thereare amplification regime of emerging incident wave and regime SASE [15]. SASE(self amplified stimulated emission) develops from spontaneous noises (as well asgenerator regime in Bragg case). Since diffracted wave in Laue case propagates inthe forward direction, we have to change right difference derivative in (3.3) for Eτ

to the left one. It is clear that boundary conditions for Eτ in (2.1) should be writtenfor z = 0.

Let us examine Fig. 5. We can see dependence between value of electric fieldand current density for different amplitudes of emerging waves for both diffractiongeometries. Amplification regime corresponds to range of current density j formrange 60÷ 80 A/cm2 for Bragg geometry (Fig.5a). Current threshold is over-passedat j = 80 A/cm2. This region corresponds to regime of generation. In Fig.5b thecurve with E = 0 corresponds to SASE regime. The rest curves demonstrate regimeof amplification.

Let us consider two questions. First, where does radiation come from in the sys-tem in the absence of incident radiation? The answer is that it comes from sponta-neous noises of electron beam. The second question asks what corresponds to thisnoise in numerical realization of mathematical model (3.1)–(3.3)? The answer is thatsuch noise appears due to computational error on the right-hand side of equations. It


zL0

k

kt

t

Electro

n beam

target

zL0

k

kt

t

Electro

n beam

target

Incid

ent radia

tion

a b

Figure 4. Scheme of quasi-Cherenkov VFEL in Laue geometry.

0 20 40 60 80 100 120 140 160 180 200j, A/cm

1E-12

1E-11

1E-10

1E-9

1E-8

1E-7

1E-6

0.00001

0.0001

0.001

0.01

0.1

1

10

100

1000

| E |

E = 1

E = 0.01

E = 0.0001

E = 0

20 1000 2000 3000 4000

j, A/cm

1E-11

1E-10

1E-9

1E-8

1E-7

1E-6

0.00001

0.0001

0.001

0.01

0.1

1

10

100

1000

10000

| E |

E = 0

E= 0.0001

E=0.01

E=1

2

a b

Figure 5. Dependence between computed electromagnetic wave amplitude and electron beamcurrent density in Bragg (a) and Laue (b) geometry.

is clear that at the first iteration, when E(z) = 0 all over z ∈ [0, L], we have

I = Φ

∫ 2π

0

2π − p

8π2

(

exp(

−iΘ(z, p))

+ exp(

−iΘ(z,−p))

)

dp ≡ 0 .

But in fact we obtain I ∼ 10−15, 10−14. This is an equivalent of spontaneous noises

and fuse for the beginning of generation process.

References

[1] V.N. Abrashin, A.O. Grubich and Sytova S.N. Nonlinear stage of development of therelativistic electron beam Cherenkov instability. Mathematical Modeling, 3(8), 21 – 29,1991. (in Russian)

[2] S. Andruszkow. First observation of self-amplified spontaneous emission in a free-electron laser at 109 nm wavelength. Phys. Rev. Let., 85, 3825, 2000.

[3] V. Baryshevsky, K. Batrakov and I. Dubovskaya. Parametric (quasi-Cherenkov) X-rayFEL. Journ.Phys.D, 24, 1250, 1991.

[4] V. Baryshevsky and I. Feranchuk. Parametric X-rays from ultrarelativistic electrons in acrystals: theory and possibilities of practical utilization. Journ.Phys., 44, 913, 1983.

[5] V. Baryshevsky and I. Feranchuk. Parametric beam instability of relativistic chargedparticles in a crystal. Phys.Let.A, 102, 141, 1984.


[6] V.G. Baryshevsky, K.G. Batrakov, I.Ya. Dubovskaya and S.N. Sytova. Visible surfacequasi-Cherenkov FEL. Nucl. Instr. and Meth. in Phys. Res., A358, 508 – 511, 1995.

[7] V.G. Baryshevsky, K.G. Batrakov and et all. First lasing of a volume FEL (VFEL) at awavelength range 4-6 mm. Nucl. Instr. and Meth. in Phys. Res., A483, 21, 2002.

[8] I. Dubovskaya, V. Baryshevsky, K. Batrakov and S. Sytova. The nonliner analysis ofvisible quasi-Cherenkov FEL. In: 21th Intern. Free Electron Laser Conference (FEL99),Germany, 1999.

[9] A.V. Elzhov, N.S Ginzburg, A.K. Kaminsky, E.A. Perelstein, N.Yu. Peskov, S.N. Sedykh,A.P. Sergeev and A.S. Sergeev. Features of FEM for testing of high-gradient acceleratingstructures of linear colliders. In: Proceedings of Eighth European Particle AcceleratorConference (EPAC’2002), Paris, 2308 – 2310, 2002 , June 3-7.

[10] N. Ginzburg, N. Peskov and A. Sergeev. Dynamics of free-electron lasers with two-dimensional distributed feedback. Technical Physics Letters, 18(5), 285, 1992.

[11] N. Ginzburg, R. Rosental, N. Peskov, A. Arzhannicov and S. Sinitsky. Modeling of aplanar FEL amplifier with a sheet relativistic electron beam. Nucl. Instr. and Meth. inPhys. Res., A483, 255, 2002.

[12] N.S. Ginzburg, S.P. Kuznetzov and T.N. Fedoseeva. The theory of transient processes ina relativistic BWT. Izvestiya Vuzov. Radiophysics, 21(7), 1071, 1978.

[13] N.S. Ginzburg, R.M. Rosental and A.S. Sergeev. About possibility of synthesis of radi-ation spectrum in bit-slice relativistic backward-wave tube. Pis’ma v ZhTF, 29(4), 71 –80, 2003.

[14] Y.C. Huang, C.S. Hsue, R.H. Pantell and T.I. Smith. The FEL and IFEL design studyfor the proposal NTHU photon-electron dynamics laboratory. Nucl. Instr. and Meth. inPhys. Res.A, 429, 430, 1999.

[15] K. Kim. Three-dimensional analysis of coherent amplification and self-amplified spon-taneous emission in free electron lasers. Phys. Rev. Lett., 57, 1986.

[16] M. Korfer. The TTF-FEL status and its future as X-ray user facility. Nucl. Instr. andMeth. in Phys. Res.A, 483, 34, 2002.

[17] A.A. Samarskii. Theory of finite-difference schemes. Nauka, Moscow, 1989. (in Russian)[18] S. Sytova. A numerical method for solving one problem of nuclear physics. Vesti Nat.

Acad. Sci. Belarus. Ser. Phys.-Math., N 2, 44 – 50, 1993. (in Russian)[19] S. Sytova. A numerical method for solving hyperbolic system with singularities. Differ-

ential Equations, 32(7), 995 – 998, 1996.[20] S. Sytova. On numerical methods for modelling of terahertz sources based on low energy

relativistic beams. In: Proc. of the 3rd Intern. Conference FDS2000, 237 – 244, 2000.[21] S. Sytova. Numerical methods in problems of modelling of volume free electron lasers.

Differential Equations, 37(7), 976 – 981, 2001.[22] S. Sytova. On numerical methods for one problem of mixed type. Mathematical Mod-

elling and Analysis, 6(2), 321 – 326, 2001.

Kvazi-Cerenkovo elektroninio spindulio nestabilumo modeliavimas periodinese struk-turose

K. Batrakov, S. Sytova

Modeliuojama elektronu spindulio kvazi-Cherenkovo nestabilumo netiesine faze su dvimacioir trimacio paskirstytojo grižtamojo ryšio salyga. Nagrinejama schema su grižtamuoju ryšiusu dviem susietomis stipriomis bangomis. Pateiktas kvazi-Cherenkovo elektroninio spin-dulio nestabilumo matematinis modelis. Pasiulytas veiksmingas skaitinis algoritmas, skir-tas netiesinems integro-diferencialinems lygciu sistemoms su tokio tipo nestabilumu, spresti.Apžvelgti skaitinio eksperimento rezultai.

!"#$%'&)($+*,.-0/213435/768

9–24c© 2004 Technika ISSN 1392-6292

A PREY-PREDATOR MODEL WITH DIFFUSIONAND A SUPPLEMENTARY RESOURCE FOR THEPREY IN A TWO-PATCH ENVIRONMENT 1

J. DHAR

Department of Applied Mathematics, Beant College of Engineering Technology

Gurdaspur - 143521, Punjab, India

E-mail: 9:0;<=?>@AB<:0CDE7D=F+GIH:JEReceived June 03 2003; revised January 10 2004

Abstract. In this paper, a prey-predator dynamics, where the predator species partially de-pends upon the prey species, in a two patch habitat with diffusion and there is a non-diffusingadditional resource for the prey population, is modeled and analyzed. It is shown, that thereexists a positive, monotonic, continuous steady state solution with continuous matching at theinterface for both the species separately. Further, we obtain conditions for asymptotic stabilityfor both linear and nonlinear cases.

Key words: Population diffusion, patchiness, supplementary resource, steady state solution,stability

1. Introduction

Mathematical ecology has its roots in population ecology, which treats the increaseand fluctuation of population. An interesting problem in mathematical ecology is tostudy the growth and co-existence of species with diffusion in both homogeneousand patchy habitats. As noted before the diffusion, when it occurs, plays the roleof increasing stability in a system of interacting populations [8, 10, 23, 24, 25, 27].Some researchers have given elaborate survey of models with diffusion in both ho-mogeneous and heterogeneous environment [14, 15, 16, 23] and also surveyed theliterature related to models with diffusion and reported the effects of dispersal andspatial heterogeneity on stability of both single species and for predator-prey system[4, 5, 18]. In [21] a prey-predator model with functional response and diffusion isconsidered and it is shown, that if the equilibrium state is linearly stable, a sub-region

1 This work partially was carried out at Department of Mathematics, Indian Institute of Tech-nology, Kanpur-208016, India. Author is thankful to Prof. J. B. Shukla for his valuablesuggestion.

10 J. Dhar

of the positive quadrant can be found in the phase plane where it is non-linearly stablewith or without diffusion.

It may be noted here that in the above study the role of alternative or supplemen-tary resource on equilibrium levels of populations as well as on their stability has notbeen discussed, although the study of resource-based interacting population biologyis an interesting area of research in population dynamics. Some experimental inves-tigations on micro-organisms using the chemostat [11, 22] have been conducted andperhaps the best laboratory idealization of nature for population studies has been de-scribed in [28]. Several mathematical models of such systems, involving competitionand other types of non-interacting populations, which depend upon growth limitingnutrient in a chemostat with constant input and variable washout rates have beenstudied in [1, 2, 13, 17]. Also some other mathematical investigations related to twocompeting populations which are wholly dependent on a self-renewable resource in ahabitat without diffusion have been presented [9, 12, 19]. But very little attention hasbeen given in the resource-based prey-predator system with diffusion [6]. The effectof a predator resource on a diffusive Predator-Prey system, showing the stabilizingrole of diffusion have been studied.

In this paper, therefore, a logistically growing two species prey-predator typemodel is considered. A self-renewable supplementary resource for prey populationand diffusion in a two-patch habitats is proposed and the stability of both the linearand nonlinear systems is discussed. Both the reservoir and no-flux boundary condi-tions are considered. It is shown that the effect of explicit dependence of the preypopulation on an alternative supplementary resource in the two patches may increasethe level of steady state distribution for prey in the entire habitat. The model is pro-posed by keeping in view the depletion of forest resources biomass (prey species)with partially re-plantation of forest resource (i.e. supplementary resource) due toincreased forest resource. Dependence or independence on industrialization and pop-ulation (predator species) has caused patchiness in the Doon Valley situated at thefoot hills of Himalayas in India [26].

This paper is organized as follows: first we write the prey-predator model witha self-renewable supplementary resource for the prey in a two-patch habitat. In thenext section we study our main model in a two-patch habitat for both non-uniformand uniform steady state cases under both reservoir and no-flux boundary conditions.

2. The Mathematical Model

We consider a dynamic model of two logistically growing animal (such as deer andwolf) species with prey-predator type interaction and diffusion in a two-patch foresthabitat by assuming that the second species uses the first species as an alternative re-source. In such a case the rate of change of density of the first species decreases dueincrease in the density of the second species, but the density of the second speciesincreases due to the increase in the density of the first species in both the patches.Let xi(s, t) and yi(s, t) be the densities of first and second species in the i-th patchrespectively. Now if we supply a supplementary resource Ri(s, t) for the prey popu-lation xi(s, t) in the entire habitat, then in presence of resource biomass the growth

Prey-Predator Model with Supplementary Resource 11

rates of prey populations increases. We also assume that there is no explicit diffusionin the resource biomass. Then the model can be written as the following system ofautonomous partial differential equations:

∂Ri

∂t

= aiRi(1 −

Ri

Ci

) − αiRixi, i = 1, 2 , (2.1)

∂xi

∂t

= xigi(xi) − yipi(xi) + θαiRixi + D1i

∂2xi

∂s2

, (2.2)

∂yi

∂t

= yifi(yi) + γiyipi(xi) + D2i

∂2yi

∂s2

, 0 ≤ s ≤ L2 , (2.3)

where the i-th patch is assumed to lie along the spatial length Li−1 ≤ s ≤ Li (L0 =0), Ci, i = 1, 2 are the carrying capacity of the supplementary resource in the i-thpatch and θ is the conversion rate of biomass constant by the prey populations, re-spectively. The functions gi(xi) and fi(yi) are the respective specific growth rates,pi(xi) are the interaction rates (predator response functions) and D1i, D2i are thediffusion coefficient of xi and yi in the i-th patch respectively. The constants αi,i = 1, 2 are positive interaction rate coefficients of the prey species with the supple-mentary resource and γi, i = 1, 2 are conversion rates coefficient in the i-th patch.

We assume the following assumption for gi(xi), fi(yi), and pi(xi):

AH1 :

gi(xi), fi(yi), pi(xi) ∈ C2[0,∞) ,

gi(0) > 0, fi(0) > 0, pi(0) = 0 ,

for xi > 0, g′

i(xi) ≤ 0, p′

i(xi) > 0 ,

for yi > 0, f′

i(yi) ≤ 0 .

When the environment has a carrying capacity Ki and Mi respectively for prey andpredator populations in the i-th patch, then

gi(Ki) = 0, fi(Mi) = 0, for i = 1, 2 .

Further we assume that:

AH2 :

∃R∗

i , x∗

i , y∗

i > 0, such that R∗

i = Ci[ai − αix∗

i ]/ai ,

x∗

i gi(x∗

i ) − y∗

i pi(x∗

i ) + θαiR∗

i x∗

i = 0 ,

fi(y∗

i ) + γipi(x∗

i ) = 0 .

The model is studied using one set of boundary conditions, i.e., reservoir or no-flux conditions. In the case of reservoir boundary conditions, we take

x1(0, t) = x∗

1, x2(L2, t) = x∗

2 , (2.4)

y1(0, t) = y∗

1 , y2(L2, t) = y∗

2 (2.5)

and in the case of no-flux boundary conditions we consider

∂x1(0, t)

∂s

= 0,

∂x2(L2, t)

∂s

= 0 , (2.6)

∂y1(0, t)

∂s

= 0,

∂y2(L2, t)

∂s

= 0 . (2.7)

12 J. Dhar

We also assume the continuity and flux matching conditions at the interface s =L1. The continuity conditions at the interface s = L1 are the following:

x1(L1, t) = x2(L1, t), y1(L1, t) = y2(L1, t), R1(L1, t) = R2(L1, t) . (2.8)

The continuous flux matching conditions at the interface s = L1 for xi(s, t) andyi(s, t) are given by

D11

∂x1(L1, t)

∂s

= D12

∂x2(L1, t)

∂s

, (2.9)

D21

∂y1(L1, t)

∂s

= D22

∂y2(L1, t)

∂s

. (2.10)

Finally the model is completed by assuming some positive initial distribution ofeach species, for i = 1, 2, that is,

xi(s, 0) = χi(s) > 0, Li−1 < s < Li, (2.11)

yi(s, 0) = δi(s) > 0, Li−1 < s < Li, (2.12)

Ri(s, 0) = R0i(s) > 0, Li−1 < s < Li. (2.13)

3. Analysis of the Model in Two Patch Habitat

Our aim is to analyze the long time behavior of the system in both uniform andnonuniform cases. In next two subsection we will study the model (2.1) – (2.13), inthe case of nonuniform and uniform steady state.

3.1. The Non-uniform Steady State

Let ui, vi and wi are the steady state solutions of the prey populations xi, predatorpopulations yi and the supplementary resource Ri. Then the steady state systembecomes:

wi =Ci

ai

[ai − αiui] , (3.1)

D1i

d2ui

ds2

+ uigi(ui) − vipi(ui) + θαiwiui = 0, (3.2)

D2i

d2vi

ds2

+ vifi(vi) + γivipi(ui) = 0. (3.3)

Now substituting the value of wi from (3.1) into (3.2) and (3.3), we get:

D1i

d2ui

ds2

+ uiGi(ui) − vipi(ui) = 0, (3.4)

D2i

d2vi

ds2

+ vifi(vi) + γivipi(ui) = 0, (3.5)

where


Gi(ui) = gi(ui) + θαi

Ci

ai

(ai − αiui), i = 1, 2.

Since

Gi(0) = gi(0) + Ciθαi > 0, G

′

i(ui) = g′

i(ui) −Ciθα

2i

ai

< 0, i = 1, 2 .

Hence the behavior the steady state system (3.4) and (3.5) with same set of boundaryconditions identical to the case when there is no supplementary resource for the preypopulations. Further, we assume

AH3 :

∃x∗

i , y∗

i > 0, x∗

i Gi(x∗

i ) − y∗

i pi(x∗

i ) = 0,

fi(y∗

i ) + γipi(x∗

i ) = 0.

(3.6)

Remark 1. We are only interested to find the positive steady state of the system.Therefore, it follows from (3.1), ui < ai/αi and hence Gi(ui) ≥ gi(ui), ∀ ui.Now, from (3.6) we get

y∗

i =x∗

i Gi(x∗

i )

pi(x∗

i )> y

∗∗

i , x∗

i > x∗∗

i ,

where the non-zero positive x∗∗

i and y∗∗

i are equilibrium value of the above prey-predator system without supplementary resource, given by

x∗∗

i gi(x∗∗

i ) − y∗∗

i pi(x∗∗

i ) = 0, (3.7)

fi(y∗∗

i ) + γipi(x∗∗

i ) = 0. (3.8)

Hence in presence of a supplementary resource for the prey population, the level ofsteady state distributions of both the species are higher at each location in the habitat.

Example 1. Now, we discuss a numerical example in which the behavior of the steadystate solutions of the above system is studied. The results are compared with thecase of a prey-predator system without supplementary resource. We consider thefollowing particular form of functions:

gi(ui) = ri

(

1 −

ui

Ki

)

, fi(vi) = si

(

1 −

vi

Mi

)

, pi(ui) = eiui, i = 1, 2 .

For simplicity let assume that the supplementary resource initially is distributed uni-formly, i.e. C1 = C2 = C. Then the steady state system (3.4) and (3.5), becomes

D1i

d2ui

ds2

+ ui

[

ri

(

1 −

ui

Ki

)

+θαiC

ai

(ai − αiui)

]

− eiviui = 0,

D2i

d2vi

ds2

+ sivi

(

1 −

vi

Mi

)

+ γieiviui = 0

(3.9)

with reservoir boundary conditions

14 J. Dhar

u1(0) = x∗

1, u2(L2) = x∗

2, (3.10)

v1(0) = y∗

1 , v2(L2) = y∗

2 ,

where x∗

i , y∗

i are from (3.6), and the continuity-flux matching conditions at the inter-face s = L1 are given as

D11

du1

ds

(L1) = D12

du2

ds

(L1), D21

dv1

ds

(L1) = D22

dv2

ds

(L1),

u1(L1) = u2(L1), v1(L1) = v2(L1) . (3.11)

The equations (3.9) – (3.11) are solved numerically by using finite–differencemethod, for the following set of dimensionless values

L1 = 10, L2 = 20, D11 = 0.8, D12 = 0.9, D21 = 0.8, D22 = 0.9 ,

r1 = 0.03, r2 = 0.025, a1 = 1.0, a2 = 1.0, s1 = 0.03, s2 = 0.01 ,

e1 = 0.00005, e2 = 0.00005, K1 = 100, K2 = 125, M1 = 75, M2 = 50 ,

α1 = 0.00005, α2 = 0.00002, C = 60, γ1 = 0.4, γ2 = 1.0, θ = 0.7 .

By using above values of the parameters, we get

x∗

1 = 87.2 > x∗∗

1 = 80.77, x∗

2 = 118.58 > x∗∗

2 = 109.09 ,

y∗

1 = 118.60 > y∗∗

1 = 115.385, y∗

2 = 168.58 > y∗∗

2 = 159.09 .

We can easily verify that in presence of a supplementary resource for the prey, thelevel of steady state distributions of both species are higher at each location of thehabitat compared to the case without supplementary resource for prey population(see Fig. 1). Moreover the steady state distribution is continuous and monotonicfunction.

Now, we consider the following assumptions: For every

minx∗∗

1 , x∗∗

2 ≤ ui ≤ maxx∗∗

1 , x∗∗

2 ,

min y∗∗

1 , y∗∗

2 ≤ vi ≤ max y∗∗

1 , y∗∗

2 , i = 1, 2

we have that:

(ui − x∗∗

i )[uiGi(ui) − vipi(ui)] < 0, ∀ui 6= x∗∗

i ,

(vi − y∗∗

i )[vifi(vi) + γivipi(ui)] < 0, ∀vi 6= y∗∗

i .

Under these conditions ui and vi both will be positive through out the habitat.We now consider without loss of generality 0 < x

∗∗

1 < x∗∗

2 and 0 < y∗∗

1 < y∗∗

2 .Therefore x

∗∗

1 ≤ ui ≤ x∗∗

2 and y∗∗

1 ≤ vi ≤ y∗∗

2 . Again, from (3.4) and (3.5) underreservoir boundary conditions, let pi(s, αi) and qi(s, βi) are unique solutions of ui

and vi respectively, for i = 1, 2, such that

∂p1

∂s

(0, α1) = α1, p1(0, α1) = x∗∗

1 ,

∂p2

∂s

(L2, α2) = α2, p2(L2, α2) = x∗∗

2 ,

∂q1

∂s

(0, β1) = β1, p1(0, β1) = y∗∗

1 ,

∂p2

∂s

(L2, β2) = β2, p2(L2, β2) = y∗∗

2 .


0 10 20 40

60

80

100

120

140

160

180

Patch-II Patch-I

Ste

ady

Sta

te D

istr

ibut

ions

s

Initial prey Prey with resource Initial predator Final predator Initial resource Final resource

Figure 1. The steady state solutions for both the species, with and without supplementaryresource for the prey.

Similarly, for no-flux boundary condition, let pi(s, αi) and qi(s, βi) are unique solu-tions of ui and vi respectively, for i = 1, 2, such that

∂p1

∂s

(0, α1) = 0, p1(0, α1) = α1,∂p2

∂s

(L2, α2) = 0, p2(L2, α2) = α2 ,

∂q1

∂s

(0, β1) = 0, p1(0, β1) = β1,∂p2

∂s

(L2, β2) = 0, p2(L2, β2) = β2 .

Then the existence of the monotonic solutions are established in both the reser-voir and no-flux boundary conditions, if we can show that there exists αi and βi, fori = 1, 2, such that

p1(L1, α1) = p2(L1, α2), q1(L1, β1) = q2(L1, β2) ,

D11

∂p1

∂s

(L1, α1) = D12

∂p2(L1, α2)

∂s

, D21

∂q1(L1, β1)

∂s

= D12

∂q2(L1, β2)

∂s

.

In order to construct our required solutions for reservoir boundary conditions, weneed some preliminary lemmas, in the same manner as in [3, 4].

Lemma 1. If α1, β1 > 0, then

∂p1(s, α1)

∂s

> α1,∂q1(s, β1)

∂s

> β1, on 0 < s ≤ L1.

Lemma 2. If α2, β2 > 0, 0 < p2 < x∗∗

2 and 0 < q2 < y∗∗

2 , then

∂p2(s, α2)

∂s

> α2,∂q2(s, β2)

∂s

> β2, L1 ≤ s < L2.

16 J. Dhar

Lemma 3. Let us define F1i(αi) by F1i(αi) = pi(L1, αi). Then there exists αi > 0such that

F11 : [0, α1] → [x∗∗

1 , x∗∗

2 ] , F12 : [0, α2] → [x∗∗

2 , x∗∗

1 ] .

Lemma 4. Let us define F2i(βi) by F2i(βi) = qi(L1, βi). Then there exists βi > 0such that

F21 :[

0, β1

]

→ [y∗∗

1 , y∗∗

2 ] , F22 :[

0, β2

]

→ [y∗∗

2 , y∗∗

1 ] .

Similar type of four lemmas we can established for the steady state system withthe no-flux boundary conditions. Hence we state the following theorem.

Theorem 1. (i) There exists a positive, continuous, monotonic solution of system(3.4) with continuous flux at L1.

(ii) There exists a positive, continuous, monotonic solution of system (3.5) with con-tinuous flux at L1.

Now we consider the stability analysis of the system (2.1) – (2.3), (2.8) – (2.13)with reservoir boundary conditions (2.4) and (2.5). First we state the local stabilityof the system by the following theorem.

Theorem 2. The steady-state, continuous, monotonic solutions of the system (2.1) –(2.3) with reservoir boundary conditions and continuous flux at the interface s = L1

are locally asymptotically stable provided the following conditions are satisfied:

Xi ≤ 0, Yi ≤ 0, Zi ≤ 0 , (3.12)

U

2i ≤ 4XiYi, XiYiZi ≤ YiW

2i + ZiU

2i ,

where

Xi = gi(ui) + uig′

i(ui) − vip′

i(ui) + θαiwi,

Yi = fi(vi) + vif′

i (vi) + γipi(ui), Wi =αi

2[θui − wi] ,

Zi = ai

(

1 −

2wi

Ci

)

− αiui, Ui =1

2[γivip

′

i(ui) − pi(ui)] ,

for x∗

1 ≤ ui ≤ x∗

2, y∗

1 ≤ vi ≤ y∗

2 , where x∗

i and y∗

i are given by (3.6).

Proof. We linearize (2.1), (2.2) and (2.3) by using

Ri(s, t) = wi(s) + ri(s, t) , (3.13)

xi(s, t) = ui(s) + ni(s, t), yi(s, t) = vi(s) + mi(s, t) ,

then we obtain the system


∂ri

∂t

= ri

[

ai

(

1 −

2wi

Ci

)

− αiui

]

− niαiwi,

∂ni

∂t

= ni[gi(ui) + uig′

i(ui) − vip′

i(ui) + θαiwi]

−mipi(ui) + riθαiui + D1i

∂2ni

∂s2

,

∂mi

∂t

= mi[fi(vi) + vif′

i (vi) + γipi(ui)] + niγivip′

i(ui) + D2i

∂2mi

∂s2

.

(3.14)Using (3.13) the following boundary and matching conditions are obtained:

n1(0, t) = 0 = n2(L2, t), m1(0, t) = 0 = m2(L2, t),

n1(L1, t) = n2(L1, t), m1(L1, t) = m2(L1, t),

D11

∂n1

∂s

(L1, t) = D12

∂n2

∂s

(L1, t), D21

∂m1

∂s

(L1, t) = D22

∂m2

∂s

(L1, t).

Now we consider the following positive definite function,

V (t) =

2∑

i=1

∫ Li

Li−1

1

2

(

n2i + m

2i + r

2i

)

ds . (3.15)

Differentiating (3.15) with respect to t, we get

V (t) =2

∑

i=1

∫ Li

Li−1

(

ni

∂ni

∂t

+ mi

∂mi

∂t

+ ri

∂ri

∂t

)

ds .

By using (3.14) we get

V (t) =

2∑

i=1

∫ Li

Li−1

n2i [gi(ui) + uig

′

i(ui) − vip′

i(ui) + θαiwi] ds +

2∑

i=1

×

∫ Li

Li−1

m2i [fi(vi) + vif

′

i (vi) + γipi(ui)] ds +

2∑

i=1

∫ Li

Li−1

r2i

[

ai

(

1 −

2wi

Ci

)

− αiui

]

ds +

2∑

i=1

∫ Li

Li−1

nimi[−pi(ui) + γivip′

i(ui)] ds +

2∑

i=1

∫ Li

Li−1

niriαi

× [θui − wi] ds +

2∑

i=1

D1i

∫ Li

Li−1

ni

∂2ni

∂s2

ds +

2∑

i=1

D2i

∫ Li

Li−1

mi

∂2mi

∂s2

ds.

Therefore,

V (t) =

2∑

i=1

∫ Li

Li−1

[Xin2i + Yim

2i + Zir

2i + 2Uinimi + 2Wirini] ds

−

2∑

1

D1i

∫ Li

Li−1

(

∂ni

∂s

)2

ds −

2∑

1

D2i

∫ Li

Li−1

(

∂mi

∂s

)2

ds , (3.16)

18 J. Dhar

where the functions Xi, Yi, Zi, Ui, and Wi are as follows,

Xi = gi(ui) + uig′

i(ui) − vip′

i(ui) + θαiwi,

Yi = fi(vi) + vif′

i(vi) + γipi(ui), Zi = ai

(

1 −

2wi

Ci

)

− αiui,

Ui =1

2[γivip

′

i(ui) − pi(ui)], Wi =αi

2[θui − wi]

hence V is negative definite, if conditions (3.12) of the theorem are satisfied fori = 1, 2.

Next, we state the corresponding nonlinear stability conditions of the system.

Theorem 3. The steady-state, continuous, monotonic solutions of nonlinear system(2.1) – (2.3), (2.8) – (2.13) with reservoir boundary conditions (2.4) – (2.5) areasymptotically stable in the sub-region

R = x∗

1 ≤ xi, ui ≤ x∗

2, y∗

1 ≤ yi, vi ≤ y∗

2 , i = 1, 2 ,

provided the following conditions are satisfied:

Nxi ≤ 0, Nyi ≤ 0, Nzi ≤ 0 , (3.17)

N

2ui ≤ 4NxiNyi, NxiNyiNzi ≤ NyiN

2wi + NziN

2ui ,

where

Nxi =xigi(xi) − uigi(ui)

xi − ui

− yi

pi(xi) − pi(ui)

xi − ui

+ θαiRi ,

Nyi =yifi(yi) − vifi(vi)

yi − vi

+ γipi(ui) ,

Nzi = ai

(

1 −

Ri + wi

Ci

)

− αiui, Nwi =αi

2[θui − Ri] ,

Nui =1

2

[

γiyi

pi(xi) − pi(ui)

xi − ui

− pi(ui)

]

.

Proof. By using (3.13), we get from (2.1), (2.2) and (2.3)

∂ri

∂t

= ri

[

ai

(

1 −

Ri + wi

Ci

)

− αiui

]

− niαiRi , (3.18)

∂ni

∂t

= ni

[

xigi(xi) − uigi(ui)

xi − ui

− yi

pi(xi) − pi(ui)

xi − ui

+ θαiRi

]

− mipi(ui) + riθαiui + D1i

∂2ni

∂s2

, (3.19)

∂mi

∂t

= mi

[

yifi(yi) − vifi(vi)

yi − vi

+ γipi(ui)

]

+ ni

[

γiyi

pi(xi) − pi(ui)

xi − ui

]

+ D2i

∂2mi

∂s2

. (3.20)


Here also we consider the same positive definite function as in the case of linearstability. By using (3.18), (3.19) and (3.20), we get

V (t) =2

∑

i=1

∫ Li

Li−1

n2i

[

xigi(xi) − uigi(ui)

xi − ui

− yi

pi(xi) − pi(ui)

xi − ui

+ θαiRi

]

ds

+

2∑

i=1

∫ Li

Li−1

m2i

[

yifi(yi) − vifi(vi)

yi − vi

+ γipi(ui)

]

ds

+

2∑

i=1

∫ Li

Li−1

r2i

[

ai

(

1 −

Ri + wi

Ci

)

− αiui

]

ds +

2∑

i=1

∫ Li

Li−1

nimi

×

[

γiyi

pi(xi) − pi(ui)

xi − ui

− pi(ui)

]

ds +2

∑

1

∫ Li

Li−1

riniαi[θui − Ri] ds

+2

∑

i=1

D1i

∫ Li

Li−1

ni

∂2ni

∂s2

ds +2

∑

i=1

D2i

∫ Li

Li−1

mi

∂2mi

∂s2

ds .

Therefore,

V (t) =

2∑

i=1

∫ Li

Li−1

[Nxin2i + Nyim

2i + Nzir

2i + 2Nuinimi + 2Nwirini] ds

−

2∑

i=1

D1i

∫ Li

Li−1

(

∂ni

∂s

)2

ds −

2∑

i=1

D2i

∫ Li

Li−1

(

∂mi

∂s

)2

ds , (3.21)

where the functions Nxi, Nyi, Nzi, Nui and Nwi are given by (3.18). Hence V isnegative definite if the conditions (3.17) hold for i = 1, 2.

It can be noted that, if we linearize the conditions of Thm.3, then we get theconditions of Thm. 2.

The same theorems are true for the system (2.1) – (2.3), (2.8) – (2.13) with no-flux boundary conditions (2.6) and (2.7).

3.2. The Uniform Equilibrium State

Similar as in the previous case, the main purpose of this section to find the conditionsfor local and global stability of the uniform equilibrium state of the system

xi(s, t) ≡ K∗

, yi(s, t) ≡ M∗

, Ri(s, t) ≡ C∗

, 0 ≤ s ≤ L2, t ≥ 0

under both sets of boundary conditions.

Theorem 4. The equilibrium (C∗

, K∗

, M∗) is locally asymptotically stable, if H

∗

i +θαiC

∗

≤ 0, for i = 1, 2, where H∗

i is given by

H∗

i = gi(K∗) + K

∗

g′

i(K∗) − M

∗

pi(K∗) (3.22)

and the following conditions are satisfied(

γiM∗

p′

i(K∗) − pi(K

∗))2

≤ 4(

H∗

i + θαiC∗

)

M∗

f′

i(M∗), i = 1, 2 . (3.23)

20 J. Dhar

Proof. We linearize the system (2.1) – (2.3) by using

Ri(s, t) = C∗ + ri(s, t), (3.24)

xi(s, t) = K∗ + ni(s, t), yi(s, t) = M

∗ + mi(s, t) ,

then we get

∂ri

∂t

= ri

[

−

aiC∗

Ci

]

− niαiC∗

,

∂ni

∂t

= ni[gi(K∗) + K

∗

g′

i(K∗) − M

∗

p′

i(K∗) + θαiC

∗]

−mipi(K∗) + riθαiK

∗ + D1i

∂2ni

∂s2

,

∂mi

∂t

= miM∗

f′

i (M∗) + niγiM

∗

p′

i(K∗) + D2i

∂2mi

∂s2

.

(3.25)

We consider the following positive definite function

V =1

2

2∑

i=1

∫ Li

Li−1

[

(xi − K∗)2 + (yi − M

∗)2 + di(Ri − C∗)2

]

, (3.26)

where di, i = 1, 2 are positive constants. Differentiating (3.26) and using (3.25), weget

V =

2∑

i=1

∫ Li

Li−1

n2i

(

H∗

i + θαiC∗

)

ds +

2∑

i=1

∫ Li

Li−1

m2i M

∗

f′

i(M∗) ds

+

2∑

i=1

∫ Li

Li−1

r2i

[

−

diaiC∗

Ci

]

ds +

2∑

i=1

∫ Li

Li−1

nimi

(

− pi(K∗)

+ γiM∗

p′

i(K∗)

)

ds +

2∑

i=1

∫ Li

Li−1

niri

(

αiθK∗

− diC∗

)

ds

+

2∑

i=1

∫ Li

Li−1

D1ini

∂2ni

∂s2

ds +

2∑

i=1

∫ Li

Li−1

D2imi

∂2mi

∂s2

ds . (3.27)

Using integration by parts and for both types of boundary conditions, we get

2∑

i=1

∫ Li

Li−1

D1ini

∂2ni

∂s2

ds = −

2∑

i=1

D1i

∫ Li

Li−1

(

∂ni

∂s

)2

ds,

2∑

i=1

∫ Li

Li−1

D2imi

∂2mi

∂s2

ds = −

2∑

i=1

D2i

∫ Li

Li−1

(

∂mi

∂s

)2

ds.

We choose di, i = 1, 2, such that, coefficients of niri become zero, i.e. d1 = d2 =θK

∗

/C∗. Therefore it follows from (3.27) that V is negative definite, if the condi-

tions H∗

i + θαiC∗

≤ 0 and (3.23) are satisfied.


Theorem 5. Let H∗

i + θαiC∗

> 0. Then the equilibrium (C∗

, K∗

, M∗) is locally

asymptotically stable, if the conditions (3.23) and the following inequality hold:

H∗

i + θαiC∗

≤ D1i

π2

(Li − Li−1)2, i = 1, 2 .

Proof. From (3.28) and using Poincare’s inequality we get

D1i

∫ Li

Li−1

(

∂ni

∂s

)2

ds ≤ D1i

π2

(Li − Li−1)2

∫ Li

Li−1

n2i ds .

Therefore from (3.27) we get

V ≤

2∑

i=1

∫ Li

Li−1

n2i x

(

H∗

i + θαiC∗

− D1i

π2

(Li − Li−1)2

)

ds

+2

∑

i=1

∫ Li

Li−1

m2i M

∗

f′

i(M∗) ds +

2∑

i=1

∫ Li

Li−1

r2i

(

−

diaiC∗

Ci

)

ds

+

2∑

i=1

∫ Li

Li−1

nimi

(

− pi(K∗) + γiM

∗

p′

i(K∗)

)

ds −

2∑

i=1

D2i

∫ Li

Li−1

(

∂mi

∂s

)2

ds .

Hence the theorem is proved.

We now state the global stability of the uniform steady state.

Theorem 6. The uniform steady–state (C∗,K∗,M∗) is globally asymptotically sta-ble if

Ai(xi) =xigi(xi) − M

∗

pi(xi)

xi − K∗

+ θαiC∗

< 0, ∀xi 6= K∗

, (3.28)

(

γi

pi(xi) − pi(K∗)

xi − K∗

−

pi(xi)

xi

)2

≤ 4Ai(xi)

xi

(

fi(yi) − fi(M∗)

yi − M∗

)

, (3.29)

Ai(xi)

xi

(

fi(yi) − fi(M∗)

yi − M∗

) (

ai

Ci

)

≤

(

fi(M∗) − fi(yi)

yi − M∗

)

(

α2i (θ − Ri)

2)

+ai

Ci

(

Ai(xi)

xi

)2

. (3.30)

Proof. Let us consider the following positive definite function

V (x, y, R) =2

∑

i=1

∫ Li

Li−1

(

xi − K∗

− K∗

ln

xi

K∗

)

ds +2

∑

i=1

∫ Li

Li−1

(

yi − M∗

− M∗ ln

yi

M∗

)

ds +2

∑

i=1

∫ Li

Li−1

(

Ri − C∗

− C∗

ln

Ri

C∗

)

ds. (3.31)

22 J. Dhar

Differentiating (3.31) with respect to t and using (2.1) – (2.3) we get

V (s, t) =

2∑

i=1

∫ Li

Li−1

(

xi − K∗

xi

)

∂xi

∂t

ds +

2∑

i=1

∫ Li

Li−1

(

yi − M∗

yi

)

∂yi

∂t

ds

+

2∑

i=1

∫ Li

Li−1

(

Ri − C∗

Ri

)

∂Ri

∂t

ds =

2∑

i=1

∫ Li

Li−1

(xi − K∗)2

xi

×

(

xigi(xi) − M∗

pi(xi)

xi − K∗

+ θαiC∗

)

ds +2

∑

i=1

∫ Li

Li−1

(yi − y∗)2

×

(

fi(yi) − fi(M∗)

yi − M∗

)

ds −

2∑

i=1

∫ Li

Li−1

(Ri − C∗)2

(

−

ai

Ci

)

ds

+2

∑

i=1

∫ Li

Li−1

(xi − K∗)(yi − M

∗)

(

γi

pi(xi) − pi(K∗)

xi − K∗

−

pi(xi)

xi

)

ds

+

2∑

i=1

∫ Li

Li−1

(xi − K∗)(yi − M

∗)αi (θ − Ri) ds

+2

∑

i=1

D1i

∫ Li

Li−1

xi − K∗

xi

∂2xi

∂s2

ds +2

∑

i=1

D2i

∫ Li

Li−1

yi − M∗

yi

∂2yi

∂s2

ds .

Using both set of boundary and flux matching conditions

(

x1(0, t) − K∗

)∂x1

∂s

(0, t) = 0,

(

x2(L2, t) − K∗

)∂x2

∂s

(L2, t) = 0 ,

(

y1(0, t) − M∗

)∂y1

∂s

(0, t) = 0,

(

y2(L2, t) − M∗

)∂y2

∂s

(L2, t) = 0

we get

2∑

i=1

D1i

∫ Li

Li−1

xi − K∗

xi

∂2xi

∂s2

ds = −

2∑

i=1

D1i

∫ Li

Li−1

K∗

x2i

(

∂xi

∂s

)2

ds ,

2∑

i=1

D2i

∫ Li

Li−1

yi − M∗

yi

∂2yi

∂s2

ds = −

2∑

i=1

D2i

∫ Li

Li−1

M∗

y2i

(

∂yi

∂s

)2

ds .

Now if conditions (3.28), (3.29) and (3.30) hold, then V (x, y) < 0, andV (C∗

, K∗

, M∗) = 0. Therefore V (x, y) is negative definite over R > 0, x > 0,

y > 0 with respect to R∗

i = C∗, x

∗

i = K∗, y

∗

i = M∗, proving the theorem.

Remark 2. We conclude that the role of supplementary resource is to increase thelevel of nonuniform steady state distributions of both the species at each location ofthe linear habitat. Further, the number of conditions for stability is increased com-pared to the case with no supplementary resource for prey and the role of patchinessis destabilizing in present of supplementary resource.


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[11] D. Herbert, Elsworth and R.C. Telling. The continuous culture of bacteria: a theoreticaland experimental study. J. Gen. Microbiology, 4, 601 – 622, 1956.

[12] S.B. Hsu. On a resource based ecological competition model with interference. J. Math.Biol., 12, 45 – 52, 1981.

[13] S.B. Hsu, S.P. Hubbell and P. Waltman. A mathematical theory of single-nutrient com-petitions in continuous cultures of microorganism. SIAM J. Appl. Math., 32, 366 – 383,1977.

[14] M. Kot. Elements of Mathematical Ecology. Cambridge University Press, Cambridge,2001.

[15] S.A. Levin. Dispersion and population interactions. Am. Nat., 108, 207 – 228, 1974.[16] S.A. Levin. Spatial patterning and the structure of ecological communities. Lectures on

Mathematics in the Life Sciences, 8, 1976.[17] B. Li and H. Smith. How many species can two essential resources support? SIAM J.

Appl. Math, 62(1), 336 – 366, 2001.[18] R. McMurtrie. Persistence and stability of single species and prey-predator system in

spatially heterogeneous environments. Math. Biosc., 39, 11 – 51, 1978.[19] D. Mitra, D. Mukherjee and A.B. Roy. Permanent coexistence in a resource-based com-

petition system. Ecol. Model., 60, 77 – 85, 1992.[20] R.E. Munn and V. Fedorov. The environmental assessment. In: IIASA Project Report,

Vol.1, International Institute for Applied Systems Analysis, Laxenburg, Austria, 1984.[21] R. Nallaswamy and J.B. Shukla. Effects of dispersal on the stability of a Prey-Predator

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Plešruno-aukos modelis su difuzija ir papildomu resursu aukai dvieju sriciu areale

J. Dhar

Šiame straipsnyje modeliuojama ir analizuojama plešr unu ir auku dinamika, laikant, kadplešr unu populiacija dalinai priklauso nuo auku skaiciaus. Areala sudaro dvi sritys, kuriosevyksta populiaciju individu difuzija, be to, aukoms yra išskirtas nedifunduojantis resursas.

Irodyta, kad egzistuoja teigiamas, monotoniškas, tolydus stacionarusis sprendinys, tenki-nantis tolydumo salyga abiems populiacijoms atskirai. Gautos asimptotinio stabilumo salygostiesiniu ir netiesiniu atvejais.

!"#$%'&)($+*,.-0/213435/768

25–38c© 2004 Technika ISSN 1392-6292

NUMERICAL SOLUTION OF SINGLE MODEGYROTRON EQUATION

O. DUMBRAJS1, H. KALIS2 and A. REINFELDS2

1Helsinki University of Technology

FIN-02150, Espoo, Finland

E-mail: 90:;<=>?@AB4@BDCFEG2Institute of Mathematics Latvian Academy of Sciences and University of Latvia

Akademijas laukums 1, LV-1524, Rıga, Latvia

E-mail: H>IG4@JAI4>KLMNCOIP%Q=L#GRKEAI0>M4KL0MSCOIJPReceived October 12 2003; revised December 15 2003

Abstract. In this paper we study numerical problems arising in solving the single modegyrotron equation. Using the method of finite differences analytical and numerical solutionsare obtained. Quasistationary solutions and corresponding eigenvalues and eigenfunctions ofthis problem are investigated.

Key words: the spectral problems, finite-difference schemes, stability analysis, analyticalsolutions

1. Introduction

Gyrotrons are microwave sources whose operation is based on the stimulated cy-clotron radiation of electrons oscillating in a static magnetic field. Single mode non-stationary gyrotron oscillations can be described by the following system of partialdifferential equations [2]:

∂p

∂x

+ i

(

∆ + |p|2− 1

)

p = if(t, x) ,

∂2f

∂x2− i

∂f

∂t

+ δf =I

2π

∫ 2π

0

p dθ0.

(1.1)

Here i =√

−1, x ∈ [0, L] is the normalized axial coordinate, t is the normalizedtime, ∆ is the frequency mismatch, δ describes variation of the critical frequencies,∆ and δ are real numbers, p = p(t, x, θ0) is the dimensionless complex transversemomentum of the electron, f = f(t, x) is the high-frequency field in resonator, I

26 O. Dumbrajs, H. Kalis, A. Reinfelds

is the dimensionless current, θ0 is the parameter. The system of equations has to besupplemented by the standard initial condition

p(t, 0, θ0) = exp(iθ0), 0 ≤ θ0 < 2π, f(0, x) = f0(x) ,

and by the boundary condition for the field at the entrance to the interaction spacef(t, 0) = 0, and at the exit to the interaction space

∂f(t, L)

∂x

= −iγf(t, L),

where γ is a positive parameter, f0(x) is given complex function. An efficient nu-merical method for solving this reduced system of equations was presented in [1].However, it was discovered that the results of the computations depend in a nontrivialmanner on the chosen spatial and temporal step-lengths. So, main difficulties arisesin numerical solving of Schrödingers type equation with special boundary condi-tions. The aim of this paper is to study in detail numerical problems for the secondequation of (1.1).

2. Solution of the differential problem

We begin with the homogeneous Schrödinger type partial differential equation (I =0)

∂2f

∂x2− i

∂f

∂t

+ δf = 0, (2.1)

where x ∈ (0, L), t > 0 – is time, δ = const1. Boundary conditions can be written

as

f(t, 0) = 0,

∂f(t, L)

∂x

= −iγf(t, L). (2.2)

We represent the quasi-stationary solution of the problem (2.1) and (2.2) in the form

f(t, x) = g(x) exp(iαt), (2.3)

where α is a complex number α = α1 + iα2 (α2 is a temporal damping factor:if α2 > 0, the solution (2.3) decreases, if α2 < 0, the solution increases, and forα2 = 0 the solution is oscillating in time). We now consider nontrivial solutions (2.3)of the differential problem by computing allowed values of the parameter α, as wellas the corresponding discrete problem. Substituting the solution (2.3) into equation(2.1) and boundary conditions (2.2), we obtain the Sturm-Liouville problem for theordinary differential equation

g′′(x) + λ

2g(x) = 0,

g(0) = 0, g′(L) = −iγg(L),

(2.4)

1 Using the substitution g(t, x) = f(t, x) exp(iδt) for function g we would obtain theboundary value problem (2.1) and (2.2) with δ = 0. We don’t use mentioned substitutionbecause function f and parameter δ have the physical interpretation

Numerical solution of single mode gyrotron equation 27

where λ2 = α + δ is complex value. The solution of problem (2.4) is

g(x) = C1 sin(λx),

where C1 is an arbitrary constant. From boundary conditions we obtain a transcen-dental complex equation for calculating the eigenvalue λ:

λ cos(λL) + iγ sin(λL) = 0

orz cos z = −iγ sin z, (2.5)

where z = z1 + iz2 = λL and γ = γL. It is obvious that z = 0 is a root of theequation. Moreover, if z is the root of (2.5), then also −z is the root of this equation.Therefore we can confine ourselves to consider only z1 > 0. Separating real andimaginary parts in equation (2.5), we obtain a system of two real transcendentalequations

z1 cos z1 cosh z2 + z2 sin z1 sinh z2 = γ cos z1 sinh z2,

z2 cos z1 cosh z2 − z1 sin z1 sinh z2 = −γ sin z1 cosh z2.

(2.6)

Multiplying the first equation of system (2.6) by sin z1 cosh z2 and the second equa-tion by cos z1 sinh z2 and summing, we exclude the parameter γ and obtain the rela-tion

z1 sin(2z1) + z2 sinh(2z2) = 0 .

It follows that the nontrivial roots of the last equation satisfy the inequality

sin(2z1) < 0 or tan(z1) < 0 .

Dividing the second equation of the system (2.6) by cos z1 cosh z2, we obtain

z2 − z1 tan z1 tanh z2 = −γ tan z1.

It can be seen that if z1 > 0, then also z2 > 0. Let us number the roots of (2.5) z(k),

k = 1, 2, . . ., whose real parts z(k)

1 are positive, by increasing their real parts and

take into account that (k − 1)π < z(k)

1 < kπ, 0 < z(k)

2 < γ + 1. Since λ =√

α + δ

or z2 = L

2(α + δ), we have

α1 =z21 − z

22

L2

− δ, α2 =2z1z2

L2

.

It is seen that the parameter δ affects only the values of α1 and α2 > 0. The resultsof computations performed by means of ”MAPLE” for L = 15, γ = 2, and δ = 0

are summarized in Tab. 1 for the first eight eigenvalues and numerical values of α(k)

1

and α(k)

2 , k = 1, 8. Fig. 1 shows the first fifty eigenvalues λk . It can be seen that

α(k)

2 > 0 and that all solutions

f(k)(t, x) = sin

(√

α(k)

1 + δ + iα(k)

2 x

)

exp((−α(k)

2 + iα(k)

1 )t)


Table 1. The roots of equation (2.6) and values of α(k)

.

k z(k)

1z(k)

2α

(k)

1α

(k)

2

1 3,1381 0,10498 0,0437 0,002932 6,2758 0,21232 0,1748 0,011843 9,4128 0,32466 0,3933 0,027164 12,5484 0,44547 0,6989 0,049695 15,6814 0,57970 1,0914 0,080806 18,8092 0,73539 1,5700 0,122957 21,9251 0,92732 2,1327 0,180738 25,0054 1,18594 2,7727 0,26360

Continue_eigenvalues lambda

0.02

0.04

0.06

0.08

0.1

Im(lambda)

2 4 6 8 10

Re(lambda)

Figure 1. Eigenvalues of the continuous problem λk, k = 1, 59.

monotonically decrease in time, i.e.,

f(k)(x, t) → 0, t → +∞, k = 1, 2, . . . ,

(here C1 = 1). Taking the square root in expression, we obtain two complex num-bers in the form ±(a(k) + ib

(k)), where a(k)

> 0 and b(k)

> 0, if α(k)

1 + δ > 0 and

α(k)

2 > 0. Since the functions f(k)(t, x) contain an arbitrary constant C1, the com-

plex number with the minus sign does not give us any new result and can be ignored.Separating real and imaginary parts we obtain

|f(k)(t, x)| = exp(−α

(k)

2 t)

√

sinh2(b(k)x) + sin2(a(k)

x).

Let us note that the complex eigenfunctions gk(x) = sin(λkx) (λk =√

α(k) + δ

and α(k) = α

(k)

1 + iα(k)

2 ) are orthogonal, i.e.,

〈gk, gk〉 =

∫ L

0

gk(x)gn(x) dx = 0, k 6= n.

Correspondingly


‖gk‖2 = 〈gk, gk〉 =

∫ L

0

g2k(x) dx =

1

2

(

L +iγ

λ2k − γ

2

)

.

Each continuous function f(x), x ∈ (0, L) with boundary conditions (2.4) can beexpanded in the series of the orthonormalized eigenfunctions gk(x) = gk(x)/‖gk‖

f(x) =

∞

∑

k=1

ckgk(x) ,

where the expansion coefficients can be found in the form ck = 〈f , gk〉. Calculatingby means of ”MAPLE”, we obtain that the oscillation frequency of the functionsincreases and their absolute values rapidly decrease with increasing k.

3. Solution of the discrete problem

In the finite differences method we use a uniform homogeneous spatial and temporalgrids:

ωh = xj : xj = jh, j = 1, N − 1, Nh = L , ωτ = tn : tn = nτ, n ≥ 1

(corresponding step-lengths are h and τ ). We substitute the continuous functionf = f(t, x) in these grids by the discrete grid function y = y(t, x), t ∈ ωτ , x ∈ ωh

with values y(tn, xj) ≡ ynj . The corresponding derivatives of the function we ap-

proximate by finite–differences

∂2f(tn, xj)

∂x2

≈ Λynj ≡

ynj+1 − 2y

nj + y

nj−1

h2

, (3.1)

∂f(tn, xj)

∂t

≈

yn+1

j − ynj

τ

,

∂f(tn, L)

∂x

≈

ynN − y

nN−1

h

. (3.2)

Difference (3.2) approximates the first derivative only to the first order of accuracy,i.e., O(h). To obtain the second order approximation, we must use the expression

∂f(tn, L)

∂x

≈

1.5ynN − 2y

nN−1 + 0.5y

nN−2

h

. (3.3)

Substituting differences (3.1), (3.2) into the problem (2.1) – (2.2), we obtain a two-layer finite-difference scheme with weight σ ∈ [0, 1]

i

yn+1

j − ynj

τ

= σ(Λyn+1

j + δyn+1

j ) + (1 − σ)(Λynj + δy

nj ), j = 1, N − 1 ,

yn+10 = 0,

yn+1

N − yn+1

N−1

h

= −iγyn+1

N .

(3.4)Difference equations (3.4) approximate the initial differential equation (2.1) to thesecond order both in space and time, if σ = 1/2, and to the first order in time, ifσ 6= 1/2. Boundary conditions (2.2) are approximated only to the first order. To


obtain the second order, one has to use expression (3.3). Seeking to find the discretequasi-stationary solution by analogy to (2.3) we take

ynj = gj exp(iαnτ) (gj = g(xj), tn = nτ),

then we obtain that the discrete function gj 6≡ 0 satisfies the three-point finite-difference scheme

Λgj + µ2gj = 0, j = 1, N − 1 ,

g0 = 0, gN = CgN−1,

(3.5)

which approximates the continuous problem (2.4). Here

C = (1 + iγh)−1, µ

2 = α + δ, α =(1 − exp(iατ))i

(σ exp(iατ) + 1 − σ)τ(3.6)

are complex constants (α → α, if τ → 0). Now the solution of (3.5) can be writtenas gj = C1 sin(qxj), where C1 is arbitrary constant, 1 − µ

2h

2/2 = cos(qh) and

xj = jh. It follows from boundary conditions (3.5) that the complex parameter q

has to be determined from the complex transcendental equation

sin(qL) = C sin(q(L − h)), (3.7)

where the parameter q has complex values

qk = ak + ibk, k = 1, N − 1. (3.8)

If γ = ∞ (boundary conditions of the first kind), then C = 0 and equation sin(qL) =0 is valid, if qk = kπ

L(real numbers). Then we get also real eigenvalues [4]

µ2k =

4

h2

sin2 kπh

2L

, k = 1, N − 1 .

Thereforeαk = 2h

−2(1 − cos(qkh)) − δ = Ak + iBk, (3.9)

whereAk = 2h

−2(1 − cos(akh) cosh(bkh)) − δ,

Bk = 2h−2 sin(akh) sinh(bkh), k = 1, N − 1.

(3.10)

Since C = C1 + iC2, C1 = (1 + (γh)2)−1, and C2 = −γh(1 + (γh)2)−1, weseparate in equation (3.7) real and imaginary parts and obtain the system of two realtranscendental equations

sin(akL) cosh(bkL) = C1 sin(akl1) cosh(bkl1) − C2 cos(akl1) sinh(bkl1),

cos(akL) sinh(bkL) = C1 cos(akl1) sinh(bkl1) + C2 sin(akl1) cosh(bkl1),(3.11)

where l1 = L−h. If h → 0, then akL → z1, bkL → z2 and we obtain the system ofequations (2.6). After calculation of αk, we obtain from (3.6) the approximate values

αk =1

iτ

ln

(

1 −

ταk

i + σταk

)

, k = 1, N − 1. (3.12)


It can be seen from (3.12) that the temporal step-length τ and the parameter of thescheme σ, i.e., the temporal approximation, do not affect the values of αk, theirchanges have to be taken into account only in the expression (3.12). Approximatingboundary conditions (2.4) by the second order expression (3.3), instead of equation(3.7) we obtain the complex transcendental equation

sin(qL) = C∗(2 sin(ql1) − 0, 5 sin(ql2)),

where l2 = L−2h, C∗ = (1, 5+ iγh)−1. The results of computations with L = 15,

γ = 2, δ = 0, τ = h = 0, 1, and σ = 1 are presented in Tab. 2, where ak and bk aresolutions of (3.11) and

π(k − 1)

L

< ak <

πk

L

, 0 < bk < 1 .

The values Ak and Bk were obtained from (3.9) and (3.10) k = 1, 8. The resultsdo not change much (five digits remain the same) by changing the temporal step-length τ in interval (0, 01, 0, 1). More accurate results can be obtained with σ = 0.5.Comparing the solutions of the continuous and discrete problems, we see that onlyfor the first two eigenvalues three or four digits remain the same, while for othereigenvalues the accuracy rapidly deteriorates. Using the second order approximationeven for the eighth eigenvalue two digits are correct, if σ = 1/2. Considering onlythe spatial discretization (the variable x is discretized xj = jh and the variable t iscontinuous), we obtain (by means of the method of lines) the boundary problem for

Table 2. The discrete values qkL, αk.

k akL bkL Ak Bk

1 3,1380 0,1050 0,0437 0,00292 6,2745 0,2115 0,1748 0,01183 9,4095 0,3240 0,3929 0,02714 12,5400 0,4440 0,6976 0,04945 15,6630 0,5745 1,0879 0,07986 18,7725 0,7215 1,5618 0,12017 21,8535 0,8925 2,1154 0,17288 24,8805 1,0875 2,7396 0,2393

the system of complex ordinary differential equations

i

dyj

dt

= Λyj + δyj , j = 1, N − 1 ,

y0 = 0,

yN − yN−1

h

= −iγyN ,

where yj = yj(t) are continuous functions of time, j = 0, N . Seeking the quasi-stationary solution of this system in the form

yj(t) = gj exp(iαt)


we obtain the system similar to (3.5) where µ2 = α + δ. This means that the quan-

tities Ak + iBk in expression (3.10) are approximate eigenvalues αk, k = 1, N − 1,obtained by means of the method of lines (see Tab. 2). Using in the boundary condi-tions the second order approximation, we obtain an analogous problem, which, justas the grid method, gives more accurate results. In oder to increase the accuracy ofdiscrete equation (3.5) we will use the Taylor expansion

Λg(xi) = g′′(xi) +

h2

12g(4)(xi) + . . . +

2h2m−2

(2m)!g(2m)(xi) + O(h2m),

where m = 1, 2, . . .. From equation (2.4) it follows that

Λg(xi) =2g(xi)

h2

µ2m ,

where

µ2m =

(

−

(λh)2

2!+

(λh)4

4!+ . . . + (−1)m (λh)2m

2m!+ O(h2m)

)

.

Similarly from boundary conditions (3.5) we obtain

g(xN−1) = g(xN ) − hg′(xN ) + . . . +

(−1)lh

l

l!g(l)(xN ) + O(hl+1), l ≥ 0 ,

and from the boundary condition of the problem (2.4) g(xN−1) = Ckg(xN ), where

Ck = 1 −

(hλ)2

2!+ . . . + (−1)k (hλ)2k−2

(2k − 2)!

+iγ

λ

(

hλ −

(hλ)3

3!+ . . . + (−1)k (hλ)2k−1

(2k − 1)!

)

+ O(h2k), k ≥ 1 .

It can be seen that the discrete problem (the errors are proportional to O(h2m) andO(h2k) m, k = 1, 2, . . .) is given in the form

Λgi + µ2mgi = 0, i = 1, N − 1 ,

g0 = 0, gN = C−1

k gN−1.

It can be seen that in the limit case (m → ∞, k → ∞) µ2m →

2

h2 (1 − cos(hλ))and Ck → cos(hλ) + iγλ

−1 sin(hλ) or we obtain the transcendental equation (2.5).Eigenfunctions of the discrete problem (3.5)

g(k)

j ≡ g(k)(xj) = sin(qkxj), xj = jh, j = 0, N, k = 1, N − 1

are orthogonal with respect to the scalar product 〈g(k), g

(n)〉 ≡ h

∑N

j=1g(k)

j g(n)

j ,

i.e., 〈g(k), g

(n)〉 = 0, if k 6= n. This follows from the Green formula [4]. Evaluat-

ing ‖g(k)

‖2 = 〈g

(k)g(k)

〉, we obtain orthonormalized system g(k) = g

(k)/‖g

(k)‖,

for which 〈g(k)

, g(n)

〉 = δk,n (the Kronecker symbol). Considering the second or-der approximation for the boundary condition (3.3), we cannot obtain a system oforthogonal eigenfunctions. Evaluating ‖g

(k)‖2, we obtain


‖g(k)

‖

2 =1

2

(

L −

h sin(qkL) cos(qk(L − h))

sin(qkh)

)

.

If h → 0, then qk → λk and ‖g(k)

‖2→ ‖gk‖

2. If boundary conditions (3.4) are

given as yn+1

N = 0 (γ = ∞), then qk =kπ

L

and ‖g(k)

‖2 =

L

2[4]. Each grid

function f(x), x ∈ ωh with boundary conditions (3.5) can be expanded as a finitesum

f(x) =

N∑

k=1

ckg(k)(x)

of orthonormalized eigenfunctions g(k)(x) = g

(k)(x)/‖g(k)‖, x ∈ ωh, where the

expansion coefficients can be found with the help of the expressions ck = 〈f , g(k)

〉.The solution of the boundary problem

Λg = −f(x), x ∈ ωh ,

g(0) = 0, g(L) = Cg(L − h)

is g(x) =∑N

k=1ckg

(k)(x)/µ2k.

4. Stability of the difference scheme

To study the stability of the discrete problem (difference scheme) (3.4), we rewritethe difference equations with respect to the difference zj = y

nj − f(xj , tn) in the

matrix operator form

(E + iτσ(Λ + δ))zn+1 = (E − iτ(1 − σ)(Λ + δ))zn,

where zn = (zn

1 , zn2 , . . . , z

nN−1)

T is the error vector-column and E is the unit oper-ator. Hence z

n+1 = Gzn, where

G = (E + iτσ(Λ + δ))−1(E − iτ(1 − σ)(Λ + δ))

is the transition operator with the eigenvalues

λk =1 + iτ(1 − σ)(µ2

k − δ)

1 − iτσ(µ2k − δ)

, k = 1, N − 1,

where µ2k are eigenvalues of the difference operator (−Λ) to be determined from the

boundary problem (3.5) µ2k = 2h

−2(1 − cos(qkh)). If µ2k are real numbers, e.g., in

the case of the first kind boundary conditions (γ = ∞, zn+1

N = 0) qk = kπL

, thenfrom the stability condition [4]

|λk|2 = (1 + τ

2(1 − σ)2(µ2k − δ)2)(1 + τ

2σ

2(µ2k − δ)2)−1

≤ 1,

it follows that

σ ≥

1

2(4.1)


independent of the size of the temporal step-length τ . Similar problem for Schrödin-ger type differential equation was investigated in [3]. Taking the boundary conditionof the third kind in the form z

n+1

N = Czn+1

N−1and determining the complex parame-

ter qk in the form (3.8), we find from (3.11) that µ2k = αk + δ, where αk can be

determined from (3.9) and (3.10). Then from

|λk|2 =

(1 − τ(1 − σ)Bk)2 + A2kτ

2(1 − σ)2

(1 + τσBk)2 + A2kτ

2σ

2≤ 1

it follows that−2Bk + τ(1 − 2σ)(A2

k + B2k) ≤ 0.

If σ ≥

1

2and Bk ≥ 0, then this inequality holds and the difference scheme (3.4) is

stable. If σ = 1, we obtain the inequality

τ ≥ −2Bk(A2k + B

2k)−1

, (4.2)

which is important, if Bk < 0. It is seen from (3.11) that, if (ak, bk) is a solutionof this system, then also (−ak,−bk) is a solution. The values of the coefficients Ak,Bk do not change and it is sufficient to consider only ak > 0. If simultaneouslybk > 0, then also Bk > 0, and the stability condition holds in the form (4.1). Ifak = bk = 0, then Bk = 0 and the difference scheme is stable. Calculations with thehelp of ”MAPLE” show that positive variables ak correspond to positive variablesbk i.e., Bk > 0. If the parameter γ < 0, then it can be easily seen that positive ak

correspond to negative bk and the difference scheme (3.4) is absolutely unstable, ifthe temporal step-length τ is not large enough (in inequality (4.2) Bk < 0).

5. Method of separation of variables

Let us consider the inhomogeneous equation

∂2f

∂x2− i

∂f

∂t

+ δf = F (t, x), (5.1)

with a given function F (t, x). We seek the solution f = f(t, x) with the boundaryconditions (2.2) in the form of a series

f(t, x) =

∞

∑

k=1

ak(t)gk(x), (5.2)

where gk(x) are orthonormalized eigenfunctions and λkL = z(k) = z

(k)

1 + iz(k)

2 aresolutions of (2.6). To determine functions ak(t), we use the given initial conditionsf(0, x) = f0(x). Taking a scalar product of (5.2) and a fixed eigenfunction, if t = 0,we obtain ak(0) = 〈f0, gk〉. By analogy expanding the right-hand side of (5.1)

F (x, t) =

∞

∑

k=1

Fk(t)gk(x)


we obtain Fk(t) = 〈F, gk〉. Assuming that series (5.2) and the series, differentiatedtwice with respect to x and once with respect to t, uniformly converge, and substi-tuting it into (5.1), we obtain the ordinary differential equation

−λ2kak(t) − iak(t) + δak(t) = Fk(t), t > 0

and

ak(t) = ak(0) exp(iα(k)t) + i

∫ t

0

exp(iα(k)(t − ζ))Fk(ζ) dζ,

where ak(t) =dak

dt

, α(k) = λ

2k − δ. As example, if f0(x) = sin

(

πxL

)

, F = 0, then

solutions of the differential problem can be obtained in the form

f(t, x) = 2πL

∞

∑

k=1

exp(iα(k)t) sin(λkx) sin(λkL)(λ2

k − γ2)

(π2− λ

2kL

2)(λ2kL − γ

2L + iγ)

.

Solving the corresponding discrete problem with the initial condition

yj0 = sin(

π

L

xj) ≡ f0(xj), j = 0, N ,

we obtain ynj =

∑N−1

k=1a

nk g

(k)(xj), where g(k)(xj) = sin(qkxj)/‖g

(k)‖ are discrete

eigenfunctions. Determining a0k from the initial condition

a0k = (f0, g

(k)) = hbk, bk =N−1∑

s=1

sin(π

L

sh)g(k)(sh),

we find from difference equations the recurrence relation an+1

k = ρkank or a

nk =

(ρk)na0k, where ρk = 1 + iτ αk/(1 − iτσαk). Hence

ynj = h

N−1∑

k=1

bk exp(iαkτn)g(k)(xj)

‖g(k)

‖2

, (5.3)

where

bk =sin( π

Lh) sin(qkL)

2(cos(qkh) − cos( πLh))

,

αk can be determined from (3.12), and αk from (3.9). It can be easily seen thaty

nj → f(tn, xj), if h → 0, τ → 0, i.e., the solution of the discrete problem converges

to the solution of the continuous problem. Using the method of lines (only spatialdiscretization), we obtain yj(t) =

∑N−1

k=1ak(t)g(k)(xj) by analogy, where functions

ak(t) are solutions of the Cauchy problem

dak(t)

dt

= iα(k)

ak(t) ,

ak(0) = hbk ,

i.e., ak(t) = ak(0) exp(iαkt). Hence the solution can be written as

yj(t) = h

N−1∑

k=1

bk exp(iαkt) sin(qjxj)/‖g(k)

‖

2,

i.e., analogously to (5.3) where αk = 2h−2(1 − cos(qkh)) − δ.


Table 3. The values |f |, |fh|, |fl|, |f2h| for x = L, τ = 0, 01.

t |f | |fh| |fl| |f2h|

0,1 0,05087 0,05085 0,05162 0,044130,2 0,06217 0,06224 0,06269 0,057850,3 0,06871 0,06862 0,06899 0,065570,4 0,07289 0,07286 0,07311 0,070560,5 0,07574 0,07593 0,07590 0,074030,6 0,07806 0,07827 0,07808 0,076720,7 0,08041 0,08013 0,07977 0,078840,8 0,08103 0,08164 0,08225 0,080660,9 0,08214 0,08291 0,08338 0,082081,0 0,08310 0,08398 0,08403 0,08327

10,0 0,09468 0,09526 0.09457 0,0961520,0 0,08996 0,08994 0,08996 0,0912830,0 0,10289 0,10299 0,10289 0,1029440,0 0,09560 0,09561 0,09559 0,0961650,0 0,09127 0,09127 0,09125 0,09107

Abs(f_h)_solutions

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

abs(f_h)

0 100 200 300 400

t

Figure 2. Solution of the grid problem |fh|, N = 750, x = L.

6. Numerical results and conclusions

Computations were carried out with the following values of the parameters δ = 0,L = 15, γ = 2, h = 1/10; 1/50, N = 150; 750, τ = 0, 1; 0, 01, σ = 1,and f0(x) = sin

(

πxL

)

. The finite difference scheme was realized by means of theFORTRAN code and analytically by using the expansion (finite series) in the formof a sum. The results coincide up to seven digits. The discrete solutions |fh| and|f2h| for x = L were compared with the solution |f | of the continuous problemwhich was obtained from the series at fixed time moments t ≤ 50 (|f2h| is thediscrete solution obtained by means of the FORTRAN code and the second orderapproximation of the boundary condition). In computing the series the terms weresummed up to the term whose modulus was smaller than ε = 10−8 (the number of


included terms was in all cases smaller than 1000). In Tab. 3 we also present thesolution |fl|, which was obtained by means of the analytic expansion of the linemethod. It does not depend on the temporal step-length τ . It is obvious that theresults coincide up to two or three digits. In Fig. 2 we show the numerical solution inthe interval τ ∈ (0, 400). The solution oscillates up to t ≈ 50, after which it rapidlyapproaches zero. Calculations show that reducing the spatial step-length h in the gridmethod improves the accuracy. For example, if t = τ = 0, 1, then

|fh| = 0, 0480 (h = 0, 1); 0, 0520 (h = 0, 05);

0, 0510 (h = 0, 025); 0, 0509(h = 0, 02) .

It follows from the results presented in Tab. 3 that the scheme with the first orderapproximation of the boundary condition is even more accurate. This is due to theorthogonality of the corresponding eigenfunctions and this fact is important in im-proving accuracy of numerical methods.

References

[1] M.I. Airila, O. Dumbrajs, A. Reinfelds and U. Strautinš. Nonstationary oscillations ingyrotrons. Phys. Plasmas, 8(10), 4808 – 4812, 2001.

[2] N.S. Ginzburg, N.A. Zavolsky, G.S. Nusinovich and A.S. Sergeev. Onset of autooscil-lations in electron microwave generators with diffraction radiation output. Izv. Vyssh.Uchebn. Zaved. Radiofiz., 29(1), 106 – 114, 1986. (in Russian)

[3] M. Radži unas and F. Ivanauskas. The stability conditions of finite difference schemesfor Schrödinger, Kuramoto-Tsuzuki and heat equations. Mathematical modelling andanalysis, 3, 177 – 194, 1998.

[4] A.A. Samarskii. The theory of difference schemes. Marcel Dekker, New York, NY, 2001.


Girotrono lygties vienos modos skaitiniai sprendiniai

O. Dumbrajs, H. Kalis, A. Reinfelds

Straipsnyje nagrinejami skaitiniai sprendiniai gauti tiriant girotrono lygties viena moda. Anali-tiniai ir skaitiniai sprendiniai gauti taikant baigtiniu skirtumu metoda. Ištirti kvazistacionariejisprendiniai ir atitinkamos tokio uždavinio tikrines reikšmes ir tikrines funkcijos.

!"#$%'&)($+*,.-0/213435/768

39–50c© 2004 Technika ISSN 1392-6292

SPATIAL ANALYSIS AND PREDICTION OFCURONIAN LAGOON DATA WITH GSTAT

R. GARŠKA and I. KRUMINIENE

Klaipeda University faculty of Natural and Mathematical sciences

H. Manto 84, Klaipeda, Lithuania

E-mail: 9:;4<=><?A@B<09?DC<0E0C4FG@H;JI%KMLN=B97LD><4E#LCO@C0FP@H;I


Abstract. The typical goal of geostatistical analysis is to interpolate values of variable underconsideration at unobserved locations using data on observed locations because it is not fea-sible to gather all data of the observations in the study area. The second goal is to know howthey represent the study area on the basis of the sample points. Kriging is one of geostatisticalmethods for spatial interpolation. This method relies on the spatial correlation reflected in theavailable data and so represents a global view of all the data as well as the nearest neighborinfluence. Before spatial prediction using kriging can be executed, the semivariogram has tobe computed and modelled.

The objective of our work is to create maps of the Curonian lagoon using kriging andcokriging methods. Our spatial data consist of observations on sounding and bed sedimentsof different Curonian lagoon locations. For computation and simulation of semivariograms,as well as for application kriging and cokriging methods and visualization of results on mapsGstat and PCRaster are used.

Key words: variogram, semivariogram, cross semivariogram, kriging, cokriging

1. Introduction

This paper discusses the use of traditional kriging techniques when when we mapvariables from data that are collected in Curonian lagoon. For most applicationskriging is usually associated with exact interpolation, that is, the kriging predictionschange smoothly in space until they get to a location where data have been collected,at this point there is a "jump" in the prediction to the exact value that was measured.This also leads to discontinuity in the standard prediction errors, that "jump" to zeroat the measured locations. In the research presented in the paper we have studied theCuronian lagoon and have created maps in all area by using data of a sediments in213 locations and data of depth in 263 locations. In the previous publication [2] theprecision of the results obtained by two methods, i.e. kriging and cokriging, werecompared by using cross-validation method. The results have showed that precision

40 R. Garška and I. Kruminiene

of predicted values is better when cokriging method is used. In addition the presentpaper presents maps of the predicted values in the whole Curonian lagoon, where theprediction is based on measurement data that are mentioned above.

Geostatistics is the name associated with a class of specialized statistical tech-niques used to analyze and estimate values of variables which are distributed – andphysically correlated – in space and /or time. The analysis of such a correlation isusually called a "structural analysis" or "variogram modeling". From a structuralanalysis, predictions of the value of the variable can be made at unsampled loca-tions using "kriging" or "stochastic simulation". This approach is most useful whenthe processes responsible for generating the measured variable are unknown or toopoorly constrained to permit construction of a quantitative process model to makespatial interpolations or predictions. The overall sequence of steps in a typical geo-statistical study include: exploratory data analysis (to explain the spatial characterof the variable), structural analysis (to determine the spatial correlation or continu-ity of the data) and estimates (kriging or simulations to predict values at unsampledlocations).

Kriging prediction consists of three steps:

• Estimation of the semivariogram or covariance;• Choice of a model among the family of valid semivariograms or covariances;• Estimation of the semivariogram or covariance by fitting the valid model to the

empirical semivariogram or covariance and use in one of the forms of kriging(e.g. ordinary kriging, simple kriging, universal kriging, etc.).

In many environmental researches the data of more than one observation (mea-surements of more than one variable) are often obtained. If those variables are cor-related with one another and the cross covariance functions are known or can beestimated then cokriging method can be used.

More about geostatistical analysis can be found in the book of Cressie "Statis-tics for spatial data" [1]. Krivoruchko has applied a kriging method to radio ce-sium soil contamination data, collected in Belarus after the Chernobyl accident (see,e.g. the web http://www.esri.com/software/arcgis/arcgisxtensions /geostatistical/re-search_papers.html). Lophaven has computed the spatial distribution of DissolvedInorganic Nitrogen (DIN) and Dissolved Inorganic Phosphorus (DIP) by three dif-ferent variants of kriging, i.e. ordinary kriging, universal kriging and cokriging [3].In the next section the main terms, processes and formulas which are used in geosta-tistical analysis are described.

The results of the study are presented in Section 3 and conclusions are given inSection 4.

2. Spatial Data Analysis

General Spatial Model (see [1]) is described in geostatistics as Z(s) : s∈D, where

• s = (x, y) denotes the coordinates of the sample site. Here (x, y) may be Eu-clidean coordinates (e.g., UTM coordinates), or latitude and longitude. Moregenerally, we may have s = (x, y, z).

Spatial analysis and prediction of Curonian lagoon data with Gstat 41

• Z(s) denotes the variable of interest at the location s. Note that this is written asa function of the location s.

• D denotes the set of spatial locations at which data may be obtained.

For geostatistical data, the set of all multidimensional distributions of k-tuples(

Z(s1), Z(s2), . . . , Z(sk))

for all values of k all configurations of the points s1, s2, . . . , sk constitutes a stochas-tic process Z(s) : s∈D. The stochastic variable Z(s) has mean value

E[Z(s)] = µ(s), s ∈ D.

We also assume that the variable of Z(s) exists for all s ∈ D.

The process Z is said to be Gaussian if, for any k ≥ 1 and locations s1, s2, . . .,sk, the vector

(

Z(s1), Z(s2), . . . , Z(sk))

has a multivariate normal distribution.

The process Z is said to be strictly stationarity if the joint distribution of(

Z(s1), Z(s2), . . . , Z(sk))

is the same as that of(

Z(s1+h), Z(s2+h), . . . , Z(sk +

h))

for any k spatial points s1, s2, . . . , sk and any h ∈ Rd, provided only that all of

s1, s2, . . . , sk, s1 + h, s2 + h, . . . , sk + h lie within the domain D.

The process Z is said to be second -order stationarity (also called weakly

stationarity) if µ(s) = µ (i.e., the mean value is the same for all s) and

Cov(

Z(s1), Z(s2))

= C(s1 − s2), for all s1∈D, s2∈D,

where C(s) is the covariance function of an observation at location s∈D.

It can immediately be seen that with all variances assumed finite, a strictly sta-tionary process is also second-order stationary. The converse is in general false, buta Gaussian process which is second-order stationary is also strictly stationary (see[4], 35 p.). Intrinsic stationarity is a weaker property than second-order stationarity.The variogram of intrinsic random function is written as

2γ(h) = V ar[(Z(s1) − Z(s2))] .

The function 2γ(·) is called the variogram (variance) and γ(·) the semivariogram(semivariance). If the semivariogram (covariance) depends only on distance betweenlocations the process is called isotropic. The variogram is the variance of the differ-ence between Z(s1) and Z(s2). If two units are close together, their difference willtypically be small, as would the variance of the difference. As units get apart, theirdifferences get larger and usually the variance of the difference gets larger.

If second-order stationarity is assumed, the relationship between the functionsemivariogram and the covariance is given as

γ(h) = C(0) − C(h) . (2.1)

Note that C(0) = σ2, the variance of the random function when h = 0. Equation

(2.1) shows that if the covariance is known, the variogram is also known. In prac-tice the variogram (or/and semivariogram) is used more often then the covariance,because the variogram, unlike the covariance, does not require the knowledge of themean value. Also semivariogram is less sensitive to any unidentified trend.


2.1. Estimation of the semivariogram and cross semivariogram

Determination of spatial variability is often based on a semivariogram. The sam-ple estimator of the semivariogram, which is based on the method-of-moments, isgiven by

γ(h) =1

2N(|h|)

∑

(sk,sl)∈N(|h|)

[Z(sk) − Z(sl)]2,

where N (|h|) denotes all pairs (sk, sl) for which |sk − sl| = |h| ([4], 38 p.). Thespatial variability between two correlated random variables is described by the crosssemivariogram. When the intrinsic hypothesis is assumed, it is defined as

γ12(h) = γ21(h) =1

2E

[(

Z1(sk) − Z1(sl))(

Z2(sk) − Z2(sl))]

,

where Z1(s) and Z2(s) denote two different variables. An estimator of the crosssemivariogram is defined as

γ12(h) =1

2N(|h|)

∑

(sk ,sl)∈N(|h|)

(

Z1(sk) − Z1(sl))(

Z2(sk) − Z2(sl))

,

where N(|h|) is the number of pairs of observations within distance |h|. Usuallyγij(h) is called the experimental or sample cross semivariogram (see [3]).

Range

Distance between pairs

Nugget

Sil

l

a)

Distance

Variance

b)

Distance

Variance

c)

Figure 1. Variograms: a) idealized form of variogram function; b) linear variogram; c) sphe-rical variogram.

2.2. Modelling the semivariogram and the cross semivariogram

Modelling of semivariogram and cross semivariogram is done in the same way. Theestimated semivariogram (cross semivariogram) is fitted with a model, and the bestmodels are used in the kriging estimation. Several methods have been proposed forfitting semivariogram models. One relatively simple method that appears to performwell is the Weighted Least Squares. Figure 1a show representation of general vari-ogram.


The range is the distance beyond which observations are uncorrelated or at leastapproximately uncorrelated. On the semivariogram, the range is the point on the x

– axis where the curve reaches a plateau. Sill is the value of semivariogram whereobservations are uncorrelated or nearly uncorrelated. On the semivariogram shown,the sill is the height of the curve at the plateau.

The nugget variance or nugget effect is the resulting discontinuity of the semi-variogram at the origin, the difference between zero and the semivariogram at a lagdistance is some greater than zero. The nugget effect is caused by measurement er-rors and micro-variability. A variogram model can consist of pure nugget effect.

Isotropic processes are convenient to deal with because there are a number ofwidely used parametric forms for γ(·). An often used semivariogram model is thelinear and the spherical model with nugget effect. A reason for this is an easy inter-pretation of the parameters.

A linear semivariogram model (Fig.1b) in the isotropic case is defined as:

γ(h) =

0, if |h| = 0,

C0 + C1h, if 0 < |h| < R,

C0 + C1R, if |h| > R .

(2.2)

Spherical semivariogram model (Fig.1c) is defined as:

γ(h) =

0, if |h| = 0,

C0 + C1

[

3

2

h

R

−1

2

(

h

R

)3]

, if 0 < |h| =< R,

C0 + C1, if |h| > R ,

(2.3)

where C0 is the nugget effect, R is the range and C0 + C1 is the sill [3].

When two or more variables are correlated, the nature of spatial cross correlationbetween the primary variable and several secondary variables can provide valuableinformation for estimation and simulation of the primary variable. Cross semivari-ogram modelling is always done for the purpose of developing a model to be used inestimation or simulation. The models that are to be used in estimation and simulationmust obey a number of stringent constraints to ensure that the matrix solutions to thekriging equations exist and are numerically stable.

Traditionally fitting of the cross semivariogram is done by eye, because it hasbeen shown that predictions computed by kriging are reasonably insensitive to thespecification of the cross semivariogram model. The best semivariogram (cross semi-variogram) model can be found using the least squared criterion [3].

2.3. Kriging Concept

Kriging is a generic name adopted by the geostatisticians for a family of generalizedleast-squares regression algorithms that allow one to account the spatial dependencebetween observations, as revealed by the semivariogram, into spatial prediction. It isa procedure for spatial prediction at an unobserved location, using data at observedlocations, optimized with reference to a specific error criterion.


Kriging is known to be a Best Linear Unbiased Predictor (B.L.U.P.), becauseit minimizes the variance error between the true value and the predictor. Linearpredictor of the value Z1(s0) of the data at the unsampled site s0 from the dataZ(s) = Z(s1(s)), Z(s2(s)) at the sampled sites s1, s2 is:

Z1(s0) =

n∑

k=1

wkZ1(sk),

where wk is the weight for the k-th variable of observation at location sk and n is thenumber of observations. The weights wk are chosen to minimize the mean squarederror

MSE = E

[

Z1(s0) − Z1(s0)]2

.

Z(s0) is unbiased for Z(s0) if and only ifn

∑

k=1

wk = 1 .

Ordinary kriging gives the optimal predictions under the assumption that themean value is constant (but unknown) across the whole area under study. The ordi-nary kriging variance for Z1 is given by

σ2ok =

∑

k

wkγ(sk − s0) + m, (2.4)

where m is a Lagrange multiplier

m =(

1′

−1∑

z

cz − 1)

/

(

1′

−1∑

z

1)

,

∑

z is the covariance matrix among the data, cz is Cov(z, Z(s0)) (see [1], p 143).

Cokriging is prediction of a primary variable using additional information froma secondary variable. This method is used in data sets containing two or more re-gionalized variables which are correlated with one another. Suppose that q = 2.

The prediction of Z1 is done not only on the basis of Z1, but also on measurementsof Z2. Cokriging involves the prediction of Z1(s0) at an unsampled site s0 fromthe data Z(s1), Z(s2), . . . , Z(sn), Z(s)T =

(

Z1(s), Z2(s))

at the sampled sitess1, s2, . . . , sn. The linear prediction of cokriging is defined as:

Z1(s0) =

n∑

k=1

vk1Z1(sk) +

n∑

k=1

vk2Z2(sk).

To obtain an unbiased estimate the following constraints are needed:n

∑

k=1

vk1 = 1,

m∑

k=1

vk2 = 0 .

Similarly as (2.4) the variance of cokriging prediction can be written as

σ2cok =

n∑

k=1

vk1γk1(sk − s0) +

n∑

k=1

vk2γk2(sk − s0) + m1.


3. Results

The above procedure of variogram estimation, variogram model fitting, kriging andcokriging was applied to the Curonian lagoon data. The Curonian lagoon (alsoknown as Kuršiu marios, Kurshskij zaliv, Kurische Haff) is a large (length 95 km,width up to 48 km), shallow (mean depth of 3.8 m, the maximum depth - 5.8 m)coastal water body in the south-eastern part of the Baltic Sea. The outlet of the la-goon to the Baltic Sea, Klaipeda Strait, is artificially deepened down to 12 m.

The data have been collected in 1990 year by S. Gulbinskas. It consists of bedsediments and soundings of the Curonian Lagoon. Sediments where measured in213 locations, depth was measured in 263 locations. Their x coordinate values arebetween 278199 and 333376 and y coordinate values are between 6088178 and6172784. Sediments have been divided into 7 groups (granulometric fractions) de-pending on median diameter (Md) in mm: (1) more than 0.5, (2) 0.5-0.25, (3) 0.25-0.125, (4) 0.125-0.063, (5) 0.063-0.01, (6) 0.01-0.004, (7) less than 0.004.

In order to apply the above statistical methods for data analysis, and mappingwe have chosen free available software Gstat and PCRaster. Gstat is a program forthe modelling, prediction and simulation of geostatistical data in one, two or threedimensions. In Gstat geostatistical modelling comprises calculation of sample vari-ograms and cross variograms (or covariograms) and fitting models to them. In thispaper Gstat has been used for modelling semivariance of all above fractions and forsimulation cross variance between depth and sediment fractions. PCRaster has beenused for showing kriging and cokriging prediction maps.

In Gstat a simple variogram model is denoted cMod(a) with c the vertical (vari-ance) scaling factor, Mod the model type, and a the range (horizontal, distance scal-ing factor) of this simple model. Linear and spherical models defined in (2.2) and(2.3) equations, in Gstat are denoted by Lin(·) and Sph(·), respectively. The nuggeteffect is indicated by Nug(·).

0

10

20

30

40

50

60

70

80

90

0 5000 10000 15000 20000 25000 30000 35000

sem

ivar

ianc

e

distance

300

617

8641008

1201

12611202

124710941104

1064

956

934

898

807

g_-00431.3848 Nug(0) + 53.4782 Lin(32344.6)

a)

0

1

2

3

4

5

6

7

8

0 5000 10000 15000 20000 25000 30000 35000

cros

s se

miv

aria

nce

distance

4761026

1392

1638

18981996

17781840

15801622 1692

14201374

14161232

depth x g_-0041.29422 Nug(0) + 6.56577 Lin(27254.5)

b)

Figure 2. a) semivariogram of fraction (7) where Md of sediments is less than 0.004; b) crosssemivariogram between fraction (7) and depth of the Curonian lagoon.

To describe results of our research we took measurements of depth and fraction(7). Figure 2a presents semivariogram of fraction (7) where Mod of sediments is lessthan 0.004. Equation of this semivariogram is given by


1.29422Nug(0) + 6.56577Lin(27254.5),

where Lin represents model type, Nug(0) = 1.29422, when 1

2γ(h) = 0, the sill is

6.56577 and the range is 27254.5. The parameters and models of all semivariogramfractions are given in Table 1.

Table 1. Types of semivariogram models of all fractions and the parameters:range, sill and nugget effect.

Fraction Model Range Sill Nugget effect

more than 0.5 Linear 14402.5 5.05205 15.60340.5-0.25 Linear 22615.3 167.631 290.3260.25-0.125 Linear 11036.5 167.631 504.5110.125-0.063 Linear 9773.75 50.812 264.4830.063-0.01 Linear 22051.9 297.963 316.5330.01-0.004 Linear 31426.3 52.2167 15.1969less than 0.004 Linear 32344.6 53.4782 31.3848

Figure 2b presents cross semivariogram between fraction (7) and depth of theCuronian lagoon. Equation of this cross semivariogram is given by

31.3846Nug(0) + 53.4782Lin(32344.6),

where Lin represents model type, the nugget effect equals 31.3846, when 1

2γ(h) = 0,

the sill is 53.4782 and the range is 32344.6. The parameters and models of all crosssemivariogram fractions are given in Table 2.

Table 2. Types of cross semivariogram models between depths and fractionsand the parameters: range, sill and nugget effect.

Fraction Model Range Sill Nugget effect

more than 0.5 Linear 14360.4 -0.888949 -0.7187050.5-0.25 Linear 20658.5 -9.04819 -7.575070.25-0.125 Linear 30420.6 -20.0815 -3.645060.125-0.063 Linear 4.586170.063-0.01 Linear 25260.5 16.3492 5.609720.01-0.004 Linear 30887.2 6.71625 0.738019less than 0.004 Linear 27254.5 6.56577 1.29422

Figure 3 presents the linear and spherical semivariograms of the depth. In thiscase the spherical semivariogram is preferred. The parameters and models of depthare given in Table 3.

Kriging is most sensitive to the behavior of the variogram near zero. In particular,it is sensitive to the presence / absence of the nugget effect. Maps of variations andpredictions created using kriging method are shown in Figure 4 (here (a) prediction


0

0.5

1

1.5

2

2.5

3

0 5000 10000 15000 20000 25000 30000 35000

sem

ivar

ianc

e

distance

483 1004

13841596

1907

1992

1925191016681603

16781522

1463

14001307

depth1.29812 Nug(0) + 1.37065 Lin(32972.7)

a)

0

0.5

1

1.5

2

2.5

3

0 5000 10000 15000 20000 25000 30000 35000

sem

ivar

ianc

e

distance

483 1004

13841596

1907

1992

1925191016681603

16781522

1463

14001307

depth2.22773 Sph(13957.2)

b)

Figure 3. a) linear semivariogram of depth; b) spherical semivariogram of depth.

Table 3. Types of semivariogram models of depth and the parameters:range, sill and nugget effect.

Model Range Sill Nugget effect

Linear 32972.7 1.37065 1.29812Spherical 13957.2 2.22773

Concentration of (7) fraction

10 - 20

0 - 10

No data

10000 0 10000 20000 30000 40000 50000 60000 70000 80000 Meters

a)

10000 0 10000 20000 30000 40000 50000 60000 70000 80000 Meters

Variation of (7) fraction

60 - 80

40 - 60

30 - 40

No data

b)

Figure 4. Kriging method: a) prediction map of fraction (7); b) variation map of fraction (7).

map, (b) variation map). Maps obtained using cokriging method are shown in Fig-ure 5. Figure 6a presents prediction map of depth, while Figure 6b presents variationmap of depth. These maps have been created using kriging method.

In order to check which one of the kriging and cokriging maps correspond bestto true data we must first choose one point on the map, then compare these variationmaps, and finally determine which map has smaller variation for the selected point.The method containing a smaller variation has created a better prediction map.


10000 0 10000 20000 30000 40000 50000 60000 70000 80000 Meters

Concentration of (7) fraction50 - 60%

40 - 50%

30 - 40%

20 - 30%

10 - 20%

0 - 10%

No data

a)

10000 0 10000 20000 30000 40000 50000 60000 70000 80000 Meters

Variation of (7) fraction

60 - 80

40 - 60

30 - 40

20 - 30

No data

b)

Figure 5. Cokriging method: a) prediction map of fraction (7); b) variation map of fraction(7).

10000 0 10000 20000 30000 40000 50000 60000 70000 80000 Meters

Depth

9 - 10 m

8 - 9 m

7 - 8 m

6 - 7 m

5 - 6 m

4 - 5 m

3 - 4 m

2 - 3 m

1 - 2 m

0 - 1 m

No data

a)

10000 0 10000 20000 30000 40000 50000 60000 70000 80000 Meters

Variation of depth

2.0 - 2.5 m^2

1.5 - 2.0 m^2

1.0 - 1.5 m^2

0.5 - 1.0 m^2

0.0 - 0.5 m^2

No data

b)

Figure 6. Kriging method: a) prediction map of depth; b) variation map of depth.

4. Conclusions

Statistical methods for data on bed fractions percentage and soundings have beendescribed, applied and mapped. The methods are general, but in this paper they havebeen applied only to measurements of the Curonian lagoon.


Semivariogram and cross semivariogram models have been made using percent-age of fractions and soundings of the Curonian lagoon. Variance distribution anddistribution of bed fractions percentage have been mapped using kriging and cokrig-ing methods. Variance distribution and distribution of soundings have been mappedusing kriging method.

Results demonstrate that:

• Nugget and linear models best describe semivariance and cross semivariance ofpercentage of ground fractions.

• Spherical model best describes semivariance and cross semivariance of sound-ings.

• Prediction variations of percentage of bed fractions made by kriging and cokrig-ing methods are very similar.

Also cross semivariance show interdependence of parameters of models anddepths. Prediction results of bed fractions percentage made by kriging method arevery close to the mean value, while cokriging method shows that the variation ofdata are less close to the mean value.

References

[1] N. Cressie. Statistics for Spatial Data. John Wiley, New York, 1993.[2] I. Kr uminiene, K. Ducinskas and R. Garška. Applying of kriging and cokriging methods

for prediction of Curonian lagoon data. Liet. matem. rink., 43(spec. nr.), 504 – 508, 2003.[3] Soren Nymand Lophaven. Reconstruction of data from the aquatic environment. LYN-

GBY, 2001.[4] Richard L. Smith. Environmental Statistics. University of North Carolina Chapel Nill,

NC 27599-3260, USA, 2001.


Apie Kuršiu mariu duomenu erdvine analize ir prognozavima Gstat programos pagalba

I. Kr uminiene, R. Garška

Šio darbo pagrindinis tikslas - Gstat bei PCRaster programu pagalba sukurti prognozuojamuduomenu ir ju dispersiju žemelapius. Žemelapiams sudaryti pritaikyti krigingo ir kokrigingometodai. Krigingas yra vienas iš geostatistikos metodu, kuris atsižvelgdamas i erdvini dviejukintamuju ryši ir kaimyniniu tašku reikšmes atlieka erdvine interpoliacija. Tuo tarpu kok-rigingas atlieka pirminio kintamojo duomenu prognoze naudojant antriniu kintamuju duome-nis. Pagrindinis geostatistines analizes tikslas yra interpoliuoti duomenis nežinomuose sritiestaškuose, nes dažniausiai atliekant geostatistinius tyrimus naudojami daliniai stebejimai, kurieapima tik visumos dali; arba nera žinoma, ar imties duomenys pakankamai gerai atspindi visastudijuojama sriti. Rezultatu analize parode, kad tikslesne prognoze gaunama taikant kok-rigingo metoda

!"#$%'&)($+*,.-0/213435/768

51–66c© 2004 Technika ISSN 1392-6292

DYNAMICS OF MULTISECTIONSEMICONDUCTOR LASERS

J. SIEBER1, M. RADŽIUNAS2 and K. R. SCHNEIDER2

1 University of Bristol

Dept. of Eng. Math., Queen’s Building, University of Bristol, Bristol BS8 1TR,United Kingdom

E-mail: 9:;=<8>?A@B@0CDBC7?4>FEGHI<J:KI<MLN2 Weierstrass Institute for Applied Analysis and Stochastics, Berlin

Mohrenstr. 39, 10117 Berlin, Germany

E-mail: C:4OP?QL;:>DARS?A:>TUB@0CH?Q;=<O@VW>KX4;@#?YO@CDRS?A:>TQB@CH?U;=<O@Received October 10 2003; revised December 19 2003

Abstract. We investigate the longitudinal dynamics of multisection semiconductor lasersbased on a model, where a hyperbolic system of partial differential equations is nonlinearlycoupled with a system of ordinary differential equations. We present analytic results for thatsystem: global existence and uniqueness of the initial-boundary value problem, and existenceof attracting invariant manifolds of low dimension. The flow on these manifolds is approxi-mately described by the so-called mode approximations which are systems of ordinary differ-ential equations. Finally, we present a detailed numerical bifurcation analysis of the two-modeapproximation system and compare it with the simulated dynamics of the full PDE model.

Key words: laser dynamics, invariant manifold theory, hyperbolic systems of partial differ-ential equations, model reduction, bifurcation analysis

1. Motivation

In commercial and public communication, the exchange of multimedial informationgrowths rapidly. Thus, the corresponding data traffic increases exponentially and ischaracterized by the shift from voice communication to package oriented data traf-fic. This fact implies a big challenge for a strong increase of the data transmissionrate. Due to their inherent speed, semiconductor lasers are of great interest as opti-cal devices for fast data regeneration (reamplification, retiming, reshaping) in futurephotonic networks. Typically, these devices have a non-stationary working regime.As an example we mention the regime of high-frequency oscillations. Multisectionlasers allow one to generate and to control such nonlinear effects by designing thelongitudinal structure of the device (see, e.g., [16, 19, 25, 28]).

52 J. Sieber, M. Radžiunas, K. Schneider

However, prototyping of multisection semiconductor lasers is very expensive andtime consuming. The goal of this paper is to demonstrate that mathematical modelscan be used to study the longitudinal dynamics of such lasers and to optimize theirworking regime.

We focus on the traveling wave model, a linear hyperbolic system of partial dif-ferential equations (PDEs) which is nonlinearly coupled with a system of ordinarydifferential equations (ODEs). It models the longitudinal dynamics of edge emit-ting multisection semiconductor lasers by the interaction of two physical variables:the complex light amplitude (in fact, its spatially slowly varying envelope), which isspatially resolved in the longitudinal direction of the laser and described by the linearhyperbolic PDE subsystem, and the effective carrier density within the active zoneof the device, which is section-wise spatially averaged and described by the ODEsubsystem.

This model has the advantage of meeting two seemingly contradictory criteria,accuracy and simplicity (or rather accessibility to a detailed bifurcation analysis). Onone hand, it is accurate enough to describe all phenomena of interest to the engineers.Moreover, it can easily be made more realistic by gradually incorporating secondaryphysical effects that may play a role in limiting the performance of a particular de-vice. On the other hand, it allows one to reduce the model to a low-dimensional sys-tem of ODEs by exploiting the fact that the carrier density operates on a much slowertime-scale than the light amplitude. These ODEs in turn are accessible to a detailedbifurcation analysis using standard software like AUTO [10]. Only this bifurcationanalysis gives insight into the mechanisms behind many nonlinear phenomena andis able to reveal effects (for example excitability [27]) that may be invisible in pureparameter studies.

Both aspects of the traveling wave model have been implemented in the numer-ical code (Longitudinal Dynamics of Semiconductor Lasers). Hence, this nu-merical tool provides engineers, laser physicists, and mathematicians with a wholehierarchy of models allowing them to “switch on or off” physical effects to gain in-sight which of these effects causes the particular phenomenon they are interested in.Besides numerical integration of the model equations this tool solves also the spec-tral problem of the model equations, allows to analyze the dynamics of individuallongitudinal modes and in certain cases enables one effectively to compare the so-lutions provided by the PDE model and the reduced mode approximation systems.This modeling approach has been used quite successfully in the recent past to designnew devices exhibiting high-frequency oscillations [7, 8, 28].

In this paper we focus more on the aspect of model reduction than extension,mostly because this part is more thoroughly supported by mathematical theory. Thepaper is organized as follows: In section 2. we describe the traveling wave model andgive a detailed physical interpretation of all coefficients and variables. In section 3.we show that the corresponding initial-boundary value problem is well-posed. Insection 4. we introduce a small parameter exploiting the difference in the time-scalebetween light and carrier density. In section 5. we investigate the spectral propertiesof the infinite-dimensional linear part. Section 6. combines the results of the previ-ous sections to derive conditions guaranteeing that the traveling wave model can bereduced to an ODE system. In section 7. by showing a detailed two-parameter bifur-

Dynamics of multisection semiconductor lasers 53

cation diagram we demonstrate how useful the reduced model can be. We link thisbifurcation diagram to a parameter study with a more realistic version of the trav-eling wave model. In the last section we draw conclusions and give an outlook onfuture projects.

2. The coupled traveling wave model with nonlinear gaindispersion

The coupled traveling wave model, a hyperbolic system of PDEs coupled with a sys-tem of ODEs, describes the longitudinal effects in narrow edge-emitting laser diodes[1, 15, 23]. It has been derived from Maxwell’s equations for an electro-magneticfield in a periodically modulated waveguide [1, 3] assuming that transversal and lon-gitudinal effects can be separated. In this section we introduce the corresponding

S2S1 S3

n1 n2 n3

z1z2 z3 z4

Ll2 l3z

0 1l1 = 1

Figure 1. Typical geometric configuration of the domain in a laser with 3 sections.

system of differential equations, explain the physical interpretation of its coefficientsand specify some physically sensible assumptions about these coefficients.

The dynamics in a multi-section laser is described by the evolution of the fol-lowing quantities. The variable ψ(t, z) ∈ C2 describes the complex amplitude ofthe slowly varying envelope of the optical field split into a forward and a backwardtraveling wave. The variable p(t, z) ∈ C2 describes the corresponding nonlinear po-larization of the material. Both quantities depend on time and the one-dimensionalspatial variable z ∈ [0, L] (the longitudinal direction within the laser; see Fig. 1). Aprominent feature of multi-section lasers is the splitting of the overall interval [0, L]into sections, that is, m subintervals Sk that represent sections with separate electriccontacts. We treat the carrier density within the active zone of the waveguide as asection-wise spatially averaged quantity n(t) ∈ Rm (see Fig. 1). In dimensionlessform, the coupled traveling wave model can be posed as an initial-boundary valueproblem for ψ, p, and n that reads as follows


∂tψ(t, z) =

[

−∂z + β(n(t), z) −iκ(z)

−iκ(z) ∂z + β(n(t), z)

]

ψ(t, z) + ρ(n(t), z) p(t, z),

(2.1)

∂tp(t, z) = [iΩr(n(t), z) − Γ (n(t), z)] p(t, z) + Γ (n(t), z)ψ(t, z), (2.2)

d

dtnk(t) = Ik −

nk(t)

τk−P

lk[Gk(nk(t)) − ρk(nk(t))]

∫

Sk

ψ(t, z)∗ψ(t, z) dz

−P

lkρk(nk(t)) Re

(∫

Sk

ψ(t, z)∗p(t, z) dz

)

, k = 1, . . . ,m (2.3)

subject to the inhomogeneous boundary conditions for ψ

ψ1(t, 0) = r0ψ2(t, 0) + α(t), ψ2(t, L) = rLψ1(t, L) (2.4)

and the initial conditions

ψ(0, z) = ψ0(z), p(0, z) = p0(z), n(0) = n0 . (2.5)

The Hermitian transpose of the C2-vectorψ is denoted by ψ∗ in (2.3). We will definethe appropriate function spaces and discuss the possible solution concepts in section3.. The quantities and coefficients appearing above have the following meaning (seealso Tab. 1 and Fig. 1). L is the length of the laser. The laser is subdivided into

Table 1. Ranges and explanations of the variables and coefficients appearing in (2.1) – (2.4).See also [3] to inspect their relations to the originally used physical quantities and scales.

typical range explanation

ψ(t, z) C2 optical field, forward and backward traveling wave

p(t, z) C2 nonlinear polarization

nk(t) R+ spatially averaged carrier density in section Sk

Imdk R frequency detuningRe dk < 0, O(1) negative decay rate due to internal lossesαH,k (0, 10) negative of line-width enhancement factorgk ≈ 1 differential gain in active sections Sk

κk (−10, 10) real coupling coefficient for the optical field ψdue to Bragg grating in DFB sections

ρk ≥ 0, O(1) amplitude of the gain curveΓk O(10

2) half width at half maximum of the gain curve

Ωr,k O(10) central frequency of the gain curveIk O(10

−2) current injection

τk O(102) spontaneous lifetime of the carriers

P (0,∞) scale of (ψ, p) (can be chosen arbitrarily)r0 , rL C, |r0|, |rL| < 1 facet reflectivities

m sections Sk of length lk with starting points zk for k = 1, . . . ,m. We scale thesystem such that l1 = 1 and set zm+1 = L. Thus, Sk = [zk, zk+1]. All coefficientsare supposed to be spatially constant in each section and to depend only on the carrierdensity in that section, that is, for z ∈ Sk we have


κ(z) = κk, Γ (n, z) = Γk(nk), β(n, z) = βk(nk), ρ(n, z) = ρk(nk).

Tab. 1 collects the physical interpretation and the sensible ranges of all coefficientsand variables. The model for the growth coefficient βk(nk) ∈ C in section Sk is

βk(ν) = dk + (1 + iαH,k)Gk(ν) − ρk(ν),

where dk ∈ C accounts for the static internal losses (hence, Re dk < 0) and the staticfrequency detuning, and αH,k ∈ R+ is the negative of the line-width enhancement(or Henry) factor. A section Sk is either passive, then the functions Gk and ρk areidentically zero, or Sk is active. In the active case Gk : R → R is a smooth strictlymonotone increasing function satisfying Gk(1) = 0. Its limits are

limν→−∞

Gk(ν) = −∞, limν→∞

Gk(ν) = ∞ .

Typically, an affine model for Gk in active sections is reasonably accurate, that is,

Gk(ν) = gk · (ν − 1)

with a differential gain gk = G′

k(1) > 0. In active sections Sk, that is, if Gk 6≡ 0,the gain amplitude ρk(ν) is bounded for ν < 1. Moreover, we suppose that ρk, Ωr,k,and Γk : R → R are smooth and Lipschitz continuous, and Γk(ν) > 1 for all ν. Forpassive sections Sk the variable nk is decoupled from all other equations and can bedropped from the system.

The polarization function p and equation (2.2) has been included into the coupledtraveling wave model for a more realistic account of nonlinear gain dispersion effects[3, 28]. Now, the frequency dependence of waveguide material gain is modeled bya Lorentzian function with an amplitude ρ, half width at half maximum Γ , centeredat the frequencyΩr. That is, a monochromatic light-wave ψ1(t, z) = eiωtϕ(z) in anuncoupled stationary; waveguide (κ = 0, n = 0) is amplified or damped accordingto the equation

∂z |ϕ(z)|2 = 2

[

Reβ(z) +ρ(z)Γ 2(z)

(ω −Ωr(z))2 + Γ 2(z)

]

|ϕ(z)|2.

The facet reflectivities r0 and rL in (2.4) are complex with modulus less than 1.The inhomogeneityα(t) is complex. It models an optical input at the facet z = 0. Weassume it to be L2 in time on finite time intervals to permit a discontinuous opticalinput.

The form of the right-hand-side of equation (2.3) for the carrier density can beclarified by introducing the Hermitian form

gk(ν)

[(

ψp

)

,

(

ϕq

)]

=1

lk

∫

Sk

(ψ∗(z), p∗(z))(

Gk(ν)−ρk(ν) 1

2ρk(ν)

1

2ρk(ν) 0

)

(

ϕ(z)q(z)

)

dz.

Using the notation

fk(ν, (ψ, p)) = Ik −ν

τk− Pgk(ν)

[(

ψp

)

,

(

ψp

)]

(2.6)


for ν ∈ R and ψ,ϕ, p, q ∈ L2(Sk; C2) the carrier density equation (2.3) reads

d

dtnk = fk(nk, (ψ, p)) for k = 1, . . . ,m. (2.7)

Other secondary physical effects have been incorporated into the numerical code which was developed for the simulation and analysis of longitudinal dynamicsin multi-section lasers. As example we mention the effects of nonlinear gain com-pression, that is, the dependence of G on |ψ|2, and spatial hole burning, i.e., treatingn as a fully spatially resolved variable [7, 28]. The parameter study by direct simu-lations of the extended model equations shown in Fig. 3 has taken both effects intoaccount. However, even after an inclusion of these effects, the traveling wave modelcan describe the behaviour of semiconductor lasers still only approximately. Thus, inthis paper we focus on the analysis of the traveling wave model in the rather simpleform (2.1) – (2.4).

3. Existence theory

In a first step we investigate in which sense system (2.1) – (2.3) generates a semiflowdepending smoothly on its initial values and all parameters. We want to write (2.1) –(2.3) as an abstract evolution equation in the form

d

dtu = Au+ g(u), u(0) = u0 (3.1)

in a Hilbert space V,whereA is a linear differential operator that generates a stronglycontinuous semigroupS(t), and g is smooth in V . A natural space for the variables ψand p is L2([0, L]; C2), such that V could be L2([0, L]; C2) × L2([0, L]; C2) × Rm

for u = (ψ, p, n). However, the inhomogeneity α in the boundary condition (2.4)poses a conceptual difficulty in this framework. Common approaches are boundaryhomogenization (used in [18]) or appending α as an auxiliary variable and an addi-tional equation of the form

d

dtα(t) = a(t),

where a is the derivative of α (used in [12]). Then, the nonlinearity g in the evolutionequation depends explicitly on t and it has the same regularity with respect to t as thetime derivative of α. Hence, both approaches require a high degree of regularity of αin time which is quite unnatural as the laser still works with discontinuous input suchas square waves. An alternative would be the introduction of a concept of “weaklymild” solutions as it was done in [13]. However, this would require the extension ofall needed classical results of the theory of strongly continuous semigroups to thistype of solutions.

Here, we choose an approach that is similar to that in [12] but does not requireany regularity of the inhomogeneity. We introduce the auxiliary space-dependentvariable a(t, x) (x ∈ [0,∞)) satisfying the equation

∂ta(t, x) = ∂xa(t, x) (3.2)


and change the boundary condition for z = 0 in (2.4) into

ψ1(t, 0) = r0ψ2(t, 0) + a(t, 0).

One may think of an infinitely long fibre [0,∞) storing all future optical inputs andtransporting them to the laser facet z = x = 0 by the transport equation (3.2). Ifwe choose a(0, x) = α(x) as initial value for a, than the value of a at the boundaryx = 0 at time t is α(t). In this way, the initially inhomogeneous boundary conditionbecomes linear in the variables ψ and a requiring no regularity for a. We choosea weighted norm L2

η for a, that is, ‖a(t, ·)‖2 =∫

∞

0|a(t, x)|2(1 + x2)η dx with

η < −1/2. In this way, we permit the input to be L∞ but still keep V as a Hilbertspace.

With this modification we can work within the framework of the theory ofstrongly continuous semigroups [17]. The variable u has the components (ψ, p, n, a) ∈V = L

2([0, L]; C2)×L2([0, L]; C2)×R

m×L2η([0,∞); C). We have a certain free-

dom how to choose the splitting of the right-hand-side betweenA and g. We keep Aas simple as possible, including only the unbounded terms

A

ψpna

:=

[

−∂zψ1

∂zψ2

]

00∂xa

.

The domain of definition of A is

D(A) = (ψ, p, n, a) ∈ H1([0, L]; C2) × L

2([0, L]; C2) × Rm × H

1η([0,∞); C) :

ψ1(0) = r0ψ2(0) + a(0), ψ2(L) = rLψ1(L).

In this way, A generates a strongly continuous semigroup S(t) in V [22]. The non-linearity g is smooth because it is a superposition operator of smooth coefficientfunctions, and all components either depend only linearly on the infinite-dimensionalcomponents ψ and p, or map into Rm. Then, an a-priori estimate implies the follow-ing theorem.

Theorem 1 [Global existence and uniqueness]. For any T0 > 0, there exists aunique mild solution u(t) of (3.1) in [0, T0]. Furthermore, if the initial value u0 is inthe domain of definition of A, then u(t) is a classical solution of (3.1).

This theorem implies the existence of a semiflow S(t;u0) that is strongly con-tinuous in t and smooth with respect to u and parameters. The a-priori estimate hasto be slightly more subtle than in [18]. It uses the fact that the same functions Gkand ρk appear on the right-hand-side of (2.1) and on that of (2.3) but with opposingsigns. Due to this fact the function

P

2‖ψ(t)‖2 +

m∑

k=1

lk(nk(t) − n∗)

remains non-negative for sufficiently small n∗

and, hence, bounded, giving rise to abounded invariant ball in V . The value of n

∗depends on the initial value u0 (see [22]

for details).


4. Introduction of a small parameter

For all results about the long-time behavior of system (2.1) – (2.3) we restrict our-selves to autonomous boundary conditions for ψ, that is,

ψ1(t, 0) = r0ψ2(t, 0), ψ2(t, L) = rLψ1(t, L). (4.1)

The inhomogeneous case is an open question for future work. However, understan-ding the dynamics of the autonomous laser is not only an intermediate step but animportant goal in itself since many experiments and simulations focus on this case;see for example [8] for further references.

An examination of system (2.1) – (2.3) reveals that the space dependent subsys-tem is linear in ψ and p:

∂t

(

ψp

)

= H(n)

(

ψp

)

. (4.2)

The linear operator

H(n) =

[

−∂z + β(n) −iκ−iκ ∂z + β(n)

]

ρ(n)

Γ (n) iΩr(n) − Γ (n)

(4.3)

acts from

Y := (ψ, p) ∈ H1([0, L]; C2) × L

2([0, L]; C2) :

ψ1(0) = r0ψ2(0), ψ2(L) = rLψ1(L)

into X = L2([0, L]; C2) × L2([0, L]; C2). H(n) generates a C0-semigroup Tn(t)acting in X . Its coefficients κ, and, for each n ∈ Rm, β(n), Ωr(n), Γ (n) and ρ(n)are linear operators in L2([0, L]; C2) defined by the corresponding coefficients in(2.1), (2.2). The maps β, ρ, Γ,Ωr : Rm → L(L2([0, L]; C2)) are smooth.

Furthermore, we observe that Ik and τ−1

k in (2.6) are approximately two ordersof magnitude smaller than 1 (see Tab. 1). Hence, we can introduce a small parameterε and set P = ε in (2.3), such that the carrier density equation (2.7) reads as

d

dtnk = fk (nk, E) = ε(Fk(nk) − gk(nk)[E,E]) (4.4)

for E ∈ X , where the coefficients in Fk(nk) = ε−1(Ik − nkτ−1

k ) are of order 1.Although ε is not directly accessible, we treat it as a parameter and consider the limitε → 0 while keeping Fk fixed. At ε = 0, the carrier density n is constant. It entersthe linear subsystem (4.2) as a parameter. Consequently, the spectral properties ofH(n) with fixed n determine the longtime behavior of the system for ε = 0. Inparticular, we are interested in such values of n which imply an isolated non-emptybut finite set of eigenvalues of H(n) located exactly on the imaginary axis. In thiscase, we can expect a finite-dimensional invariant manifold to persist for nonzeroε in the spirit of Fenichel’s geometric singular perturbation theory [11]. Thus, wewould like to understand the spectral properties of the operator H for fixed n andtheir correspondence to the growth of the semigroup Tn generated by H in the nextstep.


5. Spectral properties of operator H

We drop the argument n in this paragraph for brevity. The long-time behavior of thesemigroup T generated by H can be described by the following theorem (see [22]for details of the proof):

Theorem 2. Let ξ0 = 1

L

∑m

k=1Reβklk < 0, denote W = iΩr,k − Γk : k =

1, . . . ,m, and let ξ be in the interval (maxReW , ξ0, 0). Then, there exists a split-ting ofX = X1⊕X2 into twoH-invariant subspaces whereX1 is finite-dimensionaland the semigroup T restricted to X2 decays according to rate ξ:

‖T (t)|X2‖ ≤Meξt for a constant M ≥ 1 and all t ≥ 0.

Since T is neither an analytical nor an eventually compact semigroup there are nogeneral theorems implying our result. However, the operator H has a characteristicfunction h(λ) defined in C \ W (note that ReW < −1). The function h is analyticin C \W and known explicitly. Hence, most questions about the spectrum of H canbe answered by finding the roots of h. In particular, the spectrum of H is discrete inC \W , that is, it consists only of eigenvalues of finite algebraic multiplicity. In orderto obtain our result, we have to distinguish two cases, r0rL = 0 and r0rL 6= 0.

It turns out that the semigroup T is eventually differentiable if r0rL = 0. Inthis case, we can split X into two H-invariant subspaces. One corresponds to thespectrum close to W . Thus, H is bounded and T exponentially decaying in thissubspace. The semigroup T restricted to the complementary invariant subspace iseventually compact. Hence, the desired result follows from the theory of eventuallycompact semigroups [9].

If r0rL 6= 0 (the hyperbolic case), we treat the operator as a perturbation of itsdiagonal part similar to [20]. Before applying the same result as [20], the invariantsubspace corresponding to the spectrum close to W has to be split off and treatedseparately in the same way as in the case r0rL = 0.

In essence, Theorem 2 implies that we can treat H like a matrix: the dominanteigenvalues determine the growth of the corresponding semigroup.

6. Model reduction

Let us assume that there exists a simple connected open set U ⊂ Rm of carrierdensities n such that H(n) has a uniform spectral gap for all n ∈ U in a strip of thenegative complex half-plane z ∈ C : ξ ≤ Re z ≤ ξ/k (ξ < 0, integer k > 2),and that the dominant part of the spectrum of H(n) is finite. Hence, the spectralprojection Pc(n) onto the H(n)-invariant subspace corresponding to the dominantpart of the spectrum has a constant rank q > 0. This spectral gap assumption isquite natural and follows (in conjunction with Theorem 2) for example from theexistence of non-trivial dynamics that is uniformly bounded for ε→ 0 (e.g., relativeequilibria, i.e., solutions of the form E(t) = E0e

iωt, n = const) if r0rL = 0. Wecan split any E ∈ X into E = B(n)Ec + Es, where B(n) is a basis of ImPc(n)depending smoothly on n, Ec ∈ Cq, and Es ∈ X is E − Pc(n)B(n)Ec. The mapR : X × U → Cq × U given by (E, n) → (B(n)−1Pc(n)E, n) is well defined,


smooth and Lipschitz continuous on any closed subset of X × U . Then, the mainmodel reduction theorem is as follows.

Theorem 3 [Model reduction]. Let ε0 > 0 be sufficiently small, ∆ ∈ (ξ, 0), andN be a closed bounded subset of Cq × U . Then, for all ε ∈ [0, ε0) there exists a Ck

manifold C ⊂ X × Rm satisfying:

i. (Invariance) C is S(t, ·)-invariant relative to R−1N . That is, if (E, n) ∈ C,t ≥ 0, and S([0, t]; (E, n)) ⊂ R−1N , then S([0, t]; (E, n)) ⊂ C.

ii. (Representation) C can be represented as the graph of a map which maps

(Ec, n, ε) ∈ N × [0, ε0) → ([B(n) + εν(Ec, n, ε)]Ec, n) ∈ X × Rm,

where ν : N × [0, ε0) → L(Cq ;X) is Ck−2 with respect to all arguments.Denote the X-component of C by

EX (Ec, n, ε) = [B(n) + εν(Ec, n, ε)]Ec ∈ X .

iii. (Exponential attraction) Let Υ ⊂ X × Rm be a bounded set with RΥ ⊂ N anda positive distance to the boundary of N . Then, there exist a constant M anda time tc ≥ 0 with the following property: For any (E, n) ∈ Υ there exists a(Ec, nc) ∈ N such that

‖S(t+ tc; (E, n)) − S(t; (EX(Ec, nc, ε), nc))‖ ≤Me∆t

for all t ≥ 0 with S([0, t+ tc]; (E, n)) ⊂ Υ .iv. (Flow) The flow on C ∩ R−1N is differentiable with respect to t and governed

by the following system of ODEs:

dEcdt

=[

Hc(n) + εa1(Ec, n, ε) + ε2a2(Ec, n, ε)ν(Ec, n, ε)]

Ec ,

dn

dt= εF (Ec, n, ε) ,

(6.1)

where

Hc(n) = B(n)−1H(n)Pc(n)B(n) ,

a1(Ec, n, ε) = −B(n)−1Pc(n)∂nB(n)F (Ec, n, ε) ,

a2(Ec, n, ε) = B(n)−1∂nPc(n)F (Ec, n, ε)(Id− Pc(n)) ,

F (Ec, n, ε) = (Fk(nk) − gk(nk)[EX (Ec, nc, ε), EX(Ec, nc, ε)])m

k=1.

The idea to choose n-dependent coordinates for E in the construction of a re-duced model was introduced already in [1] by physicists. This choice has the advan-tage that the graph of the center manifold itself enters the flow (6.1) on the centermanifold only in the formO(ε2)ν. This fact has been pointed out first in [24], wherethe same model reduction result has been proven for ODEs of similar structure (biglinear system coupled to a slow system) using Fenichel’s theorem for singularly per-turbed systems of ODEs [11]. Since Fenichel’s theorem is not available for infinite-dimensional systems, we have to adapt the proof of Fenichel [11] to our case startingfrom the general results in [4, 5, 6] about invariant manifolds of semiflows in Banach


spaces. In particular, we apply the cut-off modifications done in [11] only to thefinite-dimensional components Ec and n outside of the set N of interest. Moreover,we adapt the modifications such that the invariant manifold for ε = 0 is compactwithout boundary as required by the theorems in [4].

Truncating all terms of order O(ε2) in (6.1) gives rise to a system of ODEs inCq × Rm, where all terms in the right-hand-side can be expressed analytically asfunctions of the eigenvalues of H . The truncated system

dEcdt

= [Hc(n) + εa1(Ec, n, ε)]Ec ,

dnkdt

= ε (Fk(nk) − gk(nk)[B(n)Ec, B(n)Ec])

(6.2)

is called the mode approximation. It is an implicit system of ODEs because the eigen-values of H are given only implicitly as roots of the characteristic function h of H .The dimension of (6.1) is typically low: q is often either 1 or 2. The consideration ofmode approximations has proven to be extremely useful for numerical and analyticalinvestigations of longitudinal effects in multi-section semiconductor lasers; see forexample [2, 21, 27] and section 7. for a demonstration.

7. Parameter study and bifurcation analysis for a laser subject todelayed optical feedback

In this section we demonstrate how the traveling wave model helps to detect andunderstand nonlinear phenomena occurring in multi-section lasers by a bifurcationanalysis using the mode approximations and the subsequent systematic parameterstudy for the full model. We investigate a three-section laser, where S1 is a single-mode DFB laser (i.e., κ1 6= 0, G1 6≡ 0), S2 is a passive phase tuning section (i.e.,κ2 = G2 = ρ2 = n2 = 0), and S3 is an amplifier section (i.e., κ3 = 0, ρ3 = 0,G3 6≡ 0). Since rL 6= 0, this device resembles the classical experiment of a single-mode semiconductor laser which is subject to delayed optical feedback. Section S1

plays the role of the single-mode laser and the sections S2 and S3 form an inte-grated cavity providing delayed optical feedback from the facet at z = L. In thisthree-section setup the two most important parameters, the feedback strength and thefeedback phase ϕ∼ Im d2 can be tuned continuously in the experiment by changingthe currents I2 and I3 into the sections S2 and S3 (up to feedback strengths close to1).

Bifurcation analysis.

Since numerical bifurcation analysis tools like [10] are available for systems ofODEs only, the mode approximations justified in Theorem 3 are extremely helpful.

It turns out that the number q of critical eigenvalues of H(n) is 2 for all rel-evant carrier densities n. Thus, Theorem 3 applies with q = 2 and m = 2 (thecarrier density n2 is constant since section S2 is passive). The center manifold C

has dimension 6 as it is a graph over C2 × R2. The flow of (6.2) is still symmetricwith respect to complex rotation of Ec. Hence, we can reduce it to a 5-dimensional


CU CuspGH Degenerate HopfHSN Hopf–Saddle-node1: Strong resonancesPSN Period-doubling–

Saddle-nodeA,B Non-central saddle-

node on closed orbit

Hopf bif.Saddle-node bif.

Homoclinic bifurcationTorus bifurcationPeriod doublingSaddle-node of limit cyclesExcitabilityMode beating pulsationsUndamped relaxationoscillations

1:3

1:4

GH

1:2

PSN

1

0

2

3

4

0 1 2 3

GH

CUB

A

HSN

Figure 2. Bifurcation diagram for the two-mode approximation (truncated (6.1) with q = 2)in the parameter plane (ϕ, I3) (see [7] for the particular parameter values).

system of ODEs. In this system, equilibria correspond to relative equilibria of theoriginal traveling wave model and periodic solutions to self-pulsations, i.e., modu-lated rotating-wave solutions. Fig. 2 shows the results of two-parameter numericalcontinuations of the physically most relevant codimension-1 bifurcation curves inthe parameter plane (ϕ, I3). The two different islands of self-pulsations are clearlyvisible along with their borders. The nature of these borders and bifurcation theoryserve as a guide for experiment and simulation to investigate interesting phenomenathat otherwise could be missed due to hysteresis or limited basins of attraction. Mostnotably, there are stable invariant tori with strong resonances above the torus bifurca-tion curve, excitability above the homoclinic bifurcation curve, and period doublingand chaos at the border of the undamped relaxation oscillations.

Parameter study for the full PDE System.

Fig. 3 gives an overview over all stable stationary states and non-stationary regimesthat can be found by direct simulation in the parameter plane (ϕ, I3) in the full PDEsystem (2.1) – (2.3). For the simulation, we also included the additional physicaleffects mentioned at the end of section 2. to match the experimental results as closelyas possible. See [7] for a full description of the traveling wave model used in thesimulation.

The two large domains of periodic solutions within each period of ϕ are quiteprominent in Fig. 3 as well. The Hopf and the saddle-node curves can be recog-nized in the simulation and give a full account of the number and stability of allpresent stationary states in Fig. 3. The shadings in Fig. 3 mark the different sta-ble non-stationary regimes in the (ϕ, I3) parameter plane observed in the simulation.Single-pulse periodic solutions are typically born in Hopf bifurcations. Double-pulse


3

I

1

2

Hopf bifurcation

3

.

.

3τ

Saddle-node bifurcation

ϕ/2π 3210

single pulse periodicdouble pulse periodicirregular & multiple pulse

0

Figure 3. Parameter study of a three section laser by direct simulation of the PDE model with

-tool. Full model and used parameter values (except of l1 = 250 µm, l2 = 400 µm andα = 15 cm−1) can be found in [7].

solutions existing nearby have approximately half the frequency. Their occurrence isrelated to the period doubling bifurcations (see also Fig. 2). Finally, multiple-pulseand irregular regimes account for dynamics (and different resonances) on the tori,and chaotic attractors.

A well-known problem of direct simulations is that only one stable regime willbe observed for each parameter value depending on the choice of initial values andthe basins of attractions. However, the bifurcation analysis shows that several stableregimes may coexist in some parameter regions. We took into account this possiblehysteresis by varying the parameters in small steps in different directions from anystable non-stationary regime we found until we hit a sharp transition. In this way,we always traced the hysteresis at sharp transitions corresponding to subcritical orsaddle-node bifurcations. Fig. 3 shows the most simple non-stationary regime in hys-teresis parameter regions (that is, mostly, the single-pulse periodic solution) becausethis is the most interesting regime for potential applications.

8. Conclusions and outlook

The coupled traveling wave model has proven its value in the exploration of non-linear phenomena in multisection laser structures. This can be seen impressively inrecent results concerning delayed optical feedback effects [14, 18, 21, 26, 27], inmulti-section lasers and subsequent new device designs [7, 8, 16, 19]. The modelhas been efficiently implemented in the code which permits extensive param-eter studies. The simulation of the model equations with this code together with thebifurcation analysis of the reduced mode approximation systems gives insight intothe underlying dynamics. Moreover, it allows the user to incorporate physical effects


like spatial hole burning or nonlinear gain compression, or experimental conditionslike optical input or electric modulation. This broadens the range of applications ofthe traveling wave model toward mode-locked lasers, optical amplifiers, ring lasers,etc. However, the theory concerning some of these extensions of the traveling wavemodel is still incomplete, even concerning basic questions like the existence of asmooth strongly continuous semiflow. Thus, an urgent task is to gain a theoreticalunderstanding of these more complex models, and whether they can exhibit substan-tially more complex phenomena.

References

[1] U. Bandelow. Theorie longitudinaler Effekte in 1.55 µm Mehrsektions DFB-Laser-dioden. PhD thesis, Humboldt-Universität Berlin, 1994.

[2] U. Bandelow, L. Recke and B. Sandstede. Frequency regions for forced locking of self-pulsating multi-section DFB lasers. Opt. Comm., 147, 212–218, 1998.

[3] U. Bandelow, M. Wolfrum, M. Radzi unas and J. Sieber. Impact of gain dispersion on thespatio-temporal dynamics of multisection lasers. IEEE J. of Quant El., 37(2), 183–189,2001.

[4] P. W. Bates, K. Lu and C. Zeng. Existence and persistence of invariant manifolds forsemiflows in Banach spaces. Mem. Amer. Math. Soc., 135, 1998.

[5] P. W. Bates, K. Lu and C. Zeng. Persistence of overflowing manifolds for semiflow.Comm. Pure Appl. Math., 52(8), 983 – 1046, 1999.

[6] P. W. Bates, K. Lu and C. Zeng. Invariant foliations near normally hyperbolic invariantmanifolds for semiflows. Trans. Amer. Math. Soc., 352, 4641–4676, 2000.

[7] S. Bauer, O. Brox, J. Kreissl, B. Sartorius, M. Radzi unas, J. Sieber, H.-J. Wünsche andF. Henneberger. Nonlinear dynamics of semiconductor lasers with active optical feed-back. Phys. Rev. E, 69, 016206, 2004.

[8] O. Brox, S. Bauer, M. Radzi unas, M. Wolfrum, J. Sieber, J. Kreissl, B. Sartorius and H.-J. Wünsche. High-frequency pulsations in DFB-lasers with amplified feedback. IEEE J.Quantum Elect., 39(11), 1381 – 1387, 2003.

[9] O. Diekmann, S. van Gils, S. M. Verduyn Lunel and H.-O. Walther. Delay Equations,volume 110 of Applied Mathematical Sciences. Springer-Verlag, 1995.

[10] E. J. Doedel, A. R. Champneys, T. F. Fairgrieve, Y. A. Kuznetsov, B. Sandstede andX. Wang. AUTO97, Continuation and bifurcation software for ordinary differentialequations, 1998.

[11] N. Fenichel. Geometric singular perturbation theory for ordinary differential equations.Journal of Differential Equations, 31, 53–98, 1979.

[12] S. Friese. Existenz und Stabilität von Lösungen eines Randanfangswertproblems derHalbleiterdynamik. Master’s thesis, Humboldt-Universität Berlin, 1999.

[13] F. Jochmann and L. Recke. Existence and uniqueness of weak solutions of an initialboundary value problem arising in laser dynamics. Preprint 515, WIAS, 1999.

[14] B. Krauskopf, K. Schneider, J. Sieber, S. Wieczorek and M. Wolfrum. Excitability andself-pulsations near homoclinic bifurcations in laser systems. Opt. Comm., 215, 367–379, 2003.

[15] D. Marcenac. Fundamentals of laser modelling. PhD thesis, University of Cambridge,1993.

[16] M. Möhrle, B. Sartorius, C. Bornholdt, S. Bauer, O. Brox, A. Sigmund, R. Steingrüber,M. Radzi unas and H.-J. Wünsche. Detuned grating multisection-RW-DFB lasers forhigh speed optical signal processing. IEEE J. on Sel. Top. Quantum Electron., 7, 217–223, 2001.


[17] A. Pazy. Semigroups of Linear Operators and Applications to Partial Differential Equa-tions. Applied mathematical Sciences. Springer Verlag, New York, 1983.

[18] D. Peterhof and B. Sandstede. All-optical clock recovery using multisection distributed-feedback lasers. J. Nonlinear Sci., 9, 575–613, 1999.

[19] M. Radzi unas, H.-J. Wünsche, B. Sartorius, O. Brox, D. Hoffmann, K. Schneider andD. Marcenac. Modeling self-pulsating DFB lasers with integrated phase tuning section.IEEE J. of Quant. El., 36(9), 1026–1034, 2000.

[20] L. Recke, K.R. Schneider and V.V. Strygin. Spectral properties of coupled wave equa-tions. Z. angew. Math. Phys., 50, 923–933, 1999.

[21] J. Sieber. Numerical bifurcation analysis for multi-section semiconductor lasers. SIAMJ. of Appl. Dyn. Sys., 1(2), 248–270, 2002.

[22] J. Sieber. Longtime behavior of the coupled traveling wave model for semiconductorlasers. Preprint 23, University of Bristol, Dept. of Eng. Math., 2003.

[23] B. Tromborg, H. E. Lassen and H. Olesen. Travelling wave analysis of semiconductorlasers. IEEE J. of Quant. El., 30(5), 939–956, 1994.

[24] D. Turaev. Fundamental obstacles to self-pulsations in low-intensity lasers. Preprint629, WIAS, 2001. Submitted to SIAM J. of Appl. Math

[25] H. Wenzel, U. Bandelow, H.-J. Wünsche and J. Rehberg. Mechanisms of fast self pulsa-tions in two-section DFB lasers. IEEE J. of Quant. El., 32(1), 69–79, 1996.

[26] M. Wolfrum and D. Turaev. Instabilities of lasers with moderately delayed optical feed-back. Opt. Comm., 212(1-3), 127 – 138, 2002.

[27] H. J. Wünsche, O. Brox, M. Radzi unas and F. Henneberger. Excitability of a semicon-ductor laser by a two-mode homoclinic bifurcation. Phys. Rev. Lett., 88(2), 023901,2002.

[28] H.-J. Wünsche, M. Radzi unas, S. Bauer, O. Brox and B. Sartorius. Simulation of phase-controlled mode-beating lasers. IEEE J. Selected Topics of Quantum Electron, 9(3), 857– 864, 2003.


Daugiasekcijiniu puslaidininkiniu lazeriu dinamika

J. Sieber, M. Radži unas, K.R. Schneider

Mes nagrinejame išilgine daugiasekcijiniu puslaidininkiniu lazeriu dinamika, kuri yra nusa-koma netiesiškai susietomis hiperboline diferencialiniu lygciu dalinemis išvestinemis bei pa-prastuju diferencialiniu lygciu sistemomis. Mes pateikiame sekancias šios sistemos savybes:globalaus pradinio-kraštinio uždavinio sprendinio egzistavimas bei vienatis; mažos dimensijospritraukianciojo invariantinio hiperpaviršiaus egzistavimas. Modelio dinamika šiame hiper-paviršiuje yra apytiksliai nusakoma paprastuju diferencialiniu lygciu sistema. Pabaigoje mespateikiame detalia skaitine šios paprastuju diferencialiniu lygciu sistemos bifurkacine anali-ze ir lyginame ja su skaitiškai nustatyta pilnos diferencialiniu lygciu dalinemis išvestinemissistemos dinamika.

!"#$%'&)($+*,.-0/213435/768

67–78c© 2004 Technika ISSN 1392-6292

ON THE EXISTENCE AND UNIQUENESS OFTWO-FLUID CHANNEL FLOWS

J. SOCOLOWSKY

University of Applied Sciences, Engineering Department – Mathematics Group

PSF 2132, D-14737 Brandenburg/Havel, GERMANY

E-mail: 9:;:<0:=79?>@A4B0CEDGF4HIJKLMJF4NHO%PQKL


Abstract. Viscous two-fluid channel flows arise in different kinds of coating technologies.The corresponding mathematical models represent two-dimensional free boundary value prob-lems for the Navier-Stokes equations. In this paper the solvability of the related stationaryproblems is discussed and computational results are presented.Furthermore, it is shown that depending on the flow parameters like viscosity or density ratiosand on the fluxes there can happen nonexistence of steady-state solutions. For other parametersets the solution is even unique.

Key words: Free boundary value problems, viscous channel flows, two-fluid flows, Navier-Stokes equations

1. Introduction

In this contribution we consider the plane stationary flow of two viscous incompress-ible fluids (with kinematic viscosities ν i > 0 and densities %i > 0, i = 1, 2) througha special uniform channel (cf. Fig.1). Emphasize that the corresponding problemwill be formulated in dimensionless form. The concrete transition to that formulationcan be found in [11]. The flow is steady-state and has some features of a slot coat-ing process. The channel is horizontal, unbounded in both directions and contains asemi-infinite inner wall (cf. Fig.1). The lower wall S1 := x ∈ R2 : −∞ < x1 <

+∞, x2 = 0 is moving with constant velocity R = (R, 0)T (R > 0). The upperwall S2 := x ∈ R2 : −∞ < x1 < +∞, x2 = 1 is at rest. Furthermore, the partialinner wall S3 := x ∈ R2 : −∞ < x1 < 0, x2 = h1 (0 < h1 < 1) is given. Thus,in fact we have two separated parallel channels for negative values of x1. Both vis-cous fluids are flowing out of the two channels and behind the pointQ(0, h 1) they arejoining and creating a free interface Γ := x ∈ R2 : 0 < x1 < +∞, x2 = ψ(x1)whereψ is unknown a priori and has to be found. It is supposed that the free interfaceΓ separates from the inner wall S3 at its endpointQ.

68 J. Socolowsky

By G1 := x ∈ R2 : 0 < x2 < h1, if − ∞ < x1 6 0, and 0 < x2 <

ψ(x1), if 0 < x1 < +∞ we denote the flow domain of the lower fluid. By G 2 wedenote the flow domain of the upper fluidG 2 := x ∈ R

2 : h1 < x2 < 1, if −∞ <

x1 6 0, and ψ(x1) < x2 < 1, if 0 < x1 < +∞. Finally, by G := G1 ∪ G2 wemean the union of both fluid layers.

Figure 1. Two–fluid channel flow with partial inner wall.

The direction eg of the gravitational force is the vector eg = (0,−1)T . We studythe two-fluid flow within the channel G caused by pressure gradients downstreamand by the motion of the lower channel wall. This means mathematically that thepositive flux F i in each liquid layer Gi (i = 1, 2) is prescribed and the final fluidlayer thicknesses h

∞and (1− h

∞) are to be determined. Note, that our (mathemat-

ical) fluxes F i are in fact the real physical fluxes divided by the constant densities ofthe fluids.

An interpretation of such a flow could be the flow of two liquids coming fromdifferent reservoirs (i.e. slots or chambers) and flowing commonly in one channelafter their unification. In slot coaters such flows occur on some parts of the coater.The corresponding motion as well as the final layer thicknesses are important there.

Let h∞

with 0 < h∞

< 1 be the constant limit of ψ(x1) at infinity. Theproblem under consideration has the following form: find a vector of velocityv = (v1(x1, x2), v2(x1, x2))

T , a pressure p(x1, x2) and a function ψ(x1) satisfy-ing in the domain G the Navier-Stokes system of equations

(v · ∇)v − ν4v + 1

%∇p = g eg ,

∇ · v = 0 ,(1.1)

and the boundary and integral conditions

v|S0= R = (R, 0)T

, v|S2= 0, v|S±

3

= 0 , (1.2)

[v]|Γ = 0, v · n|Γ− = 0, [t · S(v)n]|Γ = 0,

d

dx1

ψ′(x1)

√

1 + ψ′(x1)

2

=1

σ

[−p+ n · S(v)n]|Γ ,

limx1→+∞ψ(x1) = h

∞,

∫

δ1(bq)v1(q, x2) dx2 = F1,

(1.3)

Two-Fluid Channel Flows 69

∫

δ2(bq)

v1(q, x2) dx2 = F2. (1.4)

In problem (1.1) – (1.4) the symbol δi(q) denotes the intersection of Gi with thevertical line x1 = q and σ > 0 is the surface tension at Γ . We further emphasize thatfrom a physical point of view in (1.3), (1.4) only positive values of Fi make sense.

In problem (1.1) – (1.4) the following notations have been used: n and t are unitvectors normal and tangential to Γ and oriented as x2, x1, respectively. By a · b

we mean the inner product of a,b ∈ R2, ∇ = (∂/∂x1, ∂/∂x2)

T is the gradientoperator, ∇p = grad p, ∇ · v = div v, %|Gm

= %m (m = 1, 2) is the restrictionof % to Gm (analogously for ν). 4 = ∇2 denotes the Laplace operator. By S(v) wedenote the deviatoric stress tensor, i.e. a matrix with elements

Sij(v) = %ν

(

∂vi

∂xj

+∂vj

∂xi

)

, i, j = 1, 2 .

The symbol [w]|Γ expresses the jump of w crossing the free interface Γ , i.e.

[w(x0)] |Γ := limy→x0

w(y) − limx→x0

w(x), (x0 ∈ Γ, y ∈ G1, x ∈ G2),

and the symbolw|Γ− denotes the limit from below at the interface Γ , more precisely

w(x0)|Γ− := limy→x0

w(y), (x0 ∈ Γ, y ∈ G1).

An analogous statement is true for S±

3 (and also for Σ±

3 in Section 4). Note that theleft-hand side of (1.3)2 (i.e. of the second equation in (1.3)) is equal to the curvatureK(x1) of Γ . The fluid layer thickness h

∞has also to be determined. Obviously, it

should hold 0 < h∞< 1.

Mathematical problems for the stationary flows of a viscous incompressible fluidwith a free boundary were investigated by many authors. Numerous references onthis field can be found, e.g., in the bibliographies of [4, 6, 12, 13]. Coating flowswith the static or dynamic contact angles were studied in [1, 2, 5, 10, 11, 14, 15].In all these papers considering either compact or semi-infinite free boundary valueproblems the same general scheme developed in [3, 9] has been used.

Let us shortly recall this scheme and apply it to problem (1.1)–(1.4). The startingproblem is divided into two problems: the boundary value problem for the Navier-Stokes system of equations in a fixed domain and the problem of finding the freeboundary Γ from the equation

K(x1) = σ−1[−p(x) + n · S(v)n]|Γ (1.5)

with the corresponding boundary conditions. The solution of the free boundary prob-lem can be found by the method of successive approximations. At every step of suc-cessive approximations the Navier-Stokes system is solved in a fixed domain. Theobtained solution is substituted into the right-hand side of (1.5) and by solving thisequation one obtains the next iteration for the free boundaryΓ . Thus, one gets a newdomain in which the Navier-Stokes system has to be solved again. So, this schemecan be illustrated by the diagram

70 J. Socolowsky

Γ(0) → G

(0) → (v(1), p

(1)) → Γ(1) → G

(1) → (v(2), p

(2)) → . . .

Note that in this method at every step of successive approximations the constructionof (v, p) is separated from the construction of the free boundary Γ . On the otherhand, for free boundary problems in which the unknown flow domain is unboundedin two directions the described scheme is not applicable (cf. [4, 6, 7] and others).

2. Function Spaces

Let B be an arbitrary domain in R2 and N ⊂ B a manifold of dimension lessthan 2. The symbol %N (x) denotes (in this section only) the distance dist (x, N) :=infy∈N |x − y|. Let β = (β1, β2) be a multiindex with

|β| = β1 + β2 and Dβu =

∂|β|u

∂xβ1

1 ∂xβ2

2

(βi ∈ N ∪ 0).

The symbol brc will denote the integer part of r. Cr(B) (r > 0, non-integer) denotesthe Hölder space of functions defined in a domainB ⊂ R2 with a finite norm

|u|(r)

B =∑

|β|<r

supx∈B

|Dβu| +

∑

|β|=brc

supx,y∈B

|Dβu(x) −D

βu(y)|

|x − y|r−brc.

Let Crs (B,N) be the weighted Hölder space of functions defined inB\N and having

a finite norm

|u|Crs(B,N)

=∑

|β|<r

supx∈B\N

%|β|−s

N (x)|Dβu(x)|

+∑

|β|=brc

supx∈B\N

%r−sN (x) sup

|x−y|< 1

2%N (x)

|Dβu(x) −D

βu(y)|

|x − y|r−brc.

Crs (B,N) (r > s > 0; r, s non-integer) denotes the space of functions with a finite

norm

|u|Crs(B,N) := |u|

(s)

B +∑

s<|β|<r

supx∈B\N

%|β|−s

N (x)|Dβu(x)|

+∑

|β|=brc

supx∈B\N

%r−sN (x) sup

|x−y|< 1

2%N (x)

|Dβu(x) −D

βu(y)|

|x − y|r−brc.

Clearly, Crs (B,N) is a subspace of Cr

s (B,N) consisting of functions vanishing onN together with their derivatives of order up to bsc. For s < 0 assume Cr

s (B,N) :=C

rs (B,N).

Finally we define the weighted Hölder spaces to which the generalized solutionsto the problem (1.1)–(1.4) belong. We use the following notations:

G0 := x ∈ G : |x1| < 2, G

+ := x ∈ G : x1 > 1 ,

G− := x ∈ G : x1 < −1, J

0 := (0, 2), J+ := (1,+∞) .


For an arbitrary real number z > 0 we define the space

Crs,z(G) =

u(x), u|G0 ∈ Crs (G0

, Q∗), exp(zx1)u(x)|G+ ∈ C

r(G+),

exp(−zx1)u(x)|G− ∈ Cr(G−)

with the norm:

‖ u ‖r,zG,s:= |u|Cr

s(G0,Q) + | exp(zx1)u|

(r)

G+ + | exp(−zx1)u|(r)

G−.

For functions f(x1) defined in R1+ we introduce the space Cr

s,z(R1+) with the norm

‖ f ‖r,z

R1

+,s= |f |Cr

s(J0,0) + |f(x1) exp(zx1)|

(r)

J+ .

The spaces of vector-fields u are denoted by bold letters. The corresponding normsare the sum of the norms of the coordinate functions.

3. Analytical Results

By straightforward calculations one can determine the exact Poiseuille flows

v(−)(x), p(−)(x), x ∈ G−

i , i = 1, 2 .

The corresponding velocities do not depend on x1. In G−

1 (i.e. if 0 6 x2 6 h1) oneobtains

v(−)

1 (x) =

(

3R

h21

−6F1

h31

)

x22 +

(

−4R

h1

+6F1

h21

)

x2 +R,

v(−)

2 (x) ≡ 0,

p(−)(x1, x2) = 2ν1%1

(

3R

h21

−6F1

h31

)

x1 − %1gx2 + k1 .

(3.1)

In G−

2 (i.e., if h1 6 x2 6 1) one gets, respectively,

v(−)

1 (x1, x2) = −6F2

(1 − h1)3x

22 +

6(1 + h1)F2

(1 − h1)3x2 −

6h1F2

(1 − h1)3,

v(−)

2 (x) ≡ 0,

p(−)(x1, x2) = −

12ν2%2F2

(1 − h1)3x1 − %2gx2 + k2.

(3.2)

It is well-known that the pressure p can be determined only up to an additive constantin channel flows (cf. k 1, k2).

In [6, 7] the Poiseuille flow v (+), p

(+) for the united channel G+ was deter-mined by straightforward calculations, too. The corresponding flow fields are givenby the following formulae [cf. also equations (32) – (34) in [7] (p. 206, 207) orequations (A.11′), (A.12′) in [6] (p. 41)]

72 J. Socolowsky

v(+)

1 (x2) =

0.5a1x22 + b1x2 +R, 0 6 x2 6 h

∞

0.5a2(x22 − 1) + b2(x2 − 1), h

∞6 x2 6 1

,

v(+)

2 (x2) ≡ 0,

p(+)(x) =

p0x1 − %1g + k, 0 6 x2 6 h∞

p0x1 − %2g(x2 − 1) − %1h2g + k, h

∞6 x2 6 1

,

(3.3)

where the coefficients have the representations

a1 = −3F1−Rh∞

h2∞

− 3 F2

r(1−h2∞

), b1 = (2 + h

∞)F1−Rh∞

h2∞

+ h∞

F2

r(1−h2∞

),

a2 = −3rF1−Rh∞

h2∞

− 3 F2

1−h2∞

, b2 = r(2 + h∞

)F1−Rh∞

h2∞

+ h∞

F2

1−h2∞

,

and r :=%1ν1

%2ν2

in this section. For the pressure gradient, i.e.∂p

∂x1

= p0, it holds

p0 = a1ν1%1 = a2ν2%2.Note that in [6] the viscous two-fluid flow through a perturbed uniform channel

(without a partial inner wall) was studied by different functional-analytic methods(cf. also [8]). An essential part of the determination of v(+)

, p(+) consisted in

the calculation of the value h∞

from the following 5th degree polynomial equation(cf. also equation (A.14) in [6], p. 43).

r(r − 1)Rh5∞

+ [−4r(r − 1)R− r(r − 1)F1 − (r − 1)F2]h4∞

+ [r(6r − 5)R+ 2r(2r − 3)F1 − 2rF2]h3∞

+ [2r(−2r + 1)R (3.4)

+ 3r(−2r + 3)F1 + 3rF2]h2∞

+[

r2R+ 4r(r − 1)F1

]

h∞

− r2F1 = 0.

Note that the final thickness h∞

is a function of F1, F2, R and the rheological param-eters of the fluids. It can have up to three different values within (0, 1) for the sameparameter set (cf. [6, 7]). Let

h∞

be one of these values. Furthermore, by

ψ(x1)we denote the associated infinitely differentiable solution of the following boundaryvalue problem

−d

dx1

ψ′(x1)

√

1 + ψ′(x1)

2

+g(%1 − %2)

σ

ψ(x1) =g(%1 − %2)

σ

h∞,

ψ(0) = h1, limx1−→+∞

ψ(x1) =

h∞,

(3.5)

which can be obtained from the second line of (1.3) by setting v = 0 and p = constas the initial solution for F1 = F2 = R = 0. Let ξ = ξ(x1) be a smooth cut-off function vanishing for |x1| 6 1 and being equal to 1 for |x1| > 2. Finally,suppose that %1 > %2 is fulfilled. This assumption is physically sensefull. Now wecan formulate the main result of this section.

Theorem 1. There exist positive real numbers s0,M0 and z0 6

√

g (%1−%2)

σsuch

that for arbitrary s ∈ (0, s0), z ∈ (0, z0),max [F1, F2, R] < M0 and for parameters

h∞, F1, F2, R satisfying the condition


∣

∣

∣h1 −

h∞

∣

∣

∣ <

√

2σ

g (%1 − %2), (3.6)

where

h∞

is one of the roots to equation (3.4), the free boundary value problem(1.1) − (1.4) has a unique solution v, p, ψ which can be represented in the form

v = ξ(−x1)v(−) + ξ(x1)v

(+) + w, ψ(x1) =

ψ(x1) + ω(x1),

p = ξ(−x1)p(−) + ξ(x1)p

(+) + p0(x),(3.7)

where v(−), p

(−) denotes the Poiseuille flow from equations (3.1), (3.2) in bothchannels as x1 −→ −∞ and v(+)

, p(+) is the basic solution (3.3) for x1 −→

+∞. Moreover, w ∈ Cs+2s,z (G), p0 ∈ C

s+1s−1,z(G

0 ∪ G+),∇p0 ∈ C

ss−2,z(G) and

ω ∈ C3+s1+s,z(R

1+) hold.

The proof of this theorem can be realized in the same way as in [10] applying theabove mentioned scheme. We omit here the proof. The condition (3.6) is a con-sequence of solving the boundary value problem (3.5) and the physical restriction%1 > %2 is also essential for the applied method. The weight parameter s0 in Theo-rem 3.1 can be estimated studying a model problem for the Stokes system in a neigh-bourhood of Q in the same way as in [10]. The exponential behaviour of w, p0, ω atinfinity is well-known (cf. [4, 10]).

4. Computational Results

For computational purposes it was necessary to truncate the theoretical unboundedflow domain from Fig.1.

Figure 2. Computational (truncated) flow domain.

Therefore, one gets an artificial inlet Σ4 = Σ41 ∪Σ42 (i.e. an inflow region in bothchannels) and an artificial outlet Σ5 = Σ51 ∪ Σ52 far enough from the separationpoint Q. We obtain the following two free boundary value problems

(v · ∇)v − ν4v + 1

%∇p = g eg,

∇ · v = 0 ,(4.1)

v|Σ1= (1, 0)T

, v|Σ2= 0, v|Σ±

3

= 0, v|Σ4k= v

(4,k), (k = 1, 2), (4.2)

74 J. Socolowsky

[v]|Γ = 0, v · n|Γ− = 0, [t · S(v)n]|Γ = 0,

d

dx1

ψ′(x1)

√

1 + ψ′(x1)

2

=1

σ

[−p+ n · S(v)n]|Γ ,(4.3)

with eitherv1|Σ5k

= v(5)

1 , v2|Σ5k= 0, (k = 1, 2), (4.4D)

or

t · T(v)n|Σ5k= 0 =

∂v1

∂x1

∣

∣

∣

∣

Σ5k

, v2|Σ5k= 0, (k = 1, 2). (4.4N)

AtΣ4 we pose Dirichlet boundary conditions (4.24) (i.e. the fourth equation in (4.2))where v

(4) is in fact the Poiseuille flow v(−) from (3.1), (3.2). At the outlet Σ5 we

set either Dirichlet boundary conditions (4.4D) with v(5)

1 = v(+)

1 taken from (3.3) orNeumann boundary conditions (4.4N) for the downstream velocity v1.

We were especially interested in the case, when h∞

has three different valuesin (0, 1) for given fluxes F 1, F2. This happens if F1 = 0.41 and F2 = 0.01 hold(cf. [6, 7]). The associated remaining parameters for this first example are ν1 =10.0, ν2 = 2.0, %1 = 1.0, %2 = 0.5263, σ = 0.001. The partial inner wall Σ3 islocated at h1 = 0.5. The inflow region is situated at x 1 = 0.0, the separation pointQ at x1 = 4.0 and the outflow region was chosen at x 1 = 17.0.

The numerical simulations have been performed with the help of a FORTRANcode that uses both the FEM and the method of support lines (or spines) for thediscretization of the flow domain (cf. [11]). The discretization has been performedusing 643 nodes, 288 triangular elements and 19 spines. Thus, the total number ofunknowns was 2345. More details on the discretization of similar problems can befound in [6]. All computations presented below were realized on a PENTIUM IIIpersonal computer with 450 MHz. The time per iteration was about 20 seconds.

In the first computation we posed Dirichlet boundary conditions for v1 at theoutlet and the x2 - value h

∞of Γ at x1 = 17.0 has not been fixed. Its starting value

has been h(0)∞

= 0.6321, i.e. one of the three exact solutions to the problem withoutinner wall in [6]. The position of Γ after 5 iterations is presented in Fig.3.

Figure 3. Computed free interface for h(0)

∞= 0.6321.

When taking h(0)∞

= 0.8031 the following figure arises (see Fig.4) In the third


Figure 4. Computed free interface for h(0)

∞= 0.8031.

Figure 5. Computed free interface of a two-fluid channel flow with Neumann bound-ary conditions.

computation (cf. Fig.5) of the first example we have used Neumann boundary con-ditions for v1 at the outlet and the position h

∞of Γ at the outlet has also not been

fixed. Its starting value for the iteration scheme has been h(0)∞

= 0.5. Figure 5 showsthe computed position of the free interface Γ after 30 iterations.

Even if choosing the position h1 of the inner wall very close to one of the threeexact values of h

∞, namely h1 = 0.8, the computational results did not become

better. Figure 6 represents the corresponding situation.

Figure 6. Computed free interface of a two-fluid channel flow with a different innerwall.

In all these computations of the first example we could not reach convergence of theiteration scheme. Moreover, one can recognize that in Figs.3 – 6 the free interface Γ

76 J. Socolowsky

turns off in front of the outlet. It cannot find a final thickness h∞

. This was typicalfor all similar computations. Therefore, it seems to us that there is no solution ofproblems (4.1)− (4.4D) and (4.1)− (4.4N) for the above mentioned parameter set.

Let us introduce a second parameter set:

F1 = 0.534, F2 = 0.266, ν1 = 166.667, ν2 = 250.0 ,

%1 = 1.0, %2 = 0.9, σ = 0.0001 .

The location of the partial inner wall Σ3 is h1 = 0.3 (cf. Fig.7).

Figure 7. Computed free interface for a second parameter set.

It is well-known (cf. [6, 7]) that for this parameter set the uniform channel prob-lem without inner wall possesses a unique solution which leads to the layer thicknessh∞

= 0.5027. We could show that our problem (4.1) − (4.4N) has also a uniquesolution with the same h

∞. The iteration scheme converges independently of the

starting value h(0)∞

. Figure 7 shows the computed free interface after 30 iterations.One can see the uniform behaviour of Γ . The last picture (Fig.8) represents the ve-locity moduli at the nodes located on the free interface.

Figure 8. Velocities at the free interface for a two-fluid channel flow.

Note finally, that the pressure converges very well except at the neighbourhood ofthe separation point Q where the pressure admits a singularity.


References

[1] A. Friedman and J.J.L. Velazquez. The analysis of coating flows in a strip. J. Diff. Eq.,121, 134 – 182, 1995.

[2] A. Friedman and J.J.L. Velazquez. The analysis of coating flows near the contact line.J. Diff. Eq., 119, 137 – 208, 1995.

[3] O.A. Ladyzhenskaya and V.G. Osmolovskii. On the free surface of a fluid over a solidsphere. Vestnik Leningrad. Univ. Math., 13, 25 – 30, 1976.

[4] S.A. Nazarov and K. Pileckas. On noncompact free boundary problems for the planestationary Navier-Stokes equations. J. Reine u. Angewandte Mathematik, 438, 103 –141, 1993.

[5] K. Pileckas. Solvability of a problem of plane motion of a viscous incompress-ible fluid with noncompact free boundary. Diff. Equ. Appl. Inst. of Math. Cy-bern. Acad. Sci. Lit. SSR, 30, 57 – 96, 1981.

[6] K. Pileckas and J. Socolowsky. Viscous two-fluid flows in perturbed unbounded do-mains. Mathematische Nachrichten. (submitted for publication)

[7] K. Pileckas and J. Socolowsky. Analysis of the Navier-Stokes equations for some two-layer flows in unbounded domains. In: K. Pileckas H. Amann, G.P. Galdi and V.A.Solonnikov(Eds.), Navier-Stokes equations and Related Nonlinear Problems, Utrecht,Tokyo and Vilnius, VSP/TEV, 195 – 216, 1998.

[8] K. Pileckas and J. Socolowsky. Analysis of two linearized problems modeling viscoustwo-layer flows. Mathematische Nachrichten, 245, 129 – 166, 2002.

[9] V.V. Pukhnachov. Plane stationary free boundary problem for Navier-Stokes equation.Zh. Prikl. Mekh. i Tekhn. Fiz., 3, 91 – 102, 1972.

[10] J. Socolowsky. The solvability of a free boundary problem for the stationary Navier-Stokes equations with a dynamic contact line. Nonlinear Analysis, Theory, Methods &Applications (JNA – TMA), 21, 763 – 784, 1993.

[11] J. Socolowsky. On the numerical solution of heat-conducting multiple-layer coatingflows. Lietuvos Matematikos Rinkinys, 38, 125 – 147, 1998.

[12] V.A. Solonnikov. On the Stokes equation in domains with nonsmooth boundaries and ona viscous incompressible flow with a free surface. In: Nonlinear partial diff. equationsand their applications, College de France Seminar, volume 3, 340 – 423, 1980/81.

[13] V.A. Solonnikov. Solvability of the problem on the effluence of a viscous incompressiblefluid into an open bassin. Trudy Mat. Inst. Steklov, 179, 174 – 202, 1988.

[14] V.A. Solonnikov. Problems with free boundaries and with moving contact points for two-dimensional stationary Navier-Stokes equations. Zap. Nauchn. Sem. St.-Peterburg. Ot-del. Mat. Inst. Steklova (POMI), 213, 1994. Kraev. Zadachi Mat. Fiz. Smezh. VoprosyTeor. Funktsii 25, 179 – 205 (in Russian)

[15] V.A. Solonnikov. On some free boundary problems for the Navier-Stokes equations withmoving contact points and lines. Math. Annalen, 302, 743 – 772, 1995.

78 J. Socolowsky

Apie dvieju tekanciu kanale skysciu srauto egzistavima ir vienati

J. Socolowsky

Dvieju, tekanciu kanale, klampiu skysciu srauto uždavinys iškyla taikant ivairias skirtingur ušiu paviršiu padengimo technologijas. Atitinkamas matematinis modelis išreiškiamas dvi-maciu kraštiniu uždaviniu su laisvu paviršiumi Navje-Stokso lygtims.Straipsnyje nagrinejamas santykinai stacionaraus uždavinio išsprendžiamumas ir pateikiamiskaiciavimo rezultatai. Be to parodoma, kad priklausomai nuo sroves parametru kaip ir nuoklampumo ir tankio santykio stacionar us sprendiniai gali neegzistuoti. Su kitais parametraisegzistuoja tiksliai vienas sprendinys.

!"#$%'&)($+*,.-0/213435/768

79–90c© 2004 Technika ISSN 1392-6292

STABILITY OF THE SPLINE COLLOCATIONMETHOD FOR SECOND ORDER VOLTERRAINTEGRO-DIFFERENTIAL EQUATIONS1

M. TARANG

Institute of Applied Mathematics

Liivi 2, 50409, Tartu, Estonia

E-mail: 97:0;<%=?>:0;:@ABC>D=<<


Abstract. Numerical stability of the spline collocation method for the 2nd order Volterraintegro-differential equation is investigated and connection between this theory and corre-sponding theory for the 1st order Volterra integro-differential equation is established. Resultsof several numerical tests are presented.

Key words: The 2nd order Volterra integro-differential equation, stability of the spline col-location method

1. Introduction

We study the numerical stability of the spline collocation method for the 2nd orderVolterra integro-differential equation (VIDE). Stability means here the boundednessof approximate solutions in the uniform norm in case of the test equation when thenumber of knots increases. Basic ideas in the numerical solution of Volterra inte-gral equation (VIE) and VIDE are given in [2]. First results about stability of thecollocation method by polynomial splines for Volterra integral equation are given in[3] and the most adequate ones seems to be given in [5]. A special case of smoothsplines is treated in [4] and special case of piecewise polynomial splines, i.e. splineswith possible discontinuities in knots, is presented in [6]. There is a standard reduc-tion of the 1st order VIDE to VIE considering the derivative of the solution as anew unknown solution. But then the test equation with constant kernel transformsinto an equation with nonconstant kernel and the results obtained for VIE are notdirectly extendable to the 1st order VIDE. Similar phenomena takes place if we tryto reduce the problem of stability for the 2nd order VIDE to that for the 1st orderVIDE. Another possibility is to present the 2nd order VIDE as a system consisting

1 Research supported by the Estonian Science Foundation Grant No.5260

80 M. Tarang

of a first order VIDE and a first order differential equation or as a first order VIDEin product space. The components of unknown in this product space are the solutionof the initial 2nd order VIDE and its derivative. The notion of stability for this VIDEin product space means boundedness of approximate solutions for both componentswhich is, however, different from the notion of stability given in Section 4. Thesetwo circumstances motivate our investigations.

2. The spline collocation method

Consider the 2nd order Volterra integro-differential equation

y′′(t) = f(t, y′(t), y(t)) +

∫ t

0

K(t, s, y′(s), y(s))ds, t ∈ [0, T ], (2.1)

with initial conditionsy(0) = y0, y

′(0) = y1 .

The functions f : [0, T ]×R → R and K : S ×R → R (where S = (t, s) : 0 ≤ s ≤

t ≤ T), and numbers y0 and y1 are supposed to be given. In order to describe thismethod, let 0 = t0 < t1 < . . . < tN = T (with tn depending on N ) be a mesh onthe interval [0, T ]. Denote

hn = tn − tn−1, σn = (tn−1, tn], n = 1, . . . , N, ∆N = t1, . . . , tN−1 .

Let Pk denote the space of polynomials of degree not exceeding k. Then, for givenintegers m ≥ 1 and d ≥ −1, we define

Sdm+d(∆N ) = u : u|σn

= un ∈ Pm+d, n = 1, . . . , N − 1 ,

u(j)n (tn) = u

(j)

n+1(tn + 0), tn ∈ ∆N , j = 0, 1, . . . , d

to be the space of polynomial splines of degree m + d which for d ≥ 1, are d-timescontinuously differentiable on [0, T ], for d = 0 are continuous on [0, T ] and ford = −1 may have jump discontinuities at the knots ∆N .

An element u ∈ Sdm+d(∆N ) as a polynomial spline of degree not greater than

m + d for all t ∈ σn, n = 1, . . . , N , can be represented in the form

un(t) =

m+d∑

k=0

bnk(t − tn−1)k. (2.2)

To find coefficients bnk we suppose that a fixed selection of collocation parameters0 < c1 < . . . < cm ≤ 1 is given. Then we define collocation points tnj = tn−1 +cjhn, j = 1, . . . , m, n = 1, . . . , N , forming a set X(N). In order to determine theapproximate solution u ∈ S

dm+d(∆N ) of the equation (2.1) we impose the following

collocation conditions

u′′(t) = f(t, u′(t), u(t)) +

∫ t

0

K(t, s, u′(s), u(s))ds, t ∈ X(N). (2.3)

Stability of collocation method for 2nd order VIDE 81

Starting the calculations by this method we assume also that we can use the initialvalues u

(j)1 (0) = y

(j)(0), j = 0, . . . , d, that is justified by the requirement u ∈

Cd[0, T ]. Another possible approach is to use initial conditions u1(0) = y(0) and

u′

1(0) = y′(0) and more collocation points (if d ≥ 1) to determine u1. Thus, on every

interval σn we have d + 1 conditions of smoothness and m collocation conditionsto determine m + d + 1 parameters bnk. This allows us to implement the methodstep-by-step going from an interval σn to the next one.

In this paper we will analyse the stability of the collocation method where thesplines are at least continuously differentiable. Thus, we suppose in the sequel thatd ≥ 1.

3. The method in the case of a test equation

Let us consider the test equation

y′′(t) = αy(t) + βy

′(t) + λ

∫ t

0

y(s)ds + f(t), t ∈ [0, T ], (3.1)

where α, β and λ may be any complex numbers. The equation (3.1) is called thebasis test equation (see [1]) and it has been extensively used for studying stabilityproperties of several methods. Assume that the mesh sequence ∆N is uniform,i.e., hn = h = T/N for all n. Representing t ∈ σn as t = tn−1 + τh, τ ∈ (0, 1], wehave on σn the equality:

un(tn−1 + τh) =

m+d∑

k=0

ankτk, τ ∈ (0, 1], (3.2)

where we passed to the new parameters ank = bnkhk.

The smoothness conditions (for any u ∈ Sdm+d(∆N ))

u(j)n (tn − 0) = u

(j)n+1(tn + 0), j = 0, . . . , d, n = 1, . . . , N − 1,

can be expressed in the form

an+1,j =

m+d∑

k=j

k!

(k − j)!j!ank, j = 0, . . . , d, n = 1, . . . , N − 1. (3.3)

The collocation conditions (2.3), applied to the test equation (3.1), give

u′′(tnj) = f(tnj) + αy(tnj) + βu

′(tnj) + λ

∫ tnj

0

u(s)ds,

j = 1, . . . , m, n = 1, . . . , N . (3.4)

From (3.2) we get

82 M. Tarang

un(tnj) =

m+d∑

k=0

ankckj , u

′

n(tnj) =1

h

m+d∑

k=1

ankkck−1

j

and

u′′

n(tnj) =1

h2

m+d∑

k=2

k(k − 1)ankck−2

j .

Now the equation (3.4) becomes

1

h2

m+d∑

k=0

k(k − 1)ankkck−2

j = α

m+d∑

k=0

ankckj + β

1

h

m+d∑

k=0

kankck−1

j

+n−1∑

r=1

λ

∫ tr

tr−1

ur(s)ds + λ

∫ tnj

tn−1

un(s)ds + f(tnj)

= α

m+d∑

k=0

ankckj + β

1

h

m+d∑

k=0

kankck−1

j +

n−1∑

r=1

λh

∫ 1

0

(

m+d∑

k=0

arkτk)

dτ

+λh

∫ cj

0

(

m+d∑

k=0

ankτk)

dτ + f(tnj)

= α

m+d∑

k=0

ankckj + β

1

h

m+d∑

k=0

kankck−1

j +

n−1∑

r=1

λh

(

m+d∑

k=0

1

k + 1ark

)

+λh

m+d∑

k=0

ank

ck+1

j

k + 1+ f(tnj). (3.5)

Using the notation αn = (an0, . . . , an,m+d), we write (3.5) as follows:

m+d∑

k=0

ankk(k − 1)ck−2

j − αh2

m+d∑

k=0

ankckj − βh

m+d∑

k=0

ankkck−1

j

− λh3

m+d∑

k=0

ank

ck+1

j

k + 1= λh

3⟨

q,

n−1∑

r=1

αr

⟩

+ h2f(tnj), (3.6)

where q = (1, 1/2, . . . , 1/(m + d + 1)) and⟨

· , ·⟩

denotes the usual scalar productin Rm+d+1. The difference of the equations (3.6) with n and n + 1 yields


m+d∑

k=0

an+1,kk(k − 1)ck−2

j − βh

m+d∑

k=0

an+1,kkck−1

j − αh2

m+d∑

k=0

an+1,kckj

−λh3

m+d∑

k=0

an+1,k

ck+1

j

k + 1=

m+d∑

k=0

ankk(k − 1)ck−2

j − βh

m+d∑

k=0

ankkck−1

j

−αh2

m+d∑

k=0

ankckj − λh

3

m+d∑

k=0

ank

ck+1

j

k + 1+ λh

3⟨

q,

n−1∑

r=1

αr

⟩

+h2f(tn+1,j) − h

2f(tnj), j = 1, . . . , m, n = 1, . . . , N − 1. (3.7)

Now we may write equations (3.3) and (3.7) in the matrix form

(V − βhV1 − αh2V2 − λh

3V3) αn+1 =

(

V0 − βhV1 − αh2V2

−λh3(V3 − V4)

)

αn + h2gn, n = 1, . . . , N − 1 , (3.8)

with (m + d + 1) × (m + d + 1) matrices V , V0, V1, V2, V3, V4 as follows:

V =

(

E

C

)

, V0 =

(

A

C

)

, E =(

I 0)

,

I being the (d + 1) × (d + 1) unit matrix, 0 is the (d + 1) × m zero matrix,

C =

0 0 2 6c1 . . . (m + d)(m + d − 1)cm+d−2

1

. . . . . . . . . . . . . . . . . .

0 0 2 6cm . . . (m + d)(m + d − 1)cm+d−2m

,

A being a (d + 1) × (m + d + 1) matrix

A=

1 1 1 . . . . 10 1 2 . . . . m + d

. . . . . . . . . .

0 . . 1 . . .

(

m + d

d

)

, V1 =

0 0 0 . . . 0

0 1 2c1 . . . (m + d)cm+d−11

. . . . . . . . . . .

0 1 2cm. . . (m + d)cm+d−1m

,

V2 =

0

1 c1 c21 . . . c

m+d1

. . . . . . . . . . . . .

1 cm c2m . . . c

m+dm

, V3 =

0

c1 c21/2 . . . c

m+d+1

1 /(m + d + 1). . . . . . . . . . . .

cm c2m/2 . . . c

m+d+1m /(m + d + 1)

,

V4 having the first d+1 rows equal to 0 and the last m rows the vector q, and, finally,the m + d + 1 dimensional vector

gn =(

0, . . . , 0, f(tn+1,1) − f(tn1), . . . , f(tn+1,m) − f(tnm))

.

Thus gn = O(h) for f ∈ C1.

84 M. Tarang

Proposition 1. The matrix V − βhV1 − αh2V2 − λh

3V3 is invertible for sufficiently

small h.

Proof. Since

det V =det

(d + 1)dcd−1

1 . . . (m + d)(m + d − 1)cm+d−2

1

(d + 1)dcd−1

2 . . . (m + d)(m + d − 1)cm+d−2

2

. . . . . . . . .

(d + 1)dcd−1m . . . (m + d)(m + d − 1)cm+d−2

m

=(d + 1)dcd−1

1

× . . . × (m + d)(m + d − 1)cd−1m det

1 c1 . . . cm−11

. . . . . . . . . . . .

1 cdm . . . c

m−1m

6= 0,

and d ≥ 1, the matrix V is invertible. Such is also V − βhV1 − αh2V2 − λh

3V3 for

small h. Although we have supposed, in general, that d ≥ 1, let us remark that incases d = 0 and d = −1 we may argue similarly to the proof in [6] and show thatdet(V − βhV1 − αh

2V2 − λh

3V3) 6= 0, for small h.

Therefore, the equation (3.8) can be written in the form

αn+1 = (V −1V0 + W )αn + rn, n = 1, . . . , N − 1, (3.9)

where W = O(h) and rn = O(h3) for f ∈ C1.

4. Stability of the method

We have seen that the spline collocation method (2.3) for the test equation (3.1) leadsto the recursion (3.9).

We distinguish the method with initial values u(j)1 (0) = y

(j)(0), j = 0, . . . , d,and another method which uses u1(0) = y(0), u′

1(0) = y′(0) and additional collo-

cation points t0j = t0+c0jh, j = 1, . . . , d−1, with fixed c0j ∈ (0, 1]\c1, . . . , cm

on the first interval σ1. Denote, in addition, d0 = maxd−2, 0 for the method withinitial values and d0 = 0 for the method with additional initial collocation.

We say that the spline collocation method is stable if for anyα, β, λ ∈ C and any f ∈ C

d0 [0, T ] the approximate solution u remains boundedin C[0, T ] in the process h → 0.

Let us notice that the boundedness of ||u||C[0,T ] is equivalent to the boundedness of||αn|| in n and h in any fixed norm of R

m+d+1.The principle of uniform boundedness allows us to establish

Proposition 2. The spline collocation method is stable if and only if

||u||C[0,T ] ≤ c ||f ||Cd0 [0,T ] ∀f ∈ Cd0 [0, T ], (4.1)

where the constant c may depend only on T , α, β, λ and on parameters cj and c0j .


In order to formulate and prove the results concerning the numerical stability prop-erties of the polynomial spline collocation method, we need the following resultsfor VIE (see [5]) and for the 1st order VIDE (see [7]). The step-by-step collocationmethod for VIE is supposed to determine the approximate solution in S

dm+d(∆N )

by the collocation conditions similarly to (2.3) at the points tnj .

1. The stability for VIE depends on the matrix ˜M = ˜

U−1

0˜

U , where ˜U0 and ˜U are(m + d + 1) × (m + d + 1) matrices as follows:

˜

U0 =

(

E

˜

G

)

,˜

U0 =

(

A

˜

G

)

,˜

G =

1 c1 . . . cm+d1

. . . . . . . . . . . .

1 cm . . . cm+dm

,

E and A being defined as in V and V0.2. If all eigenvalues of ˜M are in the closed unit disk and if those which lie on

the unit circle have equal algebraic and geometric multiplicities, then the splinecollocation method is stable.

3. If ˜M has an eigenvalue outside of the closed unit disk, then the method is unsta-ble (u has exponential growth: ‖ u ‖

∞≥ c eKN

, for some constants K > 0 andc > 0).

4. If all eigenvalues of ˜M are in the closed unit disk and there is an eigenvalueon the unit circle with different algebraic and geometric multiplicities, then themethod is weakly unstable (u may have polynomial growth: ‖ u ‖

∞∼ c N

k, c >

0, k ∈ N).5. For fixed cj the eigenvalues of M = U

−1

0 U for the 1st order VIDE in the case m

and d + 1 and eigenvalues of ˜M for VIE in the case m and d coincide and havethe same algebraic and geometric multiplicities, except µ = 1 whose algebraicmultiplicity for VIDE is greater by one than for VIE. Here U0 and U are (m +d + 1) × (m + d + 1) matrices as follows:

U =

(

E

G

)

, U0 =

(

A

G

)

, G =

0 1 2c1 . . . (m + d)cm+d−1

1

. . . . . . . . . . . . . . .

0 1 2cm . . . (m + d)cm+d−1m

,

E and A being defined as in V and V0.

Theorem 1. For fixed cj the eigenvalues of M for the 2nd order VIDE in the casem and d + 2 and eigenvalues of M for the 1st order VIDE in the case m and d + 1coincide and have the same algebraic and geometric multiplicities, except µ = 1whose algebraic multiplicity for the 2nd order VIDE is greater by one than for the1st order VIDE.

Proof. The eigenvalue problem for M is equivalent to the generalized eigenvalueproblem for V0 and V , i.e. (M − µI)v = 0 for v 6= 0 if and only if (V0 − µV )v = 0

86 M. Tarang

and (M−µI)w = v takes place if and only if (V0−µV )w = V v. Denote ν = 1−µ.Then for the 2nd order VIDE with the parameters m and d + 2 we have

V0 − µV = (4.2)

=

ν 1 1 1 . . . . . . 1

0 ν 2 3 . . . . . . m + d + 1

0 0 ν

(

3

2

)

. . . . . .

(

m + d + 2

2

)

. . . . . . . . . . . . . . . . . . . . .

0 . . . . . . . . . ν . . .

(

m + d + 2

d + 2

)

0 0 ν · 2 ν · 6c1 . . . . . . ν(m + d + 2)(m + d + 1)cm+d1

. . . . . . . . . . . . . . . . . . . . .

0 0 ν · 2 ν · 6cm . . . . . . ν(m + d + 2)(m + d + 1)cm+dm

.

Let Ii,p be the diagonal matrix obtained from unit matrix, replacing the i-th diagonalelement by the number p. Thus, the products Ii,pA and AIi,p mean the multiplica-tion of i-th row and i-th column of A, respectively, by p. The direct calculation and

observation that

(

p

q

)

q

p

=

(

p − 1

q − 1

)

, allows us to get from (4.2)

Id+3,d+2 . . . I3,2(V0 − µV )I3,1/2 . . . Id+m+3,1/(m+d+2) =

(

ν q

0 U0 − µU

)

,

or

S(V0 − µV )S−1 = R

(

ν q

0 U0 − µU

)

, (4.3)

where S = Id+3,d+2 . . . I3,2, R = Id+m+3,d+m+2 . . . Id+4,d+3,

q =

(

1,

1

2, . . . ,

1

m + d + 2

)

.

Now (4.3) gives

det(V0 − µV ) = (d + 3) . . . (d + m + 2)ν det(U0 − µU) ,

which permits to get the assertion about algebraic multiplicities of eigenvalues of M

and M . Similarly to [5] we can prove that the eigenvalue µ = 1 of M and M hasgeometric multiplicity m and similarly to [6] that geometric multiplicities of µ 6= 1as an eigenvalue of M and M coincide.


Proposition 3. If M has an eigenvalue outside of the closed unit disk, then the splinecollocation method is not stable with possible exponential growth of approximatesolution.

Proof. The structure of the proof is similar to that of Prop. 5 in [5] and we will dealonly with main moments. Consider an eigenvalue µ of M +W such that |µ| ≥ 1+ δ

with some fixed δ > 0 for any sufficiently small h. For α1 6= 0, being an eigenvectorof M + W , we have here

(V − βhV1 − αh2V2 − λh

3V3)α1 = h

2g0, (4.4)

where g0 = (a10, . . . , a1d, f(t11), . . . , f(t1m)), a1j =h

jy(j)(0)

j!, j = 0, . . . , d.

Because of

y′′(0) = αy(0) + βy

′(0) + f(0) , (4.5)

y(j)(0) = αy

(j−2)(0) + βy(j−1)(0) + λy

(j−3)(0) + f(j−2)(0), j = 3, . . . , d,

the vector α1 determines via (4.4) and (4.5) the values

f(j)(0), j = 0, . . . , d − 1, f(t11), . . . , f(t1m).

We take f on [0, h] as the polynomial interpolating the values

f(j)(0), j = 0, . . . , d − 2, f(t1j), j = 1, . . . , m,

and f(j)(h) = 0, j = 0, . . . , d0 (if cm = 1, then f

(j)(h) = 0, j = 1, . . . , d0). In thecase of the method of additional knots let f be on [0, h] the interpolating polynomialby the data f(0), f(t0j), j = 0, . . . , d − 1, f(t1j), j = 1, . . . , m, and f

(j)(h) = 0,(here d0 = 0 and if cm = 1, then f(t1m) = f(h) is already given and we dropthe requirement f(h) = 0). In both cases we ask f to be on [nh, (n + 1)h], n ≥ 1,the interpolating polynomial by the values f

(j)(nh) = 0 and f(j)((n + 1)h) = 0,

j = 0, . . . , d0 (if cm = 1, then for j = 1, . . . , d0), and also f(tn+1,j) = f(t1j), j =1, . . . , m. This guarantees that f ∈ C

d0 [0, T ] and rn = 0, n ≥ 1. The interpolant f

can be represented on [tn, tn+1] by the formula:

f(t) = f(tn + τh) =κ∑

i=0

( ki∑

l=0

hsl

pilf(sl)(ξl)

) i−1∏

r=0

(τ − br) (4.6)

with br being cj or c0j , ξl being tnj or tj , 0 ≤ sl ≤ d1, ki ≤ i, constants pil

depending on cj and c0j . In the case of initial conditions κ = m + d + d0 − 1(κ = m + d + d0 − 2 if cm = 1) and in the case of additional knots κ = m + d + 1(κ = m + d, if cm = 1) on the interval [0, h] and κ = m + 2d0 + 1 (κ = m + 2d0 ifcm = 1) on the interval [nh, (n + 1)h], n ≥ 1.

Replacing h by h/k, k = 1, 2, . . . , and keeping ||α1|| = h2/k

2, we have ||g0||∞bounded which means that f(t1j), j = 1, . . . , m, and h

jy(j)(0)/k

j , j = 0, . . . , d,or h

jf

(j)(0)/kj , j = 0, . . . , d0, are bounded too in the process k → ∞. Thus (4.6)

gives

88 M. Tarang

||f ||Cd0 [0,T ] ≤ c kd0

. (4.7)

On the other hand, ||αn+1|| ≥ (1 + δ)n||α1|| yields

||αkN || ≥h

k

(1 + δ)kN−1 (4.8)

and (4.1) cannot be satisfied. The inequalities (4.7) and (4.8) mean also the expo-nential growth of approximate solution if we keep the norm of f bounded in C

d0 .

5. Examples and numerical tests

Let us consider some special cases of d and m.Case d = 1, m ≥ 1 being arbitrary. We have

V =

(

1 0 . . . 0

C

)

, V0 =

(

1 1 . . . 1

C

)

and det(V0 −µV ) = (1−µ)m+2 det C0 where C0 is obtained from C omitting firsttwo columns. This means that the method is always stable.

Case d = 2, m = 1 (cubic splines). The equation det(V0 − µV ) = 0 besidesµ = 1 has the solution µ = 1−1/c1. The method is stable if and only if 1/2 ≤ c1 ≤

1.

Case d = 2, m = 2. Now the equation det(V0 − µV ) = 0 has the root µ = 1with geometric multiplicity 2. From the solution

µ(c1, c2) = 1 −c1 + c2 + 1

c1c2

it follows that the method is stable if and only if c1 + c2 ≥ 1. In numerical tests weexplored the 2nd order integro-differential equation

y′′(t) = y(t) + y

′(t) +∫ t

0y(s)ds − sin(t) − cos(t) − e

t,

y(0) = 1, y′(0) = 1, t ∈ [0, 1] .

This equation has the exact solution y(t) = (sin t+cos t+ et)/2. As an approximate

value of ||u||∞

we actually calculated max1≤n≤N max0≤k≤10

∣

∣

un(tn−1 +kh/10)∣

∣.The results are presented in Tables 1–4. From these numerical examples we canobserve a good conformity of theoretical results presented in the proceeding sectionsand numerical results given in this section.


Table 1. Case d = 1, m = 1 (quadratic splines).

N 4 16 64 256 4096

c1 = 0.5 2.053593 2.050242 2.050041 2.050028 2.050028c1 = 1.0 2.112955 2.060136 2.052332 2.050591 2.050062

Table 2. Case d = 1, m = 2 (Hermite cubic splines).

N 4 16 64 256 4096

c1 = 0.4

c2 = 0.62.047625 2.049880 2.050018 2.050027 2.050028

c1 = 0.7

c2 = 1.02.042264 2.049630 2.050004 2.050026 2.050028

Table 3. Case d = 2, m = 1 (cubic splines).

N 4 16 64 256 512

c1 = 0.4 2.047252 2.049817 61.720406 1.60 · 10331.20 · 1077

c1 = 0.5 2.047590 2.049861 2.050017 2.050027 2.050027

c1 = 1.0 2.055555 2.050364 2.050048 2.050028 2.050028

Table 4. Case d = 2, m = 2.

N 4 64 256 512

c1 = 0.2

c2 = 0.52.049254 7.65 · 1026

2.89 · 101391.21 · 10292

c1 = 0.3

c2 = 0.72.049935 2.050027 2.050028 2.050028

c1 = 0.5

c2 = 1.02.050015 2.050028 2.050028 2.050028

References

[1] H. Brunner and J. D. Lambert. Stability of numerical methods for Volterra integro-differential equation. Computing, 12, 75 – 89, 1974.

[2] H. Brunner and P.J. van der Houwen. The Numerical Solution of Volterra Equations.North-Holland, Amsterdam, 1986. CWI Monographs, Vol. 3

[3] H.-S. Hung. The numerical solution of differential and integral equations by spline func-tions, 1970.

[4] P. Oja. Stability of collocation by smooth splines for Volterra integral equations. Pro-ceedings of the Internat. Conf. on Mathemathical Methods for Curves and Curfaces (Oslo2000), 405–412, 2001.

[5] P. Oja. Stability of the spline collocation method for Volterra integral equations. J.Integral Equations Appl., 13, 141–155, 2001.

[6] P. Oja and M. Tarang. Stability of piecewise polynomial collocation for Volterra integro-differential equations. Mathematical Modelling and Analysis, 6, 310–320, 2001.

90 M. Tarang

[7] P. Oja and M. Tarang. Stability of the spline collocation method for Volterra integro-differential equations. Commentationes Universitatis Tartuensis de Mathematica, 6, 37–49, 2002.

Antros eiles Volterra integro-diferencialiniu lygciu splainu kolokacijos metodo stabilu-mas

M. Tarang

Straipsnyje nagrinejamas antros eiles Volteros integro-diferencialiniu lygciu splainu kolokaci-jos metodo skaitinis stabilumas ir nustatytas ryšys tarp šios teorijos ir atitinkamos pirmos eilesVolterra integro-diferencialiniu lygciu teorijos. Pateikti keleto skaitiniu eksperimentu rezul-tatai.

!"#$%'&)($+*,-./-0102.435

91–98c© 2004 Technika ISSN 1392-6292

THE FIRST BASIC BOUNDARY VALUE PROBLEMOF RIEMANN’S TYPE FOR BIANALYTICALFUNCTIONS IN A PLANE WITH SLOTS

I. B. BOLOTIN and K .M. RASULOV

Smolensk State Pedagogical University

Przevalskogo 4, 214000 Smolensk, Russia

E-mail: 6789#:<;=>1=@?6A9B>6C<?EDGF1H%IJ61KCLM1HBCK6ND5CPO4=>@Q9#C<RSDF@H

Received October 13, 2003; revised February 5, 2004

Abstract. The paper is devoted to the investigation of one of the basic boundary value prob-lems of Riemann’s type for bianalytical functions. In the course of work there was made outa constructive method for solution of the problem given in a plane with slots. There was alsofound out that the solution of the problem under consideration consists of consequent solu-tions of two Riemann’s boundary value problems for analytical functions in a plane with slots.Besides, a picture of solvability of the problem is being searched and its Noether is identified.

Key words: bianalytical function, boundary value problem, plane with slots, index

1. Statement of the problem

Let us exclude from a full complex plane segments of a real axis Lm = [am, bm](m = 1, 2..., n), and let D be the remaining domain. The boundary L of the domain

D is understood as the thrown out segments (slits). Thus, L =

n⋃

m=1

Lm and D =

C\L. Further we shall use terms and definitions accepted in [3].

As it is known, a function F (z) = U(x, y) + iV (x, y) is called bianalytical inthe domain D if it belongs to the class C

2(D) and satisfies in D the condition

∂2F (z)

∂z2

= 0,

where∂

∂z

=1

2

(

∂

∂x

+ i

∂

∂x

)

92 I. B. Bolotin, K .M. Rasulov

is the Cauchy–Riemann operator in D.

A bianalytical function F (z), which is defined in domain D, be-longs to the class A2(D)

⋂

I(2)(L) if it can be prolonged to the contour L together

with the partial derivatives∂

α+βF (z)

∂zα∂z

β( α = 0, 1; β = 0, 1) so that boundary values

of this function and all specified derivatives satisfy the Holder condition everywhere,may be except for points am, bm (m = 1, . . . , n), where the reversion at infinity ofthe integrable order is possible, when α + β < 2.

Let us consider the following boundary value problem.

It is required to find all bianalytical functions F (z), belonging to the classA2(D)

⋂

I(2)(L), vanishing on infinity, limited near the extremities of the contour

and satisfying at all internal points of L the following boundary conditions:

∂F+(t)

∂x

= G1(t)∂F

−(t)

∂x

+ g1(t), (1.1)

∂F+(t)

∂y

= G2(t)∂F

−(t)

∂y

+ ig2(t), (1.2)

where

∂F+(t)

∂x

= limz→t,Imz>0

F (z),∂F

−(t)

∂x

= limz→t,Imz<0

F (z)

∂F+(t)

∂y

= limz→t,Imz>0

F (z),∂F

−(t)

∂y

= limz→t,Imz<0

F (z)

and Gk(t), gk(t) (k = 1, 2) are given on L functions of the class H(3−k)(L), and

Gk(t) 6= 0 on L.

Here, in equality (1.2), the factor i at g2(t) is put for convenience of notation. Theformulated problem is called the first basic boundary value problem of Riemann’stype for bianalytical functions in the plane with slots or, shortly, the problem R1,2.The appropriate homogeneous problem (g1(t) ≡ g2(t) ≡ 0) will be denoted asproblem R

01,2.

Let us notice, that the problem R1,2 represents one of the basic boundary valueproblems of Riemann’s type for bianalytical functions. It was formulated in the well-known monograph of Gakhov (see, for example, [1], p. 316). In case of arbitrarysmooth closed loops the considered problem was explicitly investigated in the workof Rasulov (see, for example, [3]).

In the present work for the first time we investigate a more general problem R1,2.

2. On the solution of the problem R1,2

It is known (see, for example, [1, 3]), that any vanishing on infinity bianalyticalfunction F (z) with a line of saltuses L can be represented as:

The first basic boundary value problem of the Riemann type 93

F (z) = ϕ0(z) + zϕ1(z), (2.1)

where ϕk(z) are analytical functions in domain D (analytical components of thebianalytical function), for which the following conditions are fulfilled:

Πϕk,∞ ≥ 1 + k, k = 0, 1;

here Πϕk,∞ means the order of the function ϕk(z) at the point z = ∞.Let us search for the solution of the problem R1,2 given by:

F (z) = f0(z) + (z − z)f1(z). (2.2)

Then the functions fk(z) (k = 0, 1) will be connected with analytical compo-nents of the required bianalytical function F (z) by formulas:

ϕ0(z) = f0(z) − zf1(z), ϕ1(z) = f1(z) . (2.3)

As known (see, for example, [1], p. 301)

∂

∂x

=∂

∂z

+∂

∂z

,

∂

∂y

= i

(

∂

∂z

−

∂

∂z

)

,

then taking into account (2.2) and the fact that the equality t = t is fulfilled on L, theboundary conditions (1.1) and (1.2) can be written as:

Φ+

0 (t) = G1(t)Φ−

0 (t) + g1(t), (2.4)

f+

1 (t) = G2(t)f−

1 (t) + Q2(t), (2.5)

where Φ0(z) =df0(z)

dz

, Q2(t) =1

2

(

Φ+

0 (t) − G2(t)Φ−

0 (t) − g2(t))

.

The equalities (2.4) and (2.5) represent boundary conditions of usual Riemann’sproblems for analytical functions in a plane with slots (see, for example, [1] or [2]).Thus, as a matter of fact, the solution of the initial problem R1,2 is reduced to se-quential solution of two auxiliary problems of Riemann (2.4) and (2.5) in classes ofanalytical functions in domain D with a line of saltuses L. But as in the problem R1,2

we search for the solutions, limited close to extremities of the contour and vanishingon infinity. There arises the necessity to define classes of analytical solutions of aux-iliary problems (2.4) and (2.5). Therefore, at first we shall find out in what classes itis necessary to search for solutions of boundary value problems (2.4) and (2.5).

From equalities (2.3) we can see, that the functions Φ0(z) and f1(z) on infinityshould have zero not below than the second order. Let us study the behaviour of thefunction F (z) near the extremities of the contour L. Let c be any of extremities, thenc = c. We have the following series of inequalities:

|F (z)| = |f0(z) + (z − z)f1(z)| ≤ |f0(z)| + |f1(z)||z − z|

= |f0(z)| + |f1(z)||z − c + c − z|

≤ |f0(z)| + |f1(z)||z − c| + |f1(z)||z − c|

= |f0(z)| + 2|f1(z)||z − c| . (2.6)


For function F (z) to be limited close to the extremities of the contour L, it isnecessary and sufficient, that the functions Φ0(z) and f1(z) satisfy the estimates:

|Φ0(z)| ≤const

|z − c|α0

, |f1(z)| ≤const

|z − c|α1

, 0 ≤ α0, α1 < 1. (2.7)

Really, if functions Φ0(z) and f1(z) satisfy the condition (2.7), then the requiredbianalytical function F (z) will be limited in a neighbourhood of c.

And if the function F (z) of the class A2(D)⋂

I(2)(L) is limited close to the

extremity c, then the functions Φ0(z) and f1(z) satisfy (2.7) (otherwise all solutionsof the problem R1,2 will not be found).

Thus it is required to find a solution of boundary value problems (2.4) and (2.5),belonging to a class of functions, having on infinity a zero of the second order andinfinity of the integrable order on extremities of the contour L.

Let us solve the boundary value Riemann problem (2.4). Let

G1(am) = ra,1meiθ1m

, 0 ≤ θ1m < 2π,

G1(bm) = rb,1mei(θ1m+∆θ1m)

, ∆θ1m = [argG1(t)]Lm,

then, following Gakhov (see, for example, [1], p. 448), we define integer numbersκ1m by the following formulas:

κ1m =

[

θ1m + ∆θ1m

2π

]

+ 1 .

The index of the problem (2.4) is represented by the following formula

κ1 =

n∑

m=1

κ1m .

Hence, if κ1 ≥ 2, a general solution of problem (2.4) is set by the formula (see,for example, [1, 2]):

Φ0(z) = X1(z)

1

2πi

∫

L

g1(τ)

X+

1 (τ)

dτ

τ − z

+ Pκ1−2(z)

, (2.8)

where X1(z) is a canonical function of the problem (2.4), Pκ1−2(z) is the polyno-mial of a degree not higher than (κ1 − 2) with arbitrary complex coefficients.

In case when κ1 ≤ 1, the solution of problem (2.4) also will be expressed byformula (2.8) with only one modification, that Pκ1−2(z) ≡ 0, and if κ1 ≤ 0 then|κ1| + 1 conditions of solvability should be satisfied:

∫

L

g1(τ)

X+

1 (τ)τ

k−1dτ = 0, k = 1, . . . , |κ1| + 1. (2.9)

Using the function Φ0(z) =df0(z)

dz

, after integration we obtain


f0(z) =

∫

γ

Φ0(ζ)dζ,

where γ is an arbitrary smooth curve, completely laying in the domain D and con-necting the infinite point with arbitrary point z of the domain D.

Let us take in expression (2.8) z → t ∈ L. Then, using the formulas ofSokhotzky-Plemelj (see, for example, [1, 2]), we get

Φ+

0 (t) = X+

1 (t)

1

2πi

∫

L

g1(τ)

X+1 (τ)

dτ

τ − t

+ Pκ1−2(t)

+g1(t)

2, (2.10)

Φ−

0 (t) = X−

1 (t)

1

2πi

∫

L

g1(τ)

X+

1 (τ)

dτ

τ − t

+ Pκ1−2(t)

−

g1(t)

2G1(t), (2.11)

here in equality (2.11) we have taken into account, thatX

+1 (t)

X−

1 (t)= G1(t).

Remark 1. Functions Φ±

0 (t), given by formulas (2.10) and (2.11) satisfy the Holdercondition everywhere on L, may be except the extremities, where they may have asingularity of the integrable order.

Now we will solve the boundary value Riemann problem (2.5). Let

G2(am) = ra,2meiθ2m

, 0 ≤ θ2m < 2π,

G2(bm) = rb,2mei(θ2m+∆θ2m)

, ∆θ2m = [argG2(t)]Lm.

Then following Gakhov (see, for example., [1], p. 448) we define integer numbers:

κ2m =

[

θ2m + ∆θ2m

2π

]

+ 1. (2.12)

The index of the problem (2.5) is represented by the following formula

κ2 =

n∑

m=1

κ2m. (2.13)

As it is known (see [1, 2]), if κ2 ≥ 2, a general solution of problem (2.5) isrepresented by formula:

f1(z) = X2(z)

1

2πi

∫

L

Q2(τ)

X+

2 (τ)

dτ

τ − z

+ Pκ2−2(z)

, (2.14)

where X2(z) is a canonical function of problem (2.5), Pκ2−2(z) is a polynomial ofa degree not higher than (κ2 − 2) with arbitrary complex coefficients.

If κ2 ≤ 1, the solution of problem (2.5) can be expressed by formula (2.14) withonly one modification, that Pκ2−2(z) ≡ 0, and if κ2 ≤ 0, then |κ2|+1 conditions ofa solvability should be satisfied:

96 I. B. Bolotin, K .M. Rasulov∫

L

Q2(τ)

X+2 (τ)

τk−1

dτ = 0, k = 1, ..., |κ2| + 1 . (2.15)

Remark 2. Generally speaking, absolute term Q2(t) of problem (2.5) satisfies theHolder condition everywhere on L, except for, possibly, the extremities of the con-tour, where it may have the integrable singularity. But, as we search for the solutionof the boundary value problem (2.5) in the class of functions, having infinity of theintegrable order on the extremities of the contour L, the density of the integral inthe formula (2.14) has the singularity on the extremities of the contour, which is nothigher than the integrable one.

After finding functions f0(z) and f1(z), we use formulas (2.3) and restore an-alytical components of the required bianalytical function. Then using formula (2.1)we restore the bianalytical function F (z) itself. Thus, the following basic result isvalid.

Theorem 1. Let

L =

n⋃

m=1

[am, bm], D = C\L .

Then the solution of the problem R1,2 is reduced to a sequential solution of the twoscalar Riemann problems (2.4) and (2.5) in classes of analytical functions in theplane with slots, having a zero of the second order on infinity and infinity of theintegrable order on the extremities of the contour L. The problem R1,2 is solvableif and only if problems (2.4) and (2.5) are simultaneously solvable in the specifiedclasses of functions.

3. Investigation of a solvability of the problem R1,2

As the solution of the problem R1,2 is reduced to the sequential solution of theboundary value Riemann problems (2.4) and (2.5), solvability conditions of the prob-lem R1,2 can be developed from the solvability conditions of boundary value prob-lems (2.4) and (2.5).

The number κ = κ1 + κ2 is called the index of the problem R1,2, and numbersκ1 and κ2 are its private indexes. For a full investigation of the solvability of theproblem R1,2 it is necessary to consider 9 cases.

Case 1. Let κ1 ≥ 2, κ2 ≥ 2.In this case boundary value problems (2.4) and (2.5) are solvable and have κ1−1

and κ2 − 1 linearly independent solutions, respectively. Thus, in this case problemR1,2 is solvable and by virtue of formulas (2.1) and (2.3) its general solution linearlydepends on κ1 + κ2 − 2 arbitrary complex constants.

Case 2. Let κ1 ≥ 2, κ2 = 1.In this case boundary value problem (2.4) is solvable and has κ1 − 1 linearly

independent solutions, and boundary value problem (2.5) also is solvable and has a


unique solution. Hence, in this case problem R1,2 is solvable and its general solutionlinearly depends on κ1 − 1 arbitrary complex constants.

Case 3. Let κ1 ≥ 2, κ2 ≤ 0.In this case boundary value problem (2.4) is solvable and has κ1 − 1 linearly

independent solutions, and the boundary value problem (2.5) has a unique solutionif |κ2| + 1 conditions of the solvability (2.15) are satisfied.

Remark 3. We shall notice, that some conditions (2.15) can be satisfied at the expenseof a choice of values of arbitrary constants, which are included in expression Q2(t).

Thus, in this case problem R1,2 is solvable if |κ2|+1 conditions of the solvability(2.15) are satisfied and its general solution linearly depends on l arbitrary complexconstants, where 0 ≤ l ≤ κ1 − 1.

Case 4. Let κ1 = 1, κ2 ≥ 2.In this case boundary value problem (2.4) is solvable and has a unique solution,

and the boundary value problem (2.5) is solvable and has κ2 − 1 linearly indepen-dent solutions. Thus, in this case problem R1,2 is solvable and its common solutionlinearly depends on κ2 − 1 arbitrary complex constants.

Case 5. Let κ1 = 1, κ2 = 1.In this case boundary value problems (2.4) and (2.5) are solvable and each of

them has a unique solution. Hence, by virtue of the Theorem 1, the problem R1,2 issolvable and has a unique solution.

Case 6. Let κ1 = 1, κ2 ≤ 0.In this case boundary value problem (2.4) is solvable and has a unique solution,

and the boundary value problem (2.5) has a unique solution if |κ2|+ 1 conditions ofthe solvability (2.15) are satisfied. Thus, problem R1,2 is solvable and has a uniquesolution.

Case 7. Let κ1 ≤ 0, κ2 ≥ 2.In this case boundary value problem (2.4) has a unique solution if |κ1|+1 condi-

tions of the solvability (2.9) are satisfied. Boundary value problem (2.5) is solvableand has κ2 − 1 linearly independent solutions. Thus, in this case problem R1,2 issolvable if |κ1| + 1 conditions of the solvability (2.9) are satisfied and its generalsolution linearly depends on κ2 − 1 arbitrary complex constants.

Case 8. Let κ1 ≤ 0, κ2 = 1.In this case boundary value problem (2.4) has a unique solution if |κ1|+1 condi-

tions of the solvability (2.9) are satisfied. Boundary value problem (2.5) is solvableand has a unique solution. Thus, problem R1,2 is solvable if |κ1| + 1 conditions ofthe solvability (2.9) are satisfied and it has a unique solution.

Case 9. Let κ1 ≤ 0, κ2 ≤ 0.In this case boundary value problem (2.4) has a unique solution if |κ1| + 1 con-

ditions of the solvability (2.9) are satisfied, and boundary value problem (2.5) has aunique solution if |κ2|+1 conditions of the solvability (2.15) are satisfied. Hence, inthis case problem R1,2 is solvable if |κ1|+1 conditions (2.9) and |κ2|+1 conditions(2.15) are satisfied and it has a unique solution.


From the analysis given above we obtain the following statement.

Theorem 2. For any values of the index κ = κ1 +κ2 the number p of conditions of asolvability of inhomogeneous problem R1,2 and the number l of linearly independentsolutions corresponding to homogeneous problem R

01,2 are finite, i.e. the problem

R1,2 satisfies the Noether conditions.

References

[1] F.D. Gakhov. Boundary value problems. Nauka, Moscow, 1977. (in Russian)[2] N.I. Muskhelishvili. Singular integral equations. Nauka, Moscow, 1968. (in Russian)[3] K.M. Rasulov. Boundary value problems for polyanalytical functions and some of their

applications. Smolensk State Pedagogical University, Smolensk, 1998. (in Russian)

Apie pirmojo pagrindinio kraštinio Rimano tipo uždavinio bianalizinems funkcijomsplokštumoje su itrukiais sprendima

I.B. Bolotin, K.M. Rasulov

Šiame darbe tyrinejamas uždavinys, kai ieškoma dalimis bianaliziniu funkciju, nykstanciu be-galybeje, apribotu greta konturo trukio tašku ir šiame konture tenkinanciu dvi kraštines saly-gas. Parodoma, kad sprendžiamas uždavinys suvedamas i sprendima dvieju Rimano uždaviniuanalizinems funkcijoms.

2 S. Chakravarty, P. K. Mandal, A. Mandal

Abstract. The present paper deals with a theoretical investigation of blood flow in an arterialsegment in the presence of stenosis. The streaming blood is treated to be composed of two dif-ferent layers – the central core and the plasma. The former is considered to be non-Newtonianliquid characterised by the Power law model, while the latter is chosen to be Newtonian. Theartery is simulated as an elastic (moving wall) cylindrical tube. The unsteady flow mecha-nism of the present study is subjected to a pulsatile pressure gradient arising from the normalfunctioning of the heart. The time-variant geometry of the stenosis has been accounted for inorder to improve resemblance to the real situation. The unsteady flow mechanism, subjectedto pulsatile pressure gradient, has been solved using finite difference scheme by exploitingthe physically realistic prescribed conditions. An extensive quantitative analysis has been per-formed through numerical computations of the flow velocity, the flux, the resistive impedancesand the wall shear stresses, together with their dependence with the time, the input pressuregradient and the severity of the stenosis, presented graphically at the end of the paper in or-der to illustrate the applicability of the model under consideration. Special emphasis has beenmade to compare the existing results with the present ones and found to have a good agree-ment.Key words: Moving wall, non-Newtonian liquids, stenosis, power-law

!"#$%'&)($+*,-./-0102.435

115–126c© 2004 Technika ISSN 1392-6292

PARALLEL NUMERICAL MODELLING OFSHORT LASER PULSE COMPRESSION

R. CIEGIS1, A. DEMENT’EV2 and G. ŠILKO1

1 Vilnius Gediminas Technical University

Sauletekio al. 11, LT-10223 Vilnius, Lithuania

E-mail: 67819;:=<>?1@A<CB?%DEF8F7G<>?1@A<CB?2 Institute of Physics

Goštauto 12, LT-2600 Vilnius, Lithuania

E-mail: H1IJ:4JLK?8NM?B+<O:QPPR<SB?

Received March 2, 2004; revised April 28, 2004

Abstract. In this paper we investigate parallel numerical algorithms for solution of the tran-sient stimulated scattering processes. A new symmetrical splitting scheme is proposed and aparallel version is given. The efficiency of the parallel algorithm is investigated for two cases.The first one describes a case when the computation region is constant during the whole timeof computations. The second one describes the initial phase of the process, when the com-putational region increases linearly in time. In order to distribute more evenly jobs betweenprocessors a dynamical the grid redistribution algorithm is is used. We also give a proof ofone result about optimal static grid distribution in the case of linearly increased problem com-plexity. The results of computations are presented. They were obtained on different parallelcomputers and clusters of workstations.

Key words: finite-difference schemes, symmetrical splitting method, parallel algorithms, gridredistribution, nonlinear optics

1. Introduction

Scientific investigations in various fields and different technological applications re-quire laser systems satisfying a number of requirements. They should be able togenerate short pulses in various spectral ranges with tunable pulse duration, the gen-erated pulses must be easily synchronized with external events and have a good sta-bility and low jitter [18]. The progress of solid-state lasers with nonlinear-opticalphase conjugation and pulse compression will support these objectives for extensionof the fields of laser applications [5, 13, 22]. Using different schemes for the stim-ulated Brillouin scattering (SBS) compressor, it is possible to achieve pulses with

116 R. Ciegis, A. Dement’ev, G. Šilko

durations shorter than 100 ps [4, 5, 6, 13, 22]. Note that a consecutive cascade com-pression provides a set of precisely synchronized (with the accuracy of several ps)pulses of different wavelengths and pulse durations [13, 19].

In the SBS compression experiments, Fourier-transform-limited laser pulses withGaussian transversal intensity distribution are commonly used. Therefore, it is ofgreat practical interest to investigate the statistics (energetic, temporal and spectral)of Stokes pulses for different pump pulse parameters and optical schemes of SBScompressor and amplifier. For optimization of SBS compression schemes detailedthree-dimensional numerical simulations of the transient backward SBS process fordifferent focusing geometries of phase–modulated pump pulses with different pulseshape and durations starting from spontaneous scattering level are needed.

Performing this type of calculations is also interesting from purely scientific pointof view, because a number of works appeared lately [1, 2, 3, 20, 21], presenting someresults that contradict the results of our earlier works [7, 8, 16, 17]. It should bepointed out that calculations of the transient stimulated scattering processes, espe-cially in the three-dimensional case, require lengthy computation times. Therefore,without application of the parallel algorithms, the investigation of statistical pecu-liarities of these processes is practically impossible.

We start from the algorithm developed in [9, 12] for the solution of nonlinearproblems with strongly focused beams. It is based on the expansion of the fields ofthe interacting beams into the series of eigenfunctions of the Laguerre-Gauss type.Such algorithm can be modified into a parallel algorithm easily enough by using dataparallelization paradigm [10], see also a paper by Elisseev [15], where a parallel codeis obtained using HPF.

This work presents a novel more efficient splitting type scheme and its parallelversion. This scheme was tested for a three-dimensional problem of transient stim-ulated scattering of focused beams. Thus, our goal is to investigate the efficiencyof the parallel version of the proposed symmetrical splitting scheme. We study theinfluence of different nonlinear effects on the accuracy of the obtained numerical so-lution in order to determine a region of application of the proposed parallel numericalalgorithm.

The rest of the paper is organized as follows. In Section 2 we describe a mathe-matical model of SBS. In Section 3 the new finite difference scheme is presented. Aparallel version of this algorithm is presented in Section 4. This section also describesthe load balancing problem for the front moving case. Section 5 contains analysis ofthe obtained numerical results and the last Section 6 draws some conclusions.


A schema of the numerically modelled SBS compressor is presented in in Fig. 1. At-tempts are still being made to investigate this phenomena analytically, but inclusionof the material non-stationarity can be taken into account only numerically.

The presented work gives a new parallel numerical algorithm for solving thesystem of equations, which describes the nonlinear interaction of laser, Stokes and

Parallel Numerical Modelling of Laser Pulse 117

SBS cell Lens

Polarizer

Diaphragm

Laser pulse

Stokes pulse

Output pulse

QWP

Initial Stokes pulse

SBS cell Lens

Polarizer

Diaphragm

Laser pulse

Stokes pulse

Output pulse

QWP


SBS cell Lens

Polarizer

Diaphragm

Laser pulse

Stokes pulse

Output pulse

QWP


SBS cell Lens

Polarizer

Diaphragm

Laser pulse

Stokes pulse

Output pulse

QWP


SBS cell Lens

Polarizer

Diaphragm

Laser pulse

Stokes pulse

Output pulse

QWP


SBS cell Lens

Polarizer

Diaphragm

Laser pulse

Stokes pulse

Output pulse

QWP


Figure 1. A schema of the SBS compressor.

sound waves. In 0 ≤ z ≤ L, 0 ≤ r ≤ R, 0 ≤ t ≤ T the following system ofequations is given [7, 16]:

∂u

∂t

+∂u

∂z

− iµLAu = iγLσv + iωL

(

|u|2 + 2|v|2)

u ,

∂v

∂t

−∂v

∂z

− iµSAv = iγSσ∗

u + iωS

(

2|u|2 + |v|2)

v ,

iγ0

(

∂2σ

∂t2

+ γ1

∂σ

∂t

)

+∂σ

∂t

+ γ2σ = iγσuv∗ + γf ,

(2.1)

here A = 1

r∂∂r

(r ∂∂r

) is the transverse Laplacian, u, v and σ are slowly varying com-plex amplitudes of laser, the Stokes and the sound waves, respectively. γL, γS , γσ

are coupling constants, γf is the thermal noise parameter, γ0 and γ2 are hypersoundwave parameters (related to the period and relaxation time). System (2.1) is supple-mented with the boundary and initial conditions:

u(z = 0, r, t) = g0(r, t), v(z = L, r, t) = g1(r, t), 0 ≤ r ≤ R ,

r

∂u

∂r

∣

∣

r=0= 0, r

∂v

∂r

∣

∣

r=0= 0, 0 ≤ z ≤ L, 0 ≤ t ≤ T ,

u(z, R, t) = 0, v(z, R, t) = 0 ,

u(z, r, 0) = 0, v(z, r, 0) = 0 .

Since for focused beams the diameter of the beam waist in the cuvette is by fac-tor 20-50 and more times smaller than the diameter of input beam, adaptive gridsare required in order to solve the problem accurately and efficiently. Such adaptivestrategies have proved to reduce significantly the computational cost for obtaining anumerical solution.

The Schrödinger equation looks very similar to the heat equation, but there aregreat differences. We note mainly, that the Schrödinger equation does not have a reg-ularizing effect of the contractivity as for the heat equation. Therefore the develop-ment of adaptive algorithms for solving the Schrödinger type nonlinear equations re-quires new techniques. The most popular adaptive schemes are based on applicationof transformations, which use the properties of solutions of the linear Schrödingerequation. Extensive numerical comparison of different mesh adaptation techniques


and transformations is presented in [9]. A new adaptive transformation is proposedin [23], it was also used in [17] for numerical solution of the SBS problem using asplitting finite-difference scheme.

We note that general mesh adaptation techniques are also applied for the Schrö-dinger problem (see, e.g. [14]). The basis of such procedure is a posteriori errorestimate that has to be derived for the obtained discrete solution. Then local errorestimators indicate the regions of the computational domain where we have to refinethe mesh in order to improve the accuracy of the approximation.

In this work we propose a new symmetrical splitting scheme, in which the diffrac-tion subproblem is solved using the expansion into the Laguerre-Gaussian modes.The accuracy of such an approximation is investigated in [12].

Modelling of nonlinear effects of SBS pulse compression requires to resolve theevolution of all dynamically significant scales of motion. This can be done only viavariable mesh densities. The obtained discrete problems often are too large to fit intoserial computers, either because of computational demands or memory limitations,or both. Parallel computers and algorithms are the most effective solutions of thisproblem.

3. Finite Difference Scheme

This section contains a brief description of the numerical algorithm. We introducethe following discrete meshes:

ωz =

zn : zn = nτ, n = 0, 1, . . . , N, τ =L

N

,

ωt =

tk : t

k = kτ, k = 0, 1, . . . , K

,

ωr(z) =

rj : rj = jh, j = 0, 1, . . . , J, h =R(z)

J

,

here ωr depends adaptively on the coordinate z and generally this mesh is alsononuniform in r. We use the following notation for discrete functions:

Uknj = U(zn, rj , t

k), (zn, rj , tk) ∈ ωz × ωr(z) × ωt .

We approximate problem (2.1) by the following splitting algorithm.

Symmetrical Splitting Algorithm

for k = 0, K

/* First diffraction step */for n = 0, N

Uk+ 1

3

n+ 1

2

(r) =P∑

p=0

ckp(zn) Wp(zn+ 1

2

, r), r ∈ ωr(zn+ 1

2

)

Vk+ 1

3

n+ 1

2

(r) =P∑

p=0

dkp(zn+1) W p(zn+ 1

2

, r)


end for

/* Nonlinear Interaction */for n = 0, N

Uk+ 2

3

n+ 1

2

(r) = fu(Uk+ 1

3

n+ 1

2

, Vk+ 1

3

n+ 1

2

, σkn+ 1

2

, δkn+ 1

2

), r ∈ ωr(zn+ 1

2

)

Vk+ 2

3

n+ 1

2

(r) = fv(Uk+ 1

3

n+ 1

2

, Vk+ 1

3

n+ 1

2

, σkn+ 1

2

, δkn+ 1

2

)

σk+1

n+ 1

2

(r) = fσ(Uk+ 1

3

n+ 1

2

, Vk+ 1

3

n+ 1

2

, σkn+ 1

2

, δkn+ 1

2

)

δk+1

n+ 1

2

(r) = fδ(Uk+ 1

3

n+ 1

2

, Vk+ 1

3

n+ 1

2

, σkn+ 1

2

, δkn+ 1

2

)

end for

/* Second diffraction step */for n = 0, N

ck+1p (zn+1) =

(

Uk+ 2

3

n+ 1

2

, Wp(zn+ 1

2

))

,

dk+1p (zn) =

(

Vk+ 2

3

n+ 1

2

, W p(zn+ 1

2

))

,

Uk+1

n+1(r) =P∑

p=0

ck+1p (zn+1) Wp(zn+1, r), r ∈ ωr(zn+1)

Vk+1n (r) =

P∑

p=0

dk+1p (zn) W p(zn, r), r ∈ ωr(zn)

end forend for

The analysis of the algorithm complexity

1 step: Diffraction.

In this step the equations of wave propagation and diffraction are solved in the ele-ment [zn, zn+ 1

2

]. The total complexity of this step is O(JP ) operations.

2 step: Nonlinear Interaction.

Using predictor–corrector numerical integration scheme we solve a system of ODEsdescribing the nonlinear interaction of laser, the Stokes and the sound beams. Thecomplexity of this step is O(J) operations.

3 step: Diffraction.

We complete the diffraction step, i.e. the laser and Stokes waves again propagate inthe second part of the element. The complexity of this step is O(JP ) operations.

Thus the total complexity of the splitting numerical algorithm is O(NJP ) oper-ations.


Due to symmetrical splitting algorithm the accuracy of the finite differencescheme is O(h2 + (τ4 + P

−α)/h), where α depends on the smoothness of the exactsolution (see, [12]). We note that the error can accumulate linearly with respect toN , thus in order to reduce the global error we need to change also τ and P . Thisphenomena is investigated in detail in [12].

4. Parallel Algorithm

We use an one–dimensional mesh of virtual p processors. The finite difference gridωz is partitioned in p blocks, which are distributed among processors (see Fig. 2).

p - 1 p - 2 … 1 0 p - 1 p - 2 … 1 0

Figure 2. 1D block data distribution.

It follows from the proposed numerical algorithm, that each processor needs toexchange information (i.e., coefficients cj and dj) corresponding to the boundarypoints of its local domain. It is important to note, that communication is done onlybetween neighbour processors. After the communication step each processor has allrequired information. Now all computations can be performed efficiently in paralleland the results are also stored locally on each processor.

4.1. The analysis of algorithm complexity

First we estimate the parallel execution time of the proposed algorithm during real-ization of one time step. The discrete problem size can be expressed as follows

W = JN(C1P + C2) .

The communication step on most network architectures can be done in time Tcomm =α + βP . Thus the parallel execution time Tp on p processors is given by

Tp =JN

p

(C1P + C2) + α + βP .

The additional cost of parallel algorithm can be expressed as follows

T0(W, p) = pTp − W .

We solve the equation

W = eT0(W, p), e =Ep

1− Ep

, Ep =W

W + T0(W, p),


where Ep is a selected efficiency of the parallel algorithm. Since p ≤ N, and J =O(N), we get that the proposed parallel algorithm is highly scalable.

The code was implemented using MPI library and performed on IBM SP4 com-puter. In Table 1 we present speed-up Sp = W

Tp

and efficiency Ep =Sp

pdata obtained

for two discrete problems of different sizes:

a) N = 201, J = 101, P = 15 (denoted by S1p in Table 1),

b) N = 301, J = 151, P = 45 (denoted by S2p in Table 1).

Table 1. The speed-up and efficiency of the parallel algorithm.

p S1p E1p S2p E2p

2 1.971 0.986 1.968 0.9844 3.934 0.984 3.904 0.9768 7.591 0.949 7.692 0.962

16 14.67 0.917 15.20 0.95032 24.94 0.780 27.89 0.872

These results fully confirm our theoretical predictions.

4.2. Front moving case

If the boundary condition for the Stokes wave is equal to zero, then during initialtransition time 0 ≤ t

k ≤ L the domain involved in computations enlarges dynami-cally

ωz(tk) =

zn : 0 ≤ zn ≤ tk

and the problem size at the k-th time step is given by kJ(C1P + C2) instead ofNJ(C1P + C2). Then the static mesh partitioning among processors using a blockdistribution scheme is not optimal. It is shown in [11] that the speed-up of the parallelalgorithm is equal to Sp ≈ p

2, even when the communication costs are not taken into

account.

As it was stated in [11] the computation costs can be reduced if we decomposethe grid not uniformly. It was proposed to divide the grid into p + 1 parts and assignthe last two subdomains to the last processor. A simple analysis proved that thisheuristic gives optimal static block distributions for p ≤ 3. In the case of p ≥ 4 theefficiency of the proposed heuristic was investigated numerically.

Now we will give a proof of this statement. In fact, we will show that the firstp − 1 processors should get equal numbers of grid points D0 if computations aredone at least till time moments t ≥ pD0. Here we have assumed that the front movesone grid point per time step.

Theorem 1. Let consider the static block data distribution, when the grid is dividedinto p + 1 parts and the last processor obtains the last two subdomains. Such distri-bution scheme is optimal among static block distributions.


Proof. The proof is based on induction. It is sufficient to consider the followinggrid distribution:

• the first (p − 2) processors obtain D0 grid points;• the (p − 1)th processor gets D1 grid points;• the pth processor gets D2 grid points.

These subproblems satisfy the following relations:

M + D2 = N, (p − 2)D0 + D1 = M . (4.1)

Next we compute the complexity of the computational problem. The solution of theproblem till T1 = D0 requires

W1 =

D0∑

j=1

j =D0(D0 + 1)

2

basic operations. The complexity of the problem for D0 + 1 ≤ t ≤ M −D1 + D0 is

W2 =

M−D1+D0∑

j=D0+1

D0 = D0(M − D1)

basic operations. The last part of the problem till t = M requires

W3 =M∑

j=M−D1+D0+1

(j − M + D1)

operations. The total number of operations is given by

W = W1 + W2 + W3 =1

2

(

D21 + D1 − 2D0D1 + 2D0M

)

.

Now we can find the optimal grid distribution among (p − 1) processors, whenthe problem is solved only till t = M . Optimality in this case means that we consideronly two free parameters, i.e. D0 and D1. By using (4.1) we get

∂S

∂D0

:= (p − 2)(

pD0 − M − 0.5)

= 0 ,

D0 =M

p

+1

2p

, D1 = M − (p − 2)D0 =2M

p

−p − 2

2p

.

Thus we confirm the result, that the (p − 1)th processor should get the two lastsubproblems.

Now let consider the situation, when we still continue computations for M +1 ≤

t ≤ M + K. Then we obtain the following estimate of the algorithm complexity:

W = W +

M+K∑

j=M+1

D1 = W + K

(

M − (p − 2)D0

)


and from the optimality equation it follows that D0 = M+Kp

+ 1

2p. It is easy to find

K such, that D1(K) = D0(K):

M + K

p

=M

p − 1⇒ K =

M

p − 1.

Thus we have proved, that for any number of processors p, the first processorsshould get subproblems of the size M

p−1and only the last processor gets a subproblem

of the size 2Mp−1

. Again, if the computations are continued for t > N , then the optimalgrid distribution converges to the static block distribution scheme.

Dynamic data redistribution

In general, if we distribute the grid using the static block distribution scheme, thenthe parallel execution time Tp on p processors is given by

Tp =N

p

J

(

N −N

2p

+1

2

)

(C1P + C2) +p − 1

p

N(α + βP ) .

The following data redistribution algorithm is analyzed theoretically in [11]:

• Initial Ns points of the mesh ωz are partitioned statically among processors usinga block distribution scheme.

• Starting from the time moment tk = Nsτ after Ks steps of the algorithm data is

redistributed among processors in order to preserve a load balancing.

The algorithm for determination of Ns, Ks is given in [11].

This algorithm introduces additional communication costs, but they are compen-sated by improved load balancing and therefore a total efficiency of the parallel algo-rithm is increased. Computational experiments are performed on IBM SP4 computerat CINECA. In Table 2 we present speed-ups Sp = W

Tp

of the parallel algorithm fordifferent values of redistribution starting point Ns and the same remaining discreteparameters N = 400, P = 200, K = 400.

Table 2. Speed-up of the parallel algorithm with data redistribution.

p Ns = 400 Ns = 200 Ns = 100 Ns = p

2 1.332 1.750 1.910 1.9664 2.270 3.292 3.707 3.8168 4.143 5.928 7.021 7.26516 7.676 11.238 12.826 13.248

As predicted by theoretical analysis the adaptive redistribution algorithm in-creases essentially the efficiency of the parallel discrete algorithm even for fixedsize problems.


5. Numerical Results

These numerical experiments were performed on VGTU cluster of 10 SMP PCs.Each PC contains two 1.4 GHz Pentium III processors. All nodes run Linux. In orderto estimate the computational power of this cluster we solved the problem with thefollowing parameters: N = 1200, J = 1200, P = 100, K = 4200 .

Table 3. Execution time, speed–up and efficiency of the parallel algorithm.

p Tp Sp Ep

1 141029 1.0 1.08 17751 7.95 0.9910 14231 9.91 0.9920 7166 19.68 0.98

In Table 3 we present execution time Tp, speed-up Sp = WTp

and efficiency Ep =Sp

p. These results fully confirm our theoretical predictions, the parallel algorithm is

highly scalable.

6. Conclusions

Parallel algorithms for solution of one important problem of nonlinear optics havebeen investigated. It has been shown that the parallel domain decomposition algo-rithm for this problem is highly scalable and it’s efficiency is near to one. Detailedmodelling of practically interesting cases of pulse compression will be published inphysical journals.

Acknowledgments

This work was possible thanks to the stay of R. Ciegis at the supercomputing centerCINECA in Bologna within the project MINOS. He gratefully acknowledges thehospitality and excellent working conditions in CINECA. In particular he thanks Dr.Giovanni Erbacci for his help.

References

[1] S. Afshaarvahid, V. Devrelis and J. Munch. Nature of intensity and phase modulationsin stimulated Brillouin scattering. Phys. Rev. A, 57(5), 3961 – 3971, 1998.

[2] S. Afshaarvahid, A. Heuer, R. Menzel and J. Munch. Temporal structure of stimulated-Brilouin-scattering reflectivity considering transversal-mode development. Phys. Rev. A,64, 2001. 043803


[3] S. Afshaarvahid and J. Munch. A transient, three-dimensional model of stimulated Bril-louin scattering. J. Nonlinear Opt. Phys. and Materials, 10(1), 1 – 27, 2001.

[4] R. Buzelis, A. Dement’ev, E. Kosenko and E. Murauskas. Short pulse generation by one-step SBS. In: XVII International conference on coherent and nonlinear optics, Minsk,Belarus, June 26 - July 1, 59, 2001.

[5] R. Buzelis, A. Dement’ev., E. Kosenko, E. Murauskas, R. Ciegis and G. Kairyte. Numer-ical analysis and experimental investigation of beam quality of SBS-compressors withmultipass Nd:YAG amplifier. Proc. SPIE, 2772, 158 – 169, 1996.

[6] R. Buzelis, A. Dement’ev and E. Murauskas. Investigation of possibilities of effectiveSBS compression of nanosecond pulse to 100 ps. Lithuanian Phys. J., 39(4), 253 – 258,1999.

[7] R.R. Buzelis, V.V. Girdauskas, A.S. Dement’ev, E.K. Kosenko and R.J. Ciegis. Space-time structure of pulses of cascade SS-compressors. Izvestya AN SSSSR, Phys. Ser.,55(2), 270 – 278, 1991.

[8] R.R. Buzelis, V.V. Girdauskas, A.S. Dement’ev, E.K. Kosenko, S.A. Norvaišas andR.J. Ciegis. Mathematical modeling and experimental investigation of the fidelity ofwave-front with smooth inhomogeneities reversal for high reflection coefficients of SBS-mirror. Izvestya AN SSSSR, Phys. Ser., 54(6), 1084 – 1091, 1990.

[9] R. Ciegis and A. Dement’ev. Numerical simulation of counteracting of focused laserbeams in nonlinear optics. In: Mathematical Modelling and Applied Mathematics. Pro-ceedings of the IMACS International Conference on Mathematical Modelling and Ap-plied Mathematics, Moscow, USSR, 18-23 June, 1990. North-Holland. Amsterdam,London, New-York, Tokyo, 99 – 108 (1992)

[10] R. Ciegis, A. Dement’ev and P. Rate. A parallel algorithm for solving one problem ofnonlinear optics. Mathematical Modelling and Analysis, 4, 58 – 69, 1999.

[11] R. Ciegis and G. Šilko. A scheme for partitioning regular graphs. Lecture notes incomputer science, 2328, 404 – 409, 2002.

[12] R. Ciegis, G. Šilko and A. Dement’ev. A tool for modeling optical beam propagation.Informatica, 13, 149 – 162, 2002.

[13] A. Dement’ev, R. Buzelis, E. Kosenko, E. Murauskas and R. Navakas. Solid-state laserswith pulse compression by transient stimulated Brillouin and Raman scattering. Proc.SPIE, 4415, 92 – 97, 2001.

[14] W. Dörfler. A time- and space adaptive algorithm for the linear time-dependentSchrödinger equation. Numer. Math., 73, 419 – 448, 1996.

[15] V.V. Elisseev. Parallelization of three-dimensional spectral laser–plasma interaction codeusing High Performance Fortran. Computers in Physics, 12(2), 173 – 180, 1998.

[16] V. Girdauskas, A.S. Dement’ev, G. Kairyte and R. Ciegis. Influence of the beam aber-rations and Kerr nonlinearity of a medium on the efficiency and pulse quality of a SBS-compressor. Lithuanian Phys. J., 37(4), 269 – 275, 1997.

[17] V. Girdauskas, O. Vrublevskaja and A. Dement’ev. Numerical treatment of short laserpulse compression in transient stimulated Brillouin scattering. Nonlinear Analysis: Mod-elling and Control, 7(1), 3 – 29, 2002.

[18] Millennium Issue. IEEE J. Selected Topics Quantum Electron. 6(6), 827 – 1489, 2000.[19] K. Kuwahara, E. Takahashi, Y. Matsumoto, S. Kato and Y. Owadano. Short-pulse gener-

ation by saturated KrF laser amplification of a steep Stokes pulse produced by two-stepSBS. J. Opt. Soc. Am. B, 17, 1943 – 1947, 2000.

[20] T.R. Moore and R. W. Boyd. Three–dimensional simulations of stimulated Brillouinscattering with focused Gaussian beams. J. Nonlinear Opt. Phys. and Materials, 5(2),387 – 408, 1996.

[21] T.R. Moore, G.L. Fischer and R.W. Boyd. Measurement of the power distribution duringstimulated Brillouin scattering with focused Gaussian beams. J. Mod. Opt., 45(4), 735 –745, 1998.


[22] Phase conjugated laser optics. John Wiley&Sons, Inc., 2004.[23] L. Schoulepnikoff and V. Mitev. Numerical method for the modelling of high-gain

single-pass cascade stimulated Raman scattering in gases. J. Opt. Soc. Am. B, 14, 62– 75, 1997.

Trumpu lazerio impulsu spudos skaiciavimo lygiagretusis skaitinis algoritmas

R. Ciegis, A. Dementjevas, G. Šilko

Nagrinejamas priverstines Brijueno sklaidos fokusuotuose pluoštuose uždavinio lygiagretusisskaitinis sprendimo algoritmas. Sukonstruota simetrinio skaidymo baigtiniu skirtumu schema,kurios tikslumas yra antrosios eiles. Lygiagretusis algoritmas gautas naudojant duomenu ly-giagretumo paradigma. Detaliai nagrinejamas dinamiškai didejancio sudetingumo uždavinys,modeliuojantis Brijueno sklaidos procesa, kai neužduodama kraštine salyga Stokso ban-gai. Irodyta hipoteze apie vieno stacionaraus blokinio duomenu paskirstymo algoritmo op-timaluma. Eksperimentiškai ištirtas dinaminis duomenu perskirstymo algoritmas, patvirtintasjo efektyvumas net ir fiksuoto dydžio uždaviniams. Darbe pateikti skaitinio eksperimento, at-likto naudojant VGTU 20 procesoriu klasteri, rezultatai. Jie patvirtino gautuosius teoriniusrezultatus, išsam us fizikiniai rezultatai bus išspausdinti kituose darbuose.

!"#$%'&)($+*,-./-0102.435

127–136c© 2004 Technika ISSN 1392-6292

COMPARATIVE ANALYSIS BY MEANS OFFINITE DIFFERENCES AND DM METHODS FORLINEARIZED PROBLEM OF GYROTRONS

T. CIRULIS, H. KALIS and O. LIETUVIETIS

Institute of Mathematics of Latvian Academy of Sciencesand University of Latvia

Akademijas laukums 1, Rıga LV–1524, Latvia

E-mail: 67819#:71;<:1=?>@BADCE:GF%HJI=:7B;<:1=>@BADCE:F+H'K1L=B8;B:B<:1=>@ADCE:FReceived October 14, 2003; revised January 15, 2004

Abstract. The problem of Schrödinger equation with complex boundary conditions formodelling a motion of electrons in gyrotrons is considered. Numerical results obtained byusing Fourier, Finite Differences (FD) and Degenerate Matrices (DM) methods are comparedin the simplest case. For DM methods they are analysed also in more general cases, when FDcan not be applied because of fast oscillations of the solution.

Key words: gyrotron, Schrödinger equation, complex boundary conditions, Fourier method,finite differences, degenerate matrices method

1. Formulation of the problem

When modelling the motion of electrons in gyrotrons, it is necessary to solve thefollowing initial-boundary value problem:

i

∂u

∂t

=∂

2u

∂x2

+ δ(x)u, x ∈ (0, L), t > 0,

u|t=0 = u0(x), (initial values),

u|x=0 = 0,∂u∂x

∣

∣

x=L= −iγu

∣

∣

x=L, (boundary conditions),

(1.1)

where u = u(t, x) is an unknown complex function, δ(x) and u0(x) are given func-tions, γ > 0 is a given constant, i =

√

−1.

Quasi-stationary solutions in the case δ(x) = δ0 = const were considered in[1, 5]. They are given by:

u(t, x) = g(x) exp(

it(α − δ0))

, (1.2)

128 T. Cırulis, H. Kalis, O. Lietuvietis

where α = α1 + iα2 are complex eigenvalues and g(x) are the corresponding eigen-functions. Denoting α = λ

2, z = λL we obtain the equation

z cos z = −iγL sin z (1.3)

for finding z. Each complex root z = z1 + iz2 of (1.3) generates a solution in theform (1.2) with

Reα = α1 =z21 − z

22

L2

, Imα = α2 =2z1z2

L2

, g(x) = sin(

zx

L

)

. (1.4)

Using the argument principle for analytical complex functions it is possible to provethat all roots of equation (1.3), except z = 0, are disposed only in domains

Rez > 0, Imz > 0 , Rez < 0, Imz < 0 ,

symmetrically with respect to z = 0 [7]. Therefore the roots can be calculated onlyin the first quadrant of z-plane. We use the software package "Maple" for |z| notlarge and asymptotic formulas in the case of large |z|. These asymptotic formulascan be found by the method of indeterminate coefficients. The method is often usedto solve equations with entire analytic functions. For roots zn, n → ∞, they aregiven as follows:

zn = sn +a1

sn

+a2

s3n

+a3

s5n

+a4

s7n

+ O

( 1

s9n

)

, sn = π

(

n +1

2

)

,

a1 = iτ, a2 = τ2

(

1 +iτ

3

)

, a3 =4τ

4

3+

iτ3(τ2

− 10)

5,

a4 = τ4

(47

36τ

2− 3

)

+iτ

5

120(17τ

2− 440), τ = γL.

Let λk =z(k)

L

, where (z(k)), k = 1, 2, . . . , be the sequence of roots of (1.3)

numbered according to the increase of Re(z(k)). Then the solution of problem (1.1)can be expended into the following convergent infinite Fourier series:

u(t, x) =

∞

∑

k=1

ckgk(x) exp(

i(λ2k − δ0)t

)

, (1.5)

gk(x) = sin(λkx), ck =

L∫

0

gk(s)u0(s) ds

L∫

0

g2k(s) ds

.

Moreover, numerical results obtained by (1.5) are accurate enough only for verylarge number of the summands in (1.5), especially if t is not large. Therefore, inthe following sections we will consider other methods for numerical solving of theproblem (1.1).

Finite differences and DM methods for problem of gyrotrons 129

2. Application of the finite difference method

We consider the uniform space grid in the x-direction with the interior grid points

xj = jh, j = 1, N − 1, x0 = 0, xN = L,

and the time grid with the grid points tn = nτ, n = 1, 2, . . . , here h, τ are thesteps of the grids. We replace the continuous solution u = u(t, x) of the problem(1.1) by the discrete grid function y = y(t, x) with values y(tn, xj) = y

nj .

An approximation of the problem (1.1) is based on the following finite differencescheme:

i

yn+1

j − ynj

τ

= σΛyn+1

j + (1 − σ)Λynj , j = 1, N − 1,

yn+1

0 = 0, lyn+1

N = −iγyn+1

N ,

y0j = u

0(xj), j = 0, N,

(2.1)

where σ ∈ [0, 1] is a parameter of the scheme. Λyj denotes a central difference

expression of the second order approximation for the derivative∂

2u

∂x2

at the grid pointxj :

Λyj =yj+1 − 2yj + yj−1

h2

+ δ0yj ,

lyN denotes a difference expression of the first order approximation for the derivative∂u

∂x

in one of the following forms:

a) using the two points difference for the first order approximation

lyN =yN − yN−1

h

; (2.2)

b) using the three points difference for the second order approximation

lyN =1.5yN − 2yN−1 + 0.5yN−2

h

. (2.3)

The approximation order (AO) of the difference equations (2.1) with respect totime and space coordinates is equal to two if σ = 0.5, and equal to one with respectto time if σ 6= 0.5. Approximation order of boundary conditions is 1 (formula (2.2))and 2 (formula (2.3)).

The discrete quasi-stationary solution has the following form:

ynj = gj exp(inτα) , (2.4)

where the discrete eigenfunctions g(k)

j are given by [1]:

g(k)

j = sin(q(k)xj), k = 1, 2, . . . , N − 1.

Here q(k) are roots of one of the transcendent equations:


1) Approximation (2.2)

sin(qL) = C1 sin(

q(L − h))

, (2.5)

2) Approximation (2.3)

sin (qL) = C2

(

2 sin(

q(L − h))

− 0.5 sin(

q(L − 2h))

)

, (2.6)

C1 =1

1 + iγh

, C2 =1

1.5 + iγh

.

The parameter α in (2.4) can be obtained from expressions:

α = ln1 − τα

∗/(i + στα

∗)

iτ

, α∗

=2(

1 − cos(qh))

h2

− δ0 .

The approximate values α(k)∗

are complex, i.e., α(k)∗

= Ak + iBk:

Ak =2(

1 − cos(akh)ch(bkh))

h2

− δ0, Bk =2 sin(akh)sh(bkh)

h2

,

where q(k) = ak + ibk. Using the argument principle we can prove that all complex

roots of (2.5) or (2.6) for Req > 0 or ak > 0 are in the first quadrant of the complexq-plane, and Bk ≥ 0.

The stability conditions for finite–difference schemes (2.1) – (2.2), and (2.1)–(2.3) follow from [6]:

σ ≥ 0.5, Bk ≥ 0 .

The solution of finite-difference scheme (2.1)–(2.2) can be obtained also in thediscrete form of Fourier series:

ynj = h

N−1∑

k=1

ck sin(q(k)xj) exp(iα(k)

nτ), (2.7)

where

ck =

N−1∑

s=1

1

dk

sin(q(k)xs)u0(xs), dk =

1

2

(

L −

h sin(q(k)L) cos

(

q(k)(L − h)

)

sin(q(k)h)

)

.

In this case the discrete eigenfunctions (2.4) are orthogonal:

(g(k), g

(m)) = h

N−1∑

j=1

g(k)

j g(m)

j = 0, k 6= m.


3. Application of the Degenerate Matrix method

In this section we will consider another scheme for solving problem (1.1), whichcan be used also in the case when δ(x) is not equal to a constant. The DM method[3, 4] is based on using such differentiation matrices A for derivatives with respectto x which ensure that the approximation of the unknown function u is nonsaturated.Choosing the partitions xk, k = 0, 1, . . . , n + 1, on the interval (0, L) we form the(n + 2) × (n + 2) matrix A with elements

amk =

w′(xm)

(xm − xk)w′(xk), if m 6= k,

w′′(xk)

2w′(xk)

, if m = k ,

(3.1)

where w(x) =n+1∏

j=0

(x − xj).

Remark 1. We usually choose the nodes sk as zeroes of classical orthogonal polyno-mials on the standard interval [−1, 1]. Then the mapping xk = L

2

(

sk + 1)

gives therequired partition of (0, L), and the nonsaturatedness of approximations is ensured.

Contracting equation (1.1) on the nodes xk, k = 0, 1, . . . , n + 1, and applying thematrix A we obtain the following equation

i

d~u

dt

= (A2 + D)~u, (3.2)

where ~u and D are the column-vector and the diagonal matrix, respectively, withcorresponding components u(xk) and diagonal elements δ(xk), k = 0, 1, . . . , n+1.Matrix equation (3.2) holds only at the interior points x1, x2, . . . , xn of the interval(0, L). Therefore, we must take off its first and last rows. Then we exclude valuesu(x0) and u(xn+1) in the first and last columns from the remaining equations usingthe discretized boundary conditions

u(x0) = 0,

n+1∑

k=0

an+1,ku(xk) = −iγu(xn+1),

which are obtained after discretization of boundary conditions (1.1). This yields thesystem of n homogeneous linear differential equations

d~u

dt

= S~u (3.3)

with initial values obtained by discretization of initial value in (1.1). Here (3.3) hasconstant coefficient matrix S with elements sm,k, m, k = 1, 2, . . . , n:

smk = −i

(

a(2)

mk + µa(2)

m,n+1an+1,k + dmk

)

, (3.4)

where a(2)

mk are the elements of the matrix A2, an+1,k are the elements of the last

row of A,


µ =iγ − an+1,n+1

γ2 + a

2n+1,n+1

, dmk =

0, if m 6= k,

δ(xk), if m = k .

(3.5)

Finally, we solve system (3.3) exactly finding eigenvalues and eigenvectors forthe matrix S and using discretized initial values.

Comments

10. Now we prove that for finding eigenvalues λ and eigenfunctions u, which are

defined by the following problem:

u′′ + δ(x)u = −λu,

u(0) = 0, u′(L) = −iγu(L),

(3.6)

it is appropriate to use matrices for derivatives with nodes sk = 1, 2, . . . , N as zeroesof one of the classical orthogonal Jacobi polynomials P

(α,β)

N (s) supplemented withs0 = −1 and sN+1 = 1.

Let L be the eigenvalue xk = L2

(

sk + 1)

, k = 0, 1, . . . , N + 1. Let λ be theeigenvalue and u(x) is the eigenfunction of (3.6) corresponding to given λ . Wedenote v(s) = u

(

L2(s+1)

)

and consider the following Fourier series for s ∈ [−1, 1]:

v(s) =∞

∑

k=0

ckP(α,β)

k (s), ck =1

‖ P(α,β)

k ‖2

1∫

−1

ρ(s)v(s)P(α,β)

k (s) ds , (3.7)

where ρ(s) = (1 − s)α(1 + s)β is the weight function. Series (3.7) converges veryrapidly because v(s) is analytical. Therefore,

v(s) =N

∑

k=0

ckP(α,β)

k (s) + RN (s), RN (s) = O

(

exp(−νN))

, (3.8)

when N → +∞ with some constant ν > 0. Replacing the integral in (3.7) by theGauss-Lobatto quadrature formula, substituting it into (3.8) and using the classicalChristoffel-Darboux formula for Jacobi polynomials it is possible to prove that [2]

v(s) =

N+1∑

k=0

pN+2(s)v(sk)

(s − sk)p′N+2(sk)

+ RN (s), RN (s) = O

(

exp(−νN))

, (3.9)

where pN+2(s) = (1−s2)P

(α,β)

N (s), and the remainder in (3.9) has the same asymp-totic estimate as in (3.8). This follows from the asymptotic behaviour of a differencebetween the integral in (3.7) and its quadrature formula in the case when v(s) isan analytical function on [−1, 1]. Therefore, (3.9) gives also the nonsaturated ap-proximation of v(s). Returning to the variable x and using matrices for derivativesaccording to the scheme given at the beginning of the section, we obtain the equation

(

SN +(

λ + O(exp(−νN)))

EN

)

~uN = 0 . (3.10)


All elements of the matrix SN can be computed by analogy with (3.4) and (3.5).Therefore, the eigensystem for the matrix SN is close to the one for the matrix

SN + O

(

exp(−νN))

EN , if N → +∞ .

20. The computing scheme described above can be used not only for (1.1), but

also for solving different linear problems of heat or wave equations.

30. A possibility to compute the matrix S in (3.3) efficiently and to calculate

its eigenvalues gives very simple criterion of the stability of the DM-methods forlinear problems. They are stable if all n eigenvalues of S have negative real parts.For example, the DM-method for (1.1) with uniformly distributed nodes is unstableeven for n ≥ 5. The choice of nodes as zeroes of classical orthogonal polynomialsleads to stable schemes for very large n. For example, the method with nodes aszeroes of Chebyshev polynomials of the second kind is stable at least for n ≤ 240,for Legendre polynomials – at least for n ≤ 120.

4. Numerical results

Table 1. Values of |u(x, t)| for x = L = 15, γ = 2 and δ(x) = 0.

t F FD DM

0.1 0.05087 0.05082 0.050580.2 0.06217 0.06224 0.062090.3 0.06871 0.06862 0.068680.4 0.07289 0,07286 0.072770.5 0.07574 0.07593 0.075720.6 0.07806 0.07827 0.077990.7 0.08041 0.08013 0.080190.8 0.08103 0.08164 0.081050.9 0.08214 0.08291 0.082261.0 0.08310 0.08398 0.0830310 0.09468 0.09526 0.0946620 0.08996 0.08994 0.0899630 0.10289 0.10299 0.1028840 0.09560 0.09561 0.0956050 0.09127 0.09127 0.09127

In Table 1 we present absolute values of the numerical solutions on the boundaryx = L of problem (1.1) with δ(x) = 0 and simple initial conditions u0(x) = sin πx

L.

We set parameters γ = 2 and L = 15. Such choice is very interesting for appli-cations. In column (F ) of Table 1 we give results obtained by the classical Fouriermethod (1.5) with N = 2000. In column (FD) we present the results obtained by Fi-nite difference method (2.7) with space step h = 0.02 and time step τ = 0.01, and inthe last column (DM) the results obtained by the DM-method with 240 grid points


Table 2. Values of |u(x, t)| obtained by the DM method with the Chebyshevand Legendre nodes for x = L = 15, γ = 2 and δ(x) = tanh(7x− 3.5L).

n = 60 n = 120 n = 240

t Cheb.1 Leg. Cheb.2 Cheb.1 Leg. Cheb.2 Cheb.1 Cheb.2

0.1 0.0515 0.0498 0.0481 0.0511 0.0515 0.0514 0.0505 0.05060.2 0.0622 0.0632 0.0636 0.0628 0.0624 0.0613 0.0621 0.06210.3 0.0689 0.0698 0.0699 0.0685 0.0687 0.0683 0.0682 0.06850.4 0.0703 0.0710 0.0716 0.0729 0.0715 0.0718 0.0719 0.07180.5 0.0737 0.0734 0,0729 0.0745 0.0755 0.0746 0.0748 0.07450.6 0.0769 0.0768 0.0761 0.0762 0.0760 0.0751 0.0761 0.07590.7 0.0866 0.0858 0.0861 0.0847 0.0850 0.0848 0.0844 0.08430.8 0.0799 0.0801 0.0793 0.0789 0.0790 0.0787 0.0788 0.07880.9 0.0743 0.0757 0.0753 0.0761 0.0767 0.0760 0.0761 0.07601.0 0.0963 0.0949 0.0938 0.0939 0.0942 0.0942 0.0945 0.094410 0.1417 0.1417 0.1416 0.1422 0.1422 0.1422 0.1422 0.142220 0.1586 0.1586 0.1585 0.1577 0.1577 0.1577 0.1577 0.157730 0.1561 0.1561 0.1561 0.1549 0.1549 0.1549 0.1549 0.154940 0.1350 0.1350 0.1350 0.1339 0.1339 0.1339 0.1338 0.133850 0.0671 0.0671 0.0671 0.0684 0.0684 0.0684 0.0685 0.0685

distributed as zeroes of Chebyshev polynomials of the second kind are given. Nu-merical results were obtained by means of mathematical systems Maple-5 (Fourierseries and Finite differences) and Mathematica 2.2 (the DM-method).

As we see, Finite differences and the DM-methods give the same order of accu-racy, but such accuracy was achieved by the DM-method using approximately threetimes less grid points than by Finites differences. Moreover, the DM-method wasvery fast in calculations. It is due to the usage of the eigensystem of matrix S in(3.3) which allows us to solve (3.3) exactly. Therefore, we can easily calculate thenumerical solution of (1.1) for any t without using discrete time integration. So, allresults in column (DM) were obtained in 47 seconds on a computer with Celeron400 processor and 256 mb RAM.

In Table 2 we present numerical results obtained by the DM-method with

δ(x) = tanh(7x − 3.5L), γ = 2, L = 15

and for different sets of n grid points ( n = 60, 120, 240 ) distributed as zeroes ofChebyshev polynomials of the first and second kind and as zeroes of Legendre poly-nomials. We note that for δ(x) 6= const, the method of Finite differences (2.7) hasfailed.

It is seen from Table 2, that all distributions of grid points with fixed n givethe same accuracy which rises by increasing n. The corresponding graph of |u(L, t|

for n = 240 and nodes distributed as zeroes of Chebyshev polynomial of the secondkind is shown in Fig.1, and the graphs of |u(x, t)| at various time moments are shownin Fig.2.


10 20 30 40 50t

0.1

0.2

0.3

0.4

0.5

Figure 1. The graph of |u(L, t)| for δ(x) = tanh(7x− 3.5L).

2 4 6 8 10 12 14x

0.2

0.4

0.6

0.8

1

1.2

t=0

t=1

t=10

t=50

Figure 2. Evolution in time of |u(x, t)| for δ(x) = tanh(7x− 3.5L).

Thus we conclude that the DM method can be used efficiently to solve the prob-lem (1.1) also for δ(x) 6= const.

References

[1] M. Airila, O. Dumbrajs, A. Reinfelds and U. Strautins. Nonstationary oscillations ingyrotrons. Phys. Plasmas, 8(10), 4608 – 4612, 2001.

[2] P. Borwein and T. Erdelyi. Polynomials and Polynomial Inequalities. Springer - Verlag,1995.

[3] T. Cırulis. Nonsaturated approximation by means of Lagrange interpolation. In: Proceed-ings of the Latvian Academy of Sciences. Section B, volume 52, 234 – 244, 1998.

[4] T. Cırulis and O. Lietuvietis. Application of DM method for problems with partial differ-ential equations. Mathematical Modelling and Analysis, 7(2), 191 – 200, 2002.

[5] O. Dumbrajs, H. Kalis and A. Reinfelds. Numerical solution of single mode gyrotronequation. Mathematical Modelling and Analysis, 9(1), 25 –38, 2004.

[6] A.A. Samarskii. Theory of difference schemes. Nauka, Moscow, 1989. (In Russian)[7] B. Shabat. An Introduction to Complex Analysis, 2nd edition, volume 2. Moscow, 1976.

(In Russian)


Baigtiniu skirtumu ir DM metodo lyginamoji analize linearizuotam girotrono uždaviniui

T. Cirulis, H. Kalis, O. Lietuvietis

Nagrinejamas kraštinis uždavinys Šredingerio lygciai, aprašantis elektronu judejima girotrone.Darbe lyginami ir analizuojami Furje, baigtiniu skirtumu ir degeneruotu matricu (DM) meto-dais gauti skaitiniai rezultatai. Aptartas metodo taikymas greitu osciliaciju atveju.

!"#$%'&)($+*,-./-0102.435

137–148c© 2004 Technika ISSN 1392-6292

ON AN EXACT DESCRIPTION OF THESCHOTTKY GROUPS OF SYMMETRIES1

M.V. DUBATOVSKAYA and S.V. ROGOSIN

Belarusian State University

4, Fr. Skaryna ave, Minsk

E-mail: 6879:8;<8=>@?:1A94>7CBD9E

Received October 9, 2003; revised April 12, 2004

Abstract. Exact description of the Schottky groups of symmetries is given for certain spe-cial configurations of multiply connected circular domains. It is used in the representation ofthe solution of the Schwarz problem which is applied at the study of effective properties ofcomposite materials.

Key words: symmetries, Schottky group, Schwarz boundary value problem, composite ma-terials

1. INTRODUCTION

Description of special subgroups of the group of conformal mappings on the com-plex plane is a classical problem. The first essential results in this direction wereobtained at the end of XIX - beginning of XX centuries by F. Schottky, H. Schwarz,H. Poincaré, L. Fuchs, E. Picard, A. Hurwitz, F. Klein and others. These resultsformed the base of the theory of the groups of conformal mappings, the theory ofautomorphic functions and Poincaré θ-series (see [3]). Further results and modernview on this subject are presented in the monographs [4, 9].

The theory of conformal mapping constitutes a very suitable tool for the study oftwo-dimensional problems of mathematical physics. Recently an interest has arisedto describe special groups of conformal mappings, which belong to so called class ofthe Schottky groups. It should be noted, for instance, the application of such groupsto the constructive representation of conformal mappings of multiply connected do-mains onto canonical domains (see, e.g. [1, 2, 8]), to the analytic solution of theSchwarz boundary value problem

ReF (t) = f(t), t ∈ L, (1.1)

1 The work is partially supported by the Belarusian Fund for Fundamental Scientific Re-search

138 M. Dubatovskaya, S. Rogosin

or more general Riemann-Hilbert boundary value problem

Reλ(t)F (t) = f(t), t ∈ L, (1.2)

for a multiply connected circular domain (see [7]). They are also used for the studyof certain special cases of R-linear boundary value problem

φ+(t) = a(t)φ−(t) + b(t)φ+(t) + c(t), t ∈ L, (1.3)

for multiply connected domains. Investigations of all these problems based on themethod of functional equations are described in the recent monograph [8]. Theseresults constitute the ground for a new constructive approach to the study of boundaryvalue problems of mathematical physics.

Therefore the study of general properties of Schottky groups generated by sym-metries with respect to a number of circles becomes an actual problem. It is alsoimportant to give an exact representation of elements of such groups for certain spe-cial cases since these groups are used in formulas for the solutions of the problems(1.1), (1.2), (1.3). They can be applied for solving problems of filtration, compositematerials, porous media (see the description of these applications, e.g., in [6, 8]).

2. Notation and general results

2.1. Groups of symmetries

We consider representation of elements of so called Schottky groups (or Schottky-type groups). The formal definition of the Schottky group is as follows:

LetQ1, Q2, . . . , Qn andQ′

1, Q′

2, . . . , Q′

n be two families of circles.Let the circles of each family be situated outside each other (i.e. the circles of eachfamily are nonoverlapping). Let Tj be a (fractional)-linear transform with respect toz or z which map Qj onto Q′

j and interior of the circles of each Qj onto exteriorof Q′

j . This transform generates the group Kj . The composition of these groupsKj , j = 1, 2, . . . , n, is called Schottky group generated by the mappings Tj , j =1, 2, . . . , n.

Intensive study of such groups was done in twenties and thirties of the XX cen-tury. It appeared that in most cases the Schottky group has quite complicated struc-ture and not too many general properties can be formulated.

We consider here a special case of the Schottky groups when the generators Tj

are simply the symmetries with respect to the circles Qj (and thus Qj = Q′

j). Weobtain an exact description of the elements of the corresponding Schottky groups fora number of particular cases. Let

Qj = Qj(aj , rj) := z ∈ C : |z − aj | = rj , j = 1, 2, . . . , n,

be a family of circles on the complex plane (with centers aj and with radii rj ). Letus introduce the following mappings (see [8, p. 125]):

Schottky groups of symmetries 139

z∗

(jm,jm−1,...,j1) :=(

z∗

(jm−1,...,j1)

)

∗

(jm)

, (2.1)

where z∗(j)

=r2

j

(z−aj)+ aj is the symmetry with respect to the circle Qj . Hence,

z∗

(jm,jm−1,...,j1)is the composition of the successive symmetries with respect to the

circles Qj1 , . . .Qjm−1, Qjm

. In the sequence jm, jm−1, . . . , j1 no two neighbouringnumbers are equal. The numberm is called the level of the mapping z∗

(jm,jm−1,...,j1).

When m is even, these mappings are Möbius transformations. If m is odd then wehave anti-Möbius transformations, i.e. Möbius transformations with respect to z.Thus these mappings can be written in the form

φk(z) = (αkz + βk) / (γkz + δk) , m is even,

φk(z) = (αkz + βk) / (γkz + δk) , m is odd,

where αkδk − βkγk = 1. Here

φ0(z) = z, φ1(z) = z∗

(1), φ2(z) = z∗

(2), . . . , φm(z) = z∗

(m), (2.2)

φm+k(z) = z∗

((k+1),1), k ≥ 1 .

The functions φk generate a Schottky group K (see [3]). In the following we denoteby G the subgroup of K consisting of the mappings φk of an even order, and by F thesubgroup of K consisting of the mappings φk of an odd order. The following generalproperties of the successive symmetries are well-known (see [3]).

Properties of successive symmetries

1. Each (fractional)-linear transform w = αz+βγz+δ

of the complex plane C is equiva-lent to an even number of symmetries with respect to certain circles.

2. (Fractional)-linear transforms of the complex plane C (which are not identity,w ≡ z) have at most two fixed points. Thus it is true for the elements of thesubgroup G.

3. Any transform w = αz+β

γz+δcan be represented in one of the following forms

w − ζ1

w − ζ2

= K

z − ζ1

z − ζ2

, or w − ζ1 = K(z − ζ1),

where ζ1, ζ2 are the fixed points of the transform, and the coefficient K is acomplex number satisfying the relation

√

K +1

√

K

= α+ δ .

4. If K = Aeiθ (A > 0, θ ∈ [0, 2π)) then the transform w

a) is called hyperbolic, if K = A;a) is called elliptic, if K = e

iθ;a) is called loxodromic, if K = Ae

iθ, θ 6= 0.


5. Let the transformw = αz+β

γz+δhas two fixed points ζ1, ζ2 and is represented in the

formw − ζ1

w − ζ2

= K

z − ζ1

z − ζ2

.

Then the m-th iteration of this transform, i.e. the transform

w(m) := w w . . . w

︸︷︷︸

m− times

has the same fixed points and is represented in the form

w(m)

− ζ1

w(m)

− ζ2

= Km z − ζ1

z − ζ2

. (2.3)

It should be mentioned that the above properties hold only for the symmetries ofeven level. As for the symmetries of odd level the situation is more complicated.

6. The set of fixed points of the transform of an odd level

w =αz + β

γz + δ

can be either the whole complex plane C, or a circle, or two points, or a point, oreven an empty set. Really, fixed points (z = x+ iy) have to satisfy the followingsystem of real equations

a(x2 + y2) + bx+ cy + d = 0,

b1x+ c1y + d1 = 0 .(2.4)

Then the property 6 follows immediately. It is not difficult to see that all possi-bilities for the fixed set are achieved by certain transformations of an odd level.

2.2. Schwarz operator and groups of symmetries

Our interest to obtain an exact description of elements of Schottky groups of sym-metries is motivated by the application of such groups at the study of composite ma-terials. Thus, the properties of two-dimensional composite materials with cylindricinclusions are described in terms of the solutions of certain boundary value prob-lems for harmonic functions in a multiply connected circular domain. Such modelsare described in the monograph [8]. These solutions are represented in term of certainSchottky groups of symmetries.

To clarify this situation let us give such a formula for one of the most simpleboundary value problems, which describes the composite materials, namely, for theSchwarz boundary value problem (1.1). Let us consider mutually disjoint discs

Dj := D(aj , rj) = z ∈ C : |z − aj | < rj, j = 1, 2, . . . , n

on the complex plane C. LetD := C\

n⋃

j=1

Dj (see Fig. 1). We choose the orientation

of the boundary Q :=n⋃

j=1

Qj = ∂Dj in such a way, that the domain containing


·

·

··

z0

·&%'$

&%'$&%

'$

&%'$

D2(a2, r2)

Dn(an, rn)D

D1(a1, r1)

Figure 1. A multiply connected domain.

∞ is on the left side. We give here the formulation of the Schwarz problem in thisspecified type of domains in order to be precise at the representation of the solutions.In fact, the Schwarz problem can be posed for any Jordan domain.

The Schwarz problem for the domainD is to find a function F , analytic inD andcontinuous in cl D, such that its boundary values satisfy the relation

ReF (t) = f(t), t ∈ Q = ∂ D,

ImF (z0) = 0,(2.5)

where f is a given function on Q, z0 is a given point in D. The operator T, whichassigns to each pair (f, z0) the solution of the Schwarz problem (2.5), is called theSchwarz operator of the domain D.

In the case of the Hölder-continuous function f and the multiply connected do-main D being of the above described type the Schwarz operator is delivered by theformula [8, p. 135]

(Tf) (z) =1

2πi

n∑

j=1

∫

Qj

f(ζ)

∑

φj∈G,j 6=0

[

1

ζ − φj(z0)−

1

ζ − φj(z)

]

(2.6)

+

(

rj

ζ − aj

)2∑

φj∈F

[

1

(ζ − φj(z))−

1

(ζ − φj(z0))

]

−

1

ζ − z

dζ

+n∑

j=1

∫

Qj

f(ζ)∂A

∂ν

(ζ) dζ +n∑

m=1

Am [log(z − am) + ψm(z)] + iς,

where

Am =

n∑

j=1

∫

Qj

f(ζ)∂αj

∂ν

(ζ) dζ, j = 1, 2, . . . , n,

αj is a harmonic measure of the domain Dj , the functionsA(z), ψm(z) are uniquelydefined by certain additional relations (see [8]), ν is an external normal vector tothe corresponding circle, ς is an arbitrary real constant. This formula represents theSchwarz operator in any compact subset of the domain D.


3. Representation of elements of the Schottky groups ofsymmetries

In this Section we give a number of results concerning the representation of elementsof some special Schottky groups of symmetries. We start with the most simple caseof symmetries with respect to two circles.

3.1. Symmetries with respect to two circles

LetDj := D(aj , rj) = z ∈ C : |z − aj | < rj , j = 1, 2

be two nonoverlapping discs on the complex plane C (i.e. |a1 − a2| ≥ r1 + r2) (seeFig. 2). Then the transform w = z

∗

(1,2)can be delivered by the formula

·

·&%'$

D1(a1, r1)

D2(a2, r2)

Figure 2. Symmetries with respect to two discs.

w =Az +B

Cz +D

,

where

A = a2(a1 − a2) + r22 , B = a1a2(a1 − a2) + r

21a2 − r

22a1,

C = a1 − a2, D = a1(a2 − a1) + r21 .

(3.1)

This transform has two fixed points ζ1, ζ2 and satisfies the relation

w − ζ1

w − ζ2

= K

z − ζ1

z − ζ2

,

where

ζ1 =M +N

r1r2

, ζ2 =M −N

r1r2

, K =L−N

L+N

, (3.2)

M = r22 − r

21 +

(

a1 − a2

)

(a1 + a2) ,

N =(

(r22 − r21)

2 + |a1 − a2|4− 2|a1 − a2|

2(r21 + r22))1/2

,

L = r21 + r

22 − |a1 − a2|

2.


In the same notation the transform consisting of 2m-symmetries

w(m) := z

∗

(1,2,1,2,...,1,2)=

(

(

. . .

(

z∗

(1,2)

)

∗

(1,2). . .

)

∗

(1,2)

)

∗

(1,2)︸︷︷︸

m − times

satisfies the relationw

(m)− ζ1

w(m)

− ζ2

= Km z − ζ1

z − ζ2

.

3.2. Symmetries with respect to three circles

LetDj := D(aj , rj) = z ∈ C : |z − aj | < r, j = 1, 2, 3

be three nonoverlapping discs of equal radii r on the complex plane C (i.e. |ak −

aj | ≥ 2r, k 6= j) (see Fig.3).

·

·

·

&%'$

&%'$

&%'$

D2(a2, r)

D3(a3, r)

D1(a1, r)

Figure 3. Symmetries with respect to three discs.

Then the transform w = z∗

(1,2,3)can be delivered by the formula

w =Az +B

Cz +D

,

where

A = r2(a1 − a2 + a3) + a3(a2 − a3)(a1 − a2), (3.3)

B = r4− r

2 [a3a1 + a1(a1 − a2) − a3(a2 − a3)] − a1a3(a1 − a2)(a2 − a3),

C = r2 + (a1 − a2)(a2 − a3),

D = r2(a2 − a3 − a1) − a1(a1 − a2)(a2 − a3).

The formulae for the transforms w = z∗

(1,3,2), z

∗

(2,1,3), . . . , z

∗

(3,2,1)can be obtained

from (3.3) by interchanging of indexes.

It follows from [5] that the effective characteristics of the composites possessextreme values in the case of percolation, i.e. when the discs Dj touch each others


6

-0

x

y

·

·

·

&%'$&%

'$

&%'$

D1(−a, r)

D3(a, r)

D2(0, r)

6

-0

x

y

·

·

·&%'$&%'$

&%'$

a) b)

Figure 4. Arrangement of discs: a) three discs in line, b) three discs in line along imaginaryaxes.

and constitute a chain-type set. In the case of an external field in the direction of thereal axes the extreme configuration for three discs is the following: Dj are situatedin a line along the real or imaginary axes.

In order to present exact formulae for the corresponding transforms we considerfirstly the situation when the discs lay along certain line (see Fig.4a). Namely, let

D1 := z ∈ C : |z+a| < r,D2 := z ∈ C : |z| < r,D3 := z ∈ C : |z−a| < r,

where a ∈ C, r > 0, |a| = 2r. In this case the composition of the successivesymmetries z∗

(1,2,3)has the following representation:

z∗

(1,2,3) =a|a|

2z + r

4− |a|

2r2 + |a|

4

(r2 + |a|2)z + a|a|

2. (3.4)

The formulae for the transforms w = z∗

(1,3,2), z∗

(2,1,3), . . . , z∗

(3,2,1) can be obtainedfrom (3.3) by interchanging indexes.

Further we describe the transforms in the case of optimal effective characteristics(for the external field oriented along the real axes). Let the discs Dj , j = 1, 2, 3, besituated along the imaginary axes and touch each other (see Fig. 4b), i.e. a = 2ri.Then

z∗

(1,2,3) =8irz + 13r2

5z − 8ir, z

∗

(1,3,2) =−4irz − 7r2

−7z + 12ir, (3.5)

z∗

(2,1,3) =−12irz − 7r2

−7z + 4ir, z

∗

(2,3,1) =12irz − 7r2

−7z − 4ir,

z∗

(3,1,2) =4irz − 7r2

−7z − 12ir, z

∗

(3,2,1) =−8irz + 13r2

5z + 8ir.

Let the discs Dj , j = 1, 2, 3 be situated along the real axes and touch each other,i.e. a = 2r (see Fig. 5). Then

z∗

(1,2,3) =8rz + 13r2

5z + 8r, z

∗

(1,3,2) =−4rz − 7r2

−7z − 12r, (3.6)


6

-0

x

y

· ·· &%'$

&%'$

&%'$

Figure 5. Three discs in line along real axes.

z∗

(2,1,3) =−12rz − 7r2

−7z − 4r, z

∗

(2,3,1) =12rz − 7r2

−7z + 4r,

z∗

(3,1,2) =4rz − 7r2

−7z + 12r, z

∗

(3,2,1) =−8rz + 13r2

5z − 8r.

From the point of view of applications dealing with composite materials (see,e.g., [5, 8]) it is also interesting to consider the case of discs which constitute socalled "packages" of discs. Let us present two results for such configuration.

6

-0

x

y

·

·

·&%'$&%

'$

&%'$

6

-0

x

y

·

·

·&%'$&%

'$

&%'$

a) b)

Figure 6. Special packages: a) package of three discs I, b) package of three discs II.

Let Dj := D(aj , rj), j = 1, 2, 3, where

rj = r, a1 = 0, a2 = 2rei π

6 , a3 = 2re−i π

6 ,

i.e. the centers of the discs lay at the vertex of the right triangle (see Fig.6a). Thenthe transform w = z

∗


w =2irz − (1 + 2

√

3i)r2

(−1 + 2√

3i)z − 2ir.


Let Dj := D(aj , rj), j = 1, 2, 3, where (see Fig. 6b)

rj = r, a1 = 0, a2 = 2rei 5π

6 , a3 = 2re−i 5π

6 .

Then the transform w = z∗


w =2irz + (−1 + 2

√

3i)r2

(−1 − 2√

3i)z − 2ir.

3.3. Symmetries with respect to four circles

In the case of four discs we consider the only situation with four discs of equal radiisymmetrically situated with respect to the origin:

D1 := z ∈ C : |z − a| < r , D2 := z ∈ C : |z + a| < r ,

D3 := z ∈ C : |z + a| < r , D4 := z ∈ C : |z − a| < r ,

where a ∈ C, r > 0, |Re a| ≥ r, |Ima| ≥ r (see Fig.7).

6

-0

x

y

··

· ·

&%'$

&%'$

&%'$

&%'$

D4(a, r)

D1(a, r)

D3(−a, r)

D2(−a, r)

Figure 7. Four symmetrically situated discs.

In this case the composition of the successive symmetries w = z∗

(1,2,3,4)is deliv-

ered by the formula

w =Az +B

Cz +D

,

with

A = r4 + r

2a2− r

2a2 + aa

3− a

3aa

2a2 + a

4, (3.7)

B = r2a3 + r

2aa

2− r

2a3 + a

4a− a

2a3,

C = −a3− a

2a+ aa

2 + a3,

D = r4 + r

2a2− r

2a2 + r

2aa− a

2a2 + a

4.


The transform w = z∗

(1,2,3,4)satisfies the following relation

w − ζ1

w − ζ2

= K

z − ζ1

z − ζ2

, (3.8)

where

ζ1 =a4− 2a2

r2 + aar + 2a2

r2− aa

3− a

4 +√

F

2(a2− a

2)(a+ a), (3.9)

ζ2 =a4− 2a2

r2 + aar + 2a2

r2− aa

3− a

4−

√

F

2(a2− a

2)(a+ a),

F = (a4− 2a2

r2 + a

3a+ aar + 2a2

r2− aa

3− a

4)2

− 4(a2− a

2)(

a3r2− a

4a− aa

2r2 + a

2a3− a

3r2)

,

K =A− Cζ1

A− Cζ2

,

A, C are given in (3.7).

The most interesting case for applications is when four discs constitute the pack-age, i.e. a = r + ir. Then the transform w = z

∗

(1,2,3,4) has the following form:

w =zr(−4 + 7i) + 8r2

−8z + r(4 + 7i). (3.10)

The transform w = z∗

(1,2,3,4)satisfies the relation (3.8), its fixed points ζ1, ζ2 are

given by

ζ1 =−1 +

√

3i

2r, ζ2 =

1 −

√

3i

2r,

and the coefficient K in (3.8) is given by the formula

K = 97 + 8√

3.

References

[1] L. A. Aksent’ev. Construction of the Schwarz operator by use of symmetry method. Proc.seminar on boundary value problems (Kazan), 2; 3; 4, 3 – 11; 11 – 24; 3 – 10, 1964; 1966;1967. (in Russian)

[2] I. A. Aleksandrov and A. C. Sorokin. Schwarz problem for multiply connected circulardomains. Siberian Math. J., 13(5), 971 – 1001, 1972. (in Russian)

[3] L. R. Ford. Automorphic Functions. McGraw-Hill, New York, 1929.[4] G. M. Golusin. Geometric Theory of Functions of Complex Variable. Nauka, Moscow,

1969. (in Russian)[5] S. F. Makaruk. Boundary value problem on jump for a multiply connected domain for

optimally situated internal domains. Vesti NAN Belarusi, 3, 26 – 29, 2003. (in Russian)[6] V. V. Mityushev. Functional equations and its applications in mechanics of composites.

Demostr. Math., 30(1), 64 – 70, 1997.


[7] V. V. Mityushev. Hilbert boundary value problem for multiply connected domains. Com-plex Variables, 35, 283 – 295, 1998.

[8] V. V. Mityushev and S. V. Rogosin. Constructive Methods for Linear and NonlinearBoundary Value Problems for Analytic Functions. Theory and Applications. Chapman &

Hall /CRC Press, Boca Raton – London, 1999.[9] Ch. Pommerenke. Boundary Behaviour of Conformal Maps. Springer Verlag, Berlin,

1992.

Apie Schottky simetrijos grupiu tikslu apibrežima

M.V. Dubatovskaya, S.V. Rogosin

Darbe pateiktas Schottky simetrijos grupiu apibrežimas tam tikros specialios konfiguracijosdaugiajungems skritulinems sritims. Jis yra panaudotas gaunant Švarco uždavinio, kuris pri-taikomas nagrinejant efektyvias kompoziciju savybes, sprendinio išraiška.

!"#$%'&)($+*,-./-0102.435

149–168c© 2004 Technika ISSN 1392-6292

AN IMPROVED HYBRID OPTIMIZATIONALGORITHM FOR THE QUADRATICASSIGNMENT PROBLEM

A. MISEVICIUS

Kaunas University of Technology, Department of Practical Informatics

Studentu St. 50–400a, 3031 Kaunas, Lithuania

E-mail: 6798:;<6<>=@?BA1<C9D4AEAGF#<HIJ9FK=L7MJ

Received November 20, 2003; revised January 29, 2004

Abstract. In this paper, we present an improved hybrid optimization algorithm, which wasapplied to the hard combinatorial optimization problem, the quadratic assignment problem(QAP). This is an extended version of the earlier hybrid heuristic approach proposed by theauthor. The new algorithm is distinguished for the further exploitation of the idea of hybridiza-tion of the well-known efficient heuristic algorithms, namely, simulated annealing (SA) andtabu search (TS). The important feature of our algorithm is the so-called "cold restart mech-anism", which is used in order to avoid a possible "stagnation" of the search. This strategyresulted in very good solutions obtained during simulations with a number of the QAP in-stances (test data). These solutions show that the proposed algorithm outperforms both the"pure" SA/TS algorithms and the earlier author’s combined SA and TS algorithm.Key words: hybrid optimization, simulated annealing, tabu search, quadratic assignmentproblem, simulation

1. Introduction

The quadratic assignment problem (QAP) is the famous combinatorial optimizationproblem. It is formulated as follows. Let two matrices A = (aij)n×n and B =(bkl)n×n and the set Π of permutations of the integers from 1 to n be given. Find apermutation π =

(

π(1), π(2), . . . , π(n))

∈ Π that minimizes

z(π) =

n∑

i=1

n∑

j=1

aijbπ(i)π(j). (1.1)

One of the important applications of the QAP is computer-aided design (CAD),namely, the placement of electronic components [23, 26, 40]. In this context, the en-tries of the matrix A = (aij)n×n can be interpreted as the numbers of connections

150 A. Misevicius

(nets) between components. The entries of the matrix B = (bkl)n×n represent dis-tances between locations (positions). The permutation π =

(

π(1), π(2), . . . , π(n))

corresponds to a certain placement of components to locations (π(i) denotes the lo-cation that component i is placed into). Thus, z, or more precisely 1

2z, can be treated

as an estimation of total wire length obtained when n components are placed into n

locations (see Fig. 1). Description of the other applications of the QAP one can befound in [7, 8, 10].

Figure 1. Graphical interpretation of the quadratic assignment problem. Forgiven matrices A and B the permutation corresponding to optimal assignmentis as follows:(2,3,1,4). The connection length that corresponds this assignmentis equal to 12.

It has been proved that the QAP (like many other combinatorial optimizationproblems) is NP-hard [38]. For example, QAPs of size n > 36 are not, to this date,practically solvable in terms of obtaining exact solutions. Therefore, heuristic tech-niques have to be used for solving medium- and large-scale QAPs (see, for exam-ple, [14, 15, 16, 29, 32, 41]; for a more detailed list of heuristics for the QAP, see[8, 10, 37]).

First we introduce some basic definitions related to the combinatorial (discrete)optimization. So, let S be a set of solutions of a combinatorial optimization problemwith an objective function f : S → R

1 (without loss of generality, we assume thatf seeks a global minimum). Furthermore, let N : S → 2S be a neighbourhoodfunction which defines for each s ∈ S a set N(s) ⊆ S – a set of neighbouringsolutions of s. Each neighbouring solution s

′

∈ N(s) can be reached directly fromthe current solution s by an operation, which is called a move. Usually, the movefollows the objective function evaluation which is called a trial. An iteration is saidto be performed when |N(s)| trials are done.

Regarding the QAP, Π = π|π = (π(1), π(2), . . . , π(n)), where |Π | = n!,corresponds to S, and z (defined according to (1.1)) plays a role of the objectivefunction. In the case of the QAP, the commonly used neighbourhood function is so-called 2-exchange (pairwise exchange) function N2 which can be defined as follows:

N2(π) = π′

|π′

∈ Π, ρ(π, π′) = 2, (1.2)

An improved hybrid optimization algorithm for the QAP 151

where π ∈ Π and ρ(π, π′) is a "distance" between the current permutation π and the

neighbouring one π′ :

ρ(π, π′) =

n∑

i=1

sgn|π(i) − π′(i)| .

In this case, a move from the permutation π to the permutation π′ can formally be

defined by using a 2-way perturbation operator

pij : Π → Π

(

i, j = 1, 2, . . . , n; i 6= j

)

,

which exchanges ith and jth elements in the current permutation (notation π′ =

π ⊕ pij means that π′ is obtained from π by applying the perturbation pij). For a

permutation π and a perturbation pij , it is more efficient to compute ∆z(π, i, j) =z(π ⊕ pij)− z(π) than z(π ⊕ pij) : the direct computation of z(π ⊕ pij) needs timeO(n2), whereas ∆z(π, i, j) can be calculated in O(n) operations:

∆z (π, i, j) = (aij − aji)(

bπ(j)π(i) − bπ(i)π(j)

)

+∑

k=1,k 6=i,j

[

(aik − ajk)

×

(

bπ(j)π(k) − bπ(i)π(k)

)

+ (aki − akj)(

bπ(k)π(j) − bπ(k)π(i)

)]

, (1.3)

where aii(bii) = const, i = 1, 2, . . . , n. Moreover, for two consecutive permuta-tions π and π

′ = π ⊕ puv, if all the values ∆z(π, i, j) have been stored (i.e. alreadycalculated in previous iteration), then the values

∆z(π′

, i, j) = z(π′

⊕ pij − z(π′), i 6= u, v, j 6= (u, v)

can be computed in time O(1) [42]:

∆z (π′

, i, j) = ∆z (π, i, j) + (aiu − aiv + ajv − aju)(bπ(i)π(u)

− bπ(i)π(v) + bπ(j)π(v) − bπ(j)π(u)) + (aui − avi + avj − auj)

(bπ(u)π(i) − bπ(v)π(i) + bπ(v)π(j) − bπ(u)π(j)) . (1.4)

However, if i = u or i = v or j = u or j = v, then the formula (1.3) should beapplied.

Two main alternatives exist when exploring the neighbouring solutions. First,choose the next potential solution at random. Second, explore the neighbourhoodin a systematic way. In the case of the 2-exchange neighbourhood function N2, theorder of search can be established by a sequence pi(k)j(k). The indices i

(k), j

(k)

are easily determined by the following expression

i(k) = iif(j(k−1)

< n, i(k−1)

, iif(j(k−1)< n − 1, i

(k−1) + 1, 1)),

j(k) = iif(j(k−1)

< n, j(k−1) + 1, i

(k) + 1),

where

iif(x, y1, y2) =

y1, if x is true

y2, otherwise;

152 A. Misevicius

k is the current trial number (k = 1, 2, . . .); i(k), j

(k) are new indices; i(k−1)

, j(k−1)

are old indices (i(0) = 1, j(0) = 1), K = |N2| = n(n − 1)/2 trials are needed in

order to explore all the solutions of N2.

The remaining part of this paper is organized as follows. In Section 2 hybridoptimization strategies (paradigms) are outlined, whereas in Sections 3, 4 we surveythe simulated annealing (SA) and tabu search (TS) techniques, which were used inour hybrid approach. Section 5 describes an improved hybrid optimization algorithmfor the quadratic assignment problem. The results of simulations are presented inSection 6. Finally, Section 7 completes the paper with concluding remarks.

2. Hybrid optimization strategies

Over the last years, hybrid optimization algorithms have become very popular amongresearchers in combinatorial optimization. First of all, this is due to promising re-sults obtained by using hybrid (combined) approaches. Combinations of both single-solution algorithms (such as greedy heuristic search, simulated annealing, tabusearch) and population-based algorithms (such as genetic, evolutionary algorithms)have been proven to be extremely efficient for many combinatorial optimizationproblems [17, 19, 36]. Different hybrid meta-heuristics, i.e. paradigms of hybridiza-tion of heuristics can be proposed [43]. Further, two simple paradigms are outlinedvery roughly: first, a sequential hybridization, second, an embedded hybridization.Without loss of generality, we discuss the hybrid scheme that consists of two heuris-tics only.

procedure sequential_hybridization /* H1 + H2 + . . . */. . .apply heuristicH1;

apply heuristicH2;

. . .end

Figure 2. Pseudo-code for the framework of the sequential hybrid meta-heuristic.

So, in the first case, the self-contained heuristics H1 and H2 are executed in asequence (one after other), the heuristic H2 using the output of the heuristic H1 asits input (i.e. the heuristics act in a pipeline fashion). Here, H2 can also be thoughtof as a "post-analysis" procedure which is applied to the solution found by H1. Forexample, a greedy (or more sophisticated) heuristic can be used to generate goodinitial solutions for the genetic/evolutionary algorithm [19]. In the second variant,the heuristic H2 is embedded into heuristic H1 (i.e. heuristics act as cooperatingagents). For example, deterministic local search technique may be embedded intosimulated annealing (as proposed in [31]) or genetic algorithm (see, for example,[12, 17, 19]). For a more formal presentation of the above paradigms, see Figures2,3.


procedure embedded_hybridization /*H1(H2(. . .))∗/. . .apply heuristicH1;

. . .end

procedureH1

. . .apply heuristicH2;

. . .end

Figure 3. Pseudo-code for the framework of the embedded hybrid meta-heuristic.

However, disposing of these two hybridization paradigms only may be insuffi-cient for complex combinatorial problems, like the quadratic assignment problem.These problems can be seen as highly "discontinuous": if one "walks" in a ficti-tious solution space, the qualities of the solutions can differ dramatically, i.e. the"landscapes" of these problems are very rugged. Another distinguishing feature isa presence of a big number of local optima, which are often spread over the wholesolution space (see Figure 4).

Figure 4. Example of a complex "landscape".

In these situations, the strategies described above usually face a phenomenoncalled a "stagnation" of the search (also known as a "chaotic attractor" [5]). Thismeans that the search trajectory is confined in a limited part (region) of the solutionspace: if this part does not contain the global optimum, it will never be found.

An enhanced hybrid strategy (we call it an iterative hybridization) is designedin order to try to overcome these difficulties. In fact, this hybridization strategy isan extension of the sequential hybridization. The extension is constructed in such amanner that self-contained heuristics, say H1 and H2, are used in a cyclic (iterative)way, i.e. the heuristic H2 uses the output of the heuristic H1, and the heuristic H1

uses the output of the heuristic H2 (starting from the second iteration). The paradigmof the iterative hybrid strategy is shown in Fig. 5.

In our hybrid optimization algorithm for the QAP, which will be presented inSection 5, we use simulated annealing approach based procedure in the role of

154 A. Misevicius

procedure iterative_hybridization/*(H1 + H2 + · · · ) + (H1 + H2 + . . .) + . . .*/

. . .repeatapply heuristicH1;

apply heuristicH2;

. . .until termination criterion satisfied

end

Figure 5. Pseudo-code for the framework of the iterative hybrid meta-heuristic.

the heuristic H1, and tabu search approach based procedure as the "post-analysis"heuristic H2. Note that hybridization of SA and TS was also used for solving theother problems (see, for example, [18, 36, 45].) Before describing the details of ourhybrid algorithm, we give short overviews of SA and TS approaches.

3. Simulated annealing

Simulated annealing originated in statistical mechanics. It is based on a Monte Carlomodel that was used by Metropolis et al., 1953 [33], to simulate energy levels incooling solids. Boltzmann’s law was used to determine the probability of acceptinga perturbation resulting in a change ∆E in the energy at the current temperature t :

P =

1, ∆E < 0,

e−∆E/Ct

, ∆E ≥ 0,

where C is a Boltzmann’s constant. Cerny, 1982 [11], and Kirkpatrick et al., 1983[27] have applied firstly SA to solve combinatorial optimization problems. Severalresearchers tested SA on the QAP, as well [6, 13, 35, 44]. The principle of the sim-ulated annealing is simple: start from a random solution. Given a solution s selecta neighbouring solution s

′ and compute the difference of the objective function val-ues, ∆f = f(s′)− f(s). If the objective function value is improved (∆f < 0), thenreplace the current solution by the new one, i.e. perform a move, and use resultingconfiguration as a starting point for the next trial. If ∆f ≥ 0, then accept a movewith probability

P (∆f) = e−∆f/t

, (3.1)

where t is the current temperature (Boltzmann’s constant is not required when weapply the algorithm to combinatorial problems). Regarding the above probabilisticacceptance, it is achieved by generating a random number in [0,1] and comparing itagainst the threshold e

−∆f/t (here, the exponential function plays a role of an accep-tance function). The procedure is repeated until a termination criterion is satisfied,for example, a predefined number of trials has been performed. As a resulting solu-tion, usually the "best so far" (BSF) solution (instead of so-called "where you are"(WYA) solution) is returned by the algorithm. The paradigm of SA in a high-levelalgorithmic language form is presented in Fig. 6.


procedure simulated_annealing/* input: s

(0) – the initial solution; output: s∗ – the best solution found */

set s = s(0)

, s∗

= s;

determine the initial temperature t0, set t = t0;

repeat /* main cycle */select new solution s

′ from the neighbourhoodof the current solution s;

calculate∆f = f(s′

)− f(s);

generate uniform random number r from the interval [0,1];if (∆f < 0) or (r < e

−∆f/t) then set s = s

′

;

/* replace the current solution by the new one*/if f(s) < f(s

∗

) then set s∗

= s; /* save the best so far solution */update the temperature t;

until stopping condition is satisfiedend

Figure 6. Pseudo-code for the simulated annealing.

SA algorithms differ mainly with respect to a cooling (annealing) schedule im-plemented. The cooling schedule, in turn, is specified by:

a) an initial (and final) value of the temperature,

b) an updating function for changing the temperature.

The most important thing is how the initial temperature t0 is specified. If theinitial value of the temperature is chosen too high, then too many bad uphill movesare accepted, while if it is too low, then the search will quickly drop into a localoptimum without possibility to escape form it. Thus, an optimum initial temperaturemust be somewhere between these two extremes.

The temperature is not a constant, but changes over time according to the updat-ing function. One of the popular updating functions (known as Lundy-Mees sched-ule) is characterized by the following relation [30]:

tk+1 =tk

1 + βtk

, k = 0, 1, . . . , t0 = const, β << t0 . (3.2)

It is easy to relate the coefficient β and the number of trials, i.e. the schedule length,L, under condition that the initial and final values of the temperature (t0, tf ) arepredefined:

β =t0 − tf

Lt0tf

. (3.3)

In theory, the simulated annealing procedure should be continued until the finaltemperature tf is zero, but in practice the other stopping criteria are applied, forexample:

a) the value of the objective function has not decreased for a large number ofconsecutive trials;

b) the number of accepted moves has become less than a certain small thresholdfor a large number of consecutive trials;

156 A. Misevicius

c) a fixed a priori number of trials/iterations has been executed.

For more details about simulated annealing, the reader is referred to [1, 2, 28].

4. Tabu search

Tabu search technique was developed by Hansen and Jaumard, 1987 [24], andGlover, 1989, 1990 [20, 21]. TS has been proven to be a powerful tool for solvingmany combinatorial problems, among them the QAP (see, for example, [5, 14, 39,42]). Tabu search, like simulated annealing, is based on the neighbourhood searchwith local-optima avoidance but in a rather deterministic way. The key idea of tabusearch is allowing climbing moves when no improving neighbouring solution exists.However, some moves are to be forbidden at a present search iteration in order toavoid cycling.

TS starts from an initial solution s, maybe, randomly generated in S and movesrepeatedly from a solution to a neighbouring one. At each step of the procedure, a set(subset) N(s) of the neighbouring solutions of the current solution s is consideredand the move that improves most the objective function value f is chosen. If there areno improving moves, TS algorithm chooses one that least degrades (increases) theobjective function, i.e. a move is performed to the best neighbour s

′ in N(s) (even iff(s′) > f(s)).

In order to avoid returning to the local optimal solution just visited, the reversemove must be forbidden (prohibited). This is done by storing this move (or an at-tribute of the move) in a memory (or more precisely short-term-memory) managedlike a circular list T and called a tabu list. The tabu list keeps information on the lasth (h = |T |) moves which have been done during the search process (the parameterh is called a tabu list size). Thus, a move from s to s

′ is considered as tabu if it (orits attribute) is contained in the list T. This way of proceeding hinders the algorithmfrom returning to a solution reached in the last h iterations. However, it might beworth returning after a while to a solution visited previously to search in another di-rection. Consequently, an aspiration criterion is introduced to permit the tabu statusto be dropped under certain favourable circumstances. Typically, a tabu move froms to s

′ is permitted if f(s′) < f(s∗), where s∗ is the best solution found so far.

The resulting decision rule within TS may thus be described as follows: replace thecurrent solution s by the new solution s

′

, if

f(s′) < f(s∗) or (s′ = arg mins′′∈N(s)

f(s′′

) and s′ is not tabu). (4.1)

The whole process is stopped as soon as a termination criterion is satisfied (for exam-ple, a fixed a priori number of trials has been performed). The tabu search paradigmis shown in Figure 7.

The TS algorithms differ mainly with respect to the basic ingredients discussedabove (i.e. tabu list, aspiration criterion) and other additional features (for example,a long-term-memory, diversification mechanisms, etc.). The main forms of the tabusearch are: deterministic tabu search (strict tabu search, fixed tabu search, reactivetabu search) and stochastic tabu search (probabilistic tabu search, robust tabu search).For more details on the TS technique, the reader is addressed to [22, 25].


procedure tabu_search/* s(0) – s

∗ !#"$% */set s = s

(0), s

∗

= s;

initialize the tabu list T ;

repeat /* main cycle*/given neighbourhood function N,

tabu list T and aspiration criterion,find the best possible solution s

′ ∈ N(s);

set s = s′

; /* & ')(*(+ &&,-.,/0 +12 */insert the solution s (or its attribute)into the tabu list T ;

if f(s) < f(s∗

) then set s∗

= s /* 435* -6"$)&! */update the tabu list T (or its size) (if necessary)

until stopping condition is satisfiedend

Figure 7. Pseudo-code for the tabu search.

5. An improved hybrid simulated annealing and tabu searchalgorithm for the QAP

Now we describe details of our hybrid strategy for the QAP. It is distinguished forthe following structure: 1) simulated annealing algorithm, 2) tabu search algorithm,and 3) hybridization scheme.

5.1. Simulated annealing algorithm for the QAP (SA-QAP)

One of the important features of our implementation of the simulated annealing isthat we use an extended approach of determining the values of the initial and finaltemperatures (these values are crucial for the SA algorithm, as mentioned in Section3). Typically, the initial (and final) temperature is a function of the minimum andmaximum differences in the objective function values obtained by performing a fixednumber of moves before starting the annealing [13]. In our SA algorithm, we ignorethe maximum difference; instead, we use the average difference. The formula ofcalculating the initial and final temperatures (t0, tf ) looks, thus, as follows:

t0 = (1 − λ1)∆zmin + λ1∆zavg ,

tf = (1 − λ2)∆zmin + λ2∆zavg ,

(5.1)

where ∆zmin, ∆zavg are, respectively, the minimum and average differences in theobjective function values; λ1 ∈ (0, 1]; λ2 ∈ [0, 1); λ1 > λ2. In fact, the executionof the algorithm is controlled by operating with these factors. By choosing appropri-ate values of λ1 and λ2, one can control the cooling process flexibly. For example,having λ2 = const it is obvious that the larger the value of λ1, the higher the initialtemperature; on the other hand, the larger the difference λ1 − λ2, the more "rapid"the cooling. We use λ1 = 0.5 and λ2 = 0.05.

158 A. Misevicius

Another property of the SA algorithm is related to an intelligent annealing tech-nique. The key idea is that the temperature is not monotone decreasing, but oscil-lating; that is, a re-annealing (a repeating sequence of coolings and heatings) isconsidered instead of the straightforward annealing (see also [3, 6]). We proposethe re-annealing technique which is based on so-called dynamic cooling schedule.The parameters of this schedule (schedule length, initial and final temperatures) areadaptively changed during execution of the algorithm. We use a Lundy-Mees func-tion based temperature oscillation (LM-oscillation) that is "process-dependent", i.e.it depends upon the former "behaviour" of the (re)annealing. The schedule is as fol-lows: set the schedule length L to QSAn(n − 1)/2 (QSA ≥ 1) and start with theinitial temperature defined by the formula (5.1). The temperature is then being up-dated according to the formula (3.2), the coefficient β is known from the formula(3.3). When 0.5|N2| = n(n−1)/4 consecutive moves are rejected, stop the (prelim-inary) cooling. After cooling is stopped, the temperature is immediately increased(i.e. the system is "heated up"), and the annealing with the new parameters starts.Additionally, a deterministic downhill search procedure CRAFT [4] is applied to thebest solution found. The process is continued until a stopping criterion is satisfied,i.e. the current iteration number exceeds QSA, where QSA is the maximum numberof iterations.

The detailed template of the SA algorithm for the QAP (SA-QAP) is presentedin Figure 8 (see also [35]).

5.2. Tabu search algorithm for the QAP (TS-QAP)

Our version of the tabu search algorithm for the QAP is based on a slightly modifiedrobust tabu search (RTS) procedure due to Taillard [42]. Very roughly, our algorithmconsists of maintaining the tabu list T by constructing and updating it. The tabu list isorganized as an n × n integer matrix T = (tij)n×n, where n is the problem size. Atthe beginning, all the entries of T are set to zero. As the search progresses, the entrytij stores the current number of the iteration plus the tabu list size, h, i.e. the numberof the future iteration starting at which ith and jth elements of the permutation mayagain be interchanged. In this case, a move consisting of exchanging ith and jthelements is tabu if the value of tij is equal or greater than the current iteration number(this means that ith and jth elements were interchanged during the last h iterations).

The tabu list size h is not a constant – it is changed randomly during the searchprocess. In our implementation, h is chosen between hmin = 0.4n and hmax = 0.6n

and changed every 2hmax iterations. The standard aspiration criterion is used, i.e. thetabu status of a move is ignored (a tabu move is allowed to be selected) if the moveresults in a solution (permutation) that is better than the best one found so far.

In addition, we use the formula (1.4) to accelerate the evaluation of the neigh-bouring solutions of the current solution. Thus, the complete evaluation of the 2-exchange neighbourhood takes O(n2) operations, except the first iteration, whichtakes O(n3) operations (see formula (1.3). The run time of the tabu search proce-dure is controlled by the number of iterations, QTS.

The detailed template of the TS algorithm (TS-QAP) is presented in Figure 9.


procedure SA-QAP /* 4%#) & "$)&- *//* input: π – the current (initial) permutation, n – the problem size *//* QSA – the number of iterations (QSA ≥ 1) */

/* λ1, λ2 – the initial and final temperature factors(0 < λ1 ≤ 1, 0 ≤ λ2 < 1, λ1 > λ2) *//* output: π

∗ – the best permutation found */π∗

= π;

found ∆zmin, ∆zavg by performing n(n − 1)/2 random movesstarting from π;

M := QSAn(n − 1)/2, L0 := M ;

/* M – the number of trials, L0 – the initial cooling schedule length */initialize cooling schedule parameters L, t0, tf , β;

t := t0, i := 1, j := 1, rejected_count := 0, oscillation :=′

FALSE′

;

for k := 1 to M do begin /* main loop */i := iif(j < n, i, iif(i, n − 1, i + 1, 1)), j := iif(j < n, j + 1, i + 1);

calculate ∆ = ∆z(π, i, j);

/* ∆z(π, i, j) is the current difference of the objective function values */if ∆ < 0 then accept :=

′

TRUE′

else begingenerate uniform random number r from the interval [0,1];if r < exp(−∆/t)

then accept :=′

TRUE′ else accept :=

′

FALSE′

end /* else */if accept =

′

TRUE′ then begin

π := π ⊕ pij ; /* replace the current permutation by the new one */if z(π) < z(π

∗

) then π∗

:= π;

/* save the best permutation found so far */if ∆ 6= 0 then rejected_count := 0

endelse rejected_count := rejected_count + 1;

if (rejected_count ≥ n(n− 1)/4) or (t is at lowest point)then beginif oscillation =

′

FALSE′

then begin L∗

:= k, t∗

:= t, oscillation :=′

TRUE′ end

update cooling schedule parameters L, t0, tf , β;

t := t0; /* & ! *( &&+ 0+& )& */apply CRAFT to π

∗

endelse t := t(1 + βt) /* decrease the current temperature */

end /* main loop */end

Figure 8. Pseudo-code of the simulated annealing algorithm for the QAP. Here ∆z(π, i, j) iscalculated according to (1.3).

5.3. Hybridization scheme

The hybridization scheme used is as follows. At the beginning, an initial solution isgenerated in a random way with the subsequent improving by means of the simulatedannealing algorithm (SA-QAP). Then, an iterative hybrid process starts. It consists

160 A. Misevicius

procedure TS-QAP /* tabu search algorithm for the QAP *//*input: π – the current permutation, n – the problem size *//* QTS – the number of iterations (QTS ≥ 1), *//* hmin, hmax – lower and higher tabu list sizes (hmin < hmax) *//* output: π

∗ – the best permutation found */π∗

:= π;

for i := 1 to n− 1 do for j := i + 1 to n docalculate δij = ∆z(π, i, j);

T := 0, i := 1, j := 1;

for q := 1 to QTS do begin /* main loop */if q mod 2hmax = 1 then h := randint(hmin, hmax);

∆min :=∞;

for k := 1 to |N2| do begin /* find the best move */i := iif(j < n, i, iif(i < n− 1, i + 1, 1)), j := iif(j < n, j + 1, i + 1);

tabu := iif(tij ≥ q,′

TRUE′

,′

FALSE′

),

aspired := iif(z(π) + δij < z(π∗

),′

TRUE′

,′

FALSE′

);

if ((δij < ∆min) and (tabu =′

FALSE′)) or (aspired =

′

TRUE′

)

then beginu := i, v := j;

if aspired =′

TRUE′ then ∆min := −∞ else ∆min := δij

end /* if */end /* for */if ∆min <∞ then begin

/* perform the move: replace the current permutation by the new one */π := π ⊕ puv

for l := 1 to n − 1 do for m := l + 1 to n doupdate the difference δlm;

tuv := q + h; /* update the tabu list */if z(π) < z(π

∗

) then π∗

:= π /* save the best so far permutation */end

end /* main loop */end

Figure 9. Pseudo-code of the tabu search algorithm for the QAP, here ∆z(π, i, j) is calculatedaccording to the formula (1.3), while the difference δlm is updated according to (1.4). Thefunction "randint(x, y)" returns a random integer, uniformly distributed between x and y.

of two main phases, as mentioned in Section 2: simulated annealing (SA-QAP) andtabu search (TS-QAP). In addition, a diversification mechanism is used. The role ofsuch a mechanism play mutations that can be seen as strings of random elementaryperturbations (pairwise interchanges), like pij . The mutations may also be viewed asmoves in higher-order neighbourhoods Nµ, where 2 < µ ≤ n, here, the parameterµ is referred to as a mutation level (rate). It is obvious that the large value of µ, thestronger the mutation, and vice versa. The template of the mutation procedure basedon random interchanges is presented in Figure 10.

A corresponding example is shown in Figure 11.We can add more robustness to the mutation process by letting the parameter µ

vary in some interval, say [µmin, µmax] ⊂ [3, n]. In our implementation, µ varies inthe following way: at the beginning, µ is equal to µmin; once the maximum value


procedure ri-mutation/* random interchanges based mutation operator (ri-mutation) for the QAP *//* input: π – the current permutation, n – the problem size,µ – the mutation lavel (µ ∈ [3, n]) *//* output: π – the mutated permutation */for k := 1 to µ do beginchoose i, j, randomly, uniformly, 1 ≤ i, j ≤ n, i 6= j

π := π ⊕ pij /* interchange ith and jth elements in the current permutation */end /* for k */

end

Figure 10. Pseudo-code of the random interchanges based mutation.

Figure 11. Example of ri–mutation.

µmax has been reached (or a better locally optimum solution has been discovered),the value of µ is immediately dropped to the minimum value µmin, and so on. Notethat the mutations are to be applied to the locally optimum solutions only.

The specific feature of our hybridization scheme is that if the current locallyoptimum solution remains unchanged for a long time (a "stagnation" of the searchtakes place), then a "cold restart" of the search is carried into effect. As a "coldrestart", we use the construction (generation) of a new random solution coupled withthe simulated annealing algorithm – the same that it used at the initialization phase.The purpose of such a restart is to add more diversity to the search, more precisely,to explore new regions of the solution space: continuing the search from the newrandom solution may allow to escape from a "deep" local optimum and to find betterones. The frequency of "cold restarts" is controlled by means of a special parameter,a restart interval, v, which can be related to the problem size, n, i.e. v = ωn, where ω

is a factor of the restart frequency (0 < ω < Q/n, Q is the total number of iterationsof the hybridized algorithm.

The template of the resulting hybrid optimization algorithm entitled as IH-SA-TS-QAP (improved hybrid SA and TS algorithm for the QAP) is shown in Figure12.

6. Simulation results

We have carried out a number of simulations in order to test the performance of ourimproved hybrid algorithm IH-SA-TS-QAP. The well-known QAP instances (testdata) taken from the quadratic assignment problem library QAPLIB [9] (see also

162 A. Misevicius

procedure IH-SA-TS-QAP/* improved hybrid simulated annealing-tabu search algorithm for the QAP *//* input: A, B – the connection and distance matrices, n – the problem size *//* Q – the number of cycles (global iterations) of the hybrid algorithm *//* QSA – the number of iterations of the simulated annealing procedure *//* QTS – the number of iterations of the tabu search procedure *//* λ1, λ2, hmin, hmax, µmin, µmax, – the control parameters *//* output: π

∗ – the best permutation found */generate random (initial) permutation π;

apply SA-QAP to π with the parameters QSA, λ1, λ2, andget the resulting permutation π

•;π∗

:= π•

;

q∗

:= 0; /* q∗ is the current number of iteration at which

the new local optimum has been found */µ := µmin − 1; /* µ is the current mutation level */v := ωn; /* v is the restart interval (period) */for q := 1 to Q do begin /* main cycle */apply TS-QAP to π

• with the parameters QTS, hmin, hmax, andget the resulting permutation π

∆;

if z(π∆

) < z(π∗

) then beginπ∗

:= π∆

, q∗

:= q, µ := µmin − 1

/* save the best so far permutation and reset the control parameters*/endif q − q

∗

> v then begingenerate new random permutation π

; /* perform a "cold restart" */apply SA-QAP to π

with the parameters QSA, λ1, λ2, andget the resulting permutation π

•;if z(π

•

) < z(π∗

) then π∗

:= π•

; /* save the best so far permutation */q∗

:= q, µ := µmin − 1

endelse beginµ := iif(µ < µmax, µ + 1, µmin);

apply mutation to π∗ with the level µ,

/*i.e. perform µ random perturbations */and get the permutation π;

π•

:= π /* π• is the mutated permutation to be processed by TS procedure */

endend /* for q */

end

Figure 12. Pseudo-code of the improved hybrid simulated annealing and tabu search algo-rithm for the QAP.


http://www.seas.upenn.edu/qaplib/) were used. The following algorithms were usedfor comparison:

1) The simulated annealing algorithm by Boelte and Thonemann (coded by theauthor according to the description presented in the paper of Boelte and Thonemann[6]; the algorithm is entitled as TB2M-QAP);

2) The robust tabu search algorithm by Taillard [42] (it is entitled as RTS-QAP);

3) The combined simulated annealing and tabu search algorithm by Misevicius[34] (entitled as SA-TS-QAP).

Table 1. Comparison of the algorithms for the QAP. The values of the average deviation (Θ),the percentage of 1% optimality (P1%), and the CPU time (in seconds) are given. The valuesof the best average deviations are printed in bold face.

Instance n BKVTB2M-QAP RTS-QAP SA-TS-QAP IH-SA-TS-QAP Average

nameΘ P1% Θ P1% Θ P1% Θ P1%

CPU time

nug30 30 6124 0.94 63 0.73 72 0.70 78 0.52 90 0.12sko42 42 15812 0.66 83 1.03 55 0.55 88 0.46 90 0.29sko49 49 23386 0.67 86 0.85 64 0.54 94 0.46 97 0.45sko56 56 34458 0.66 84 0.95 55 0.53 93 0.50 96 0.65sko64 64 48498 0.57 92 0.93 61 0.48 100 0.45 99 1.00sko72 72 66256 0.60 97 0.99 53 0.52 95 0.48 98 1.34sko81 81 90998 0.46 100 0.93 60 0.41 100 0.40 99 1.87sko90 90 115534 0.49 99 0.96 60 0.43 100 0.43 100 2.55sko100b 100 153890 0.39 100 0.97 61 0.34 100 0.29 100 3.44sko100c 100 147862 0.46 98 1.15 36 0.34 99 0.32 99 3.46sko100d 100 149576 0.49 100 0.96 57 0.43 100 0.41 100 3.44sko100e 100 149150 0.52 99 1.06 45 0.45 100 0.41 100 3.43sko100f 100 149036 0.54 100 0.95 62 0.46 100 0.40 100 3.44tho30 30 149936 1.07 54 1.41 45 0.90 70 0.91 69 0.13tho40 40 240516 1.33 31 1.30 34 1.09 46 0.94 54 0.26wil50 50 48816 0.26 100 0.52 84 0.19 100 0.16 100 0.47wil100 100 273038 0.25 100 0.55 69 0.22 100 0.22 100 3.45

The performance measures used are the following:

1) the average deviation from the best known solution Θ = 100(z − z)/z[%],where z is the average objective function value over W = 1, 2, . . . restarts (i.e.single applications of the algorithm to a given instance) and z is the best knownvalue (BKV) of the objective function, BKVs are from [9];

2) the percentage of solutions that are within 1% optimality P1% = 100C1%/W,

where C1% is the total number of solutions that are within 1% optimality over W

restarts.

All the simulations were carried out on 300 MHz Pentium computer by using theoptimization package (sub-system) OPTIQAP (OPTImizer for the QAP) developed

164 A. Misevicius

by the author at Dept. of Practical Informatics of Kaunas Univ. of Technology. Thecomputations were organized in such a way that all the algorithms use identical ini-tial assignments and require similar CPU times (the execution time is controlled bythe number of iterations). The results of the comparison, i.e. the average deviationsfrom BKV and percentage of solutions that are within 1% optimality for each ofthe algorithm tested, as well as CPU times per restart are presented in Table 1. Theparameters of IH-SA-TS-QAP used in simulation are as follows:

Q = 1, QSA = 50, QTS = 5QSA = 250 ,

λ1 = 0.5, λ2 = 0.05, µmin = 0.35, µmax = 0.45 .

Let us note, that as long as the number of cycles, Q, is equal to 1, the parameter ω

can be omitted. The number of restarts, W, is equal to 100.

Table 2. Computational results of IH-SA-TS-QAP with the various numbers of iterations.The values of the average deviation (Θ), and the CPU time (in seconds) are given.In addition, in parenthesis we give the numbers of times that BKV is found.

InstanceΘ, time

name1st (W = 30) 2nd (W = 30) 3rd (W = 20) 4th (W = 2) 5th (W = 10)

nug30 0.060[9] 5.0 0.002[29] 30.0 0 88 0 150 0 360

sko42 0.075[11] 10.5 0.003[28] 63.0 0 180 0 330 0 720

sko49 0.128[2] 14.9 0.044[6] 90.0 0.009[16] 250 0 470 0 1200

sko56 0.168[1] 20.5 0.049[3] 118 0.001[19] 340 0.001[14] 570 0 1380

sko64 0.156[3] 27.4 0.020[8] 162 0.000[19] 460 0 800 0 1800

sko72 0.304[0] 35.0 0.117[0] 210 0.012[0] 570 0.012[2] 1020 0.002[6] 2250

sko81 0.191[0] 46.0 0.074[0] 275 0.016[2] 720 0.013[2] 1260 0.007[4] 2700

sko90 0.300[0] 57.0 0.133[0] 330 0.042[1] 930 0.027[0] 1440 0.004[5] 3200

sko100a 0.233[0] 72.0 0.114[0] 420 0.048[1] 1180 0.026[2] 1980 0.019[1] 4300

sko100b 0.221[0] 72.0 0.094[0] 420 0.020[0] 1200 0.011[1] 1950 0.004[1] 4200

sko100c 0.209[0] 72.0 0.061[0] 420 0.014[1] 1190 0.007[1] 1940 0.001[5] 4100

sko100d 0.299[0] 72.0 0.130[0] 420 0.042[0] 1180 0.028[0] 1960 0.011[2] 4300

sko100e 0.243[0] 72.0 0.118[0] 420 0.014[1] 1210 0.010[1] 1980 0.005[2] 4400

sko100f 0.278[0] 72.0 0.125[0] 420 0.049[0] 1180 0.020[1] 1970 0.010[2] 4300

tho30 0.074[22] 4.9 0 30.0 0 90.0 0 145 0 370

tho40 0.196[1] 9.2 0.023[6] 56.0 0.012[6] 165 0.005[8] 270 0.002[8] 660

wil50 0.054[4] 16.0 0.008[19] 92 0.002[19] 264 0.001[14] 420 0 1290

wil100a 0.175[0] 72.0 0.084[0] 420 0.011[0] 1200 0.004[2] 1990 0.001[4] 4500

It turns out that the efficiency of the algorithms depends on the QAP instance be-ing solved. Nevertheless, the results from Table 1 show that our hybrid optimizationalgorithm IH-SA-TS-QAP appears to be superior to other three efficient algorithmswith respect to both performance measures, especially, the average deviation. Thedifference in efficiency on particular instances is quite significant (see, for example,the results of RTS-QAP and IH-SA-TS-QAP obtained for the instances sko100a–sko100f, or wil50, wil100).


The results of IH-SA-TS-QAP can be improved even more by increasing thevalues of the control parameters Q and/or QSA (QTS) but at the cost of a longerprocessing time. Five long runs were carried out in order to demonstrate the im-provement of the quality of solutions. At each long run, the different values of theparameters Q, QSA, QTS are used:

1st run : Q = 30, QSA = 50, QTS = 250, ω = 0.3;

2nd run : Q = 100, QSA = 50, QTS = 500, ω = 0.3;

3rd run : Q = 200, QSA = 300, QTS = 1000, ω = 0.1;

4th run : Q = 200, QSA = 500, QTS = 1500, ω = 0.05;

5th run : Q = 300, QSA = 1000, QTS = 3000, ω = 0.03;

the values of the other control parameters remain the same, except the parameter ω.

Table 2 shows the results obtained.

These results are very promising (see 5th column of Table 2): for small andmedium instances (n ≤ 64) (except the instance tho40), the average deviation fromthe best known values of the objective function is equal to zero; while, for large in-stances (n = 100), the deviation is less than 0.02%. It can be seen from the resultsof 5th run that, for all the large instances tested, at least one restart (out of ten) of IH-SA-TS-QAP resulted in finding the best known solution. Moreover, for the instancessko100c and wil100, BKV was reached 5 and 4 times, respectively. This indicatesthat the solutions obtained for these instances are, most likely, pseudo-optimal. Toour knowledge, the pseudo-optimality of the solutions for these instances has notbeen reported yet in the literature. It also should be stressed that even finding BKVfor these instances is quite complicated task, for example, in [16], it was reportedthat to improve on the best known solutions for the instances sko100* on SPARC 10processor, it took almost 24 hours of computation time. In a more recent work [29],an efficient genetic algorithm could not find BKV for any of these instances. It tookup to 900 seconds on DEC Alpha Server 8400 to find the solutions with the averagedeviation around 0.3% only.

7. Concluding remarks

The quadratic assignment problem is one of hard combinatorial optimization prob-lems. In order to obtain near-optimal or optimal solutions for this problem withinreasonable times, heuristic techniques are to be applied. One of them, an improvedhybrid optimization algorithm, has been proposed in this paper.

Based on the well-known simulated annealing and tabu search approaches, aswell as the intelligent hybridization strategy, we have developed an effective algo-rithm for the QAP – IH-SA-TS-QAP (improved hybrid SA and TS algorithm forthe QAP), which is an extension of the earlier author’s hybrid algorithm. IH-SA-TS-QAP is distinguished for the so-called iterative hybridization scheme – a result of theelaborations of possible hybrid heuristic paradigms.

166 A. Misevicius

The additional features of our algorithm are the diversification and "cold restart"mechanisms that are used in order to try to avoid a possible "stagnation" of the search.These mechanisms and the refined hybridization scheme resulted in high qualitysolutions obtained during the simulations with a number of the QAP instances (testdata) from the QAP library – QAPLIB. These solutions indicate that, for the QAPinstances examined, the proposed algorithm appears to be superior to the "pure"simulated annealing and tabu search algorithms, as well as the earlier author’s hybrid(combined) SA and TS algorithm. Thus, it may be considered to be one of the mostefficient single-solution algorithms for the QAP.

Regarding the future work, the emphasis on the further extensions of the pro-posed hybrid approach should be made. Both the elaboration of the hybridizationscheme and improvements of its basic components (i.e. SA and TS procedures) arepossible, for example:

a) introducing new cooling schedules for the SA algorithm;

b) applying other tabu conditions/aspiration criteria for the TS algorithm;

c) trying a more accurate adjustment (tuning) of the control parameters (the initialand final temperatures, the tabu list sizes, etc).

It might also be worthy to incorporate the proposed hybrid algorithm into other(population-based, hybrid) meta-heuristics, for example, genetic and evolutionaryalgorithms, as a very efficient local search procedure.

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[34] A. Misevicius. Combining simulated annealing and tabu search for the quadratic assign-ment problem. Information Technology and Control, 3(20), 37 – 50, 2001.

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[37] P.M. Pardalos, F. Rendl and H. Wolkowicz. The quadratic assignment problem: a surveyand recent developments. In: Quadratic Assignment and Related Problems. DIMACSSeries an Discrete Mathematics and Theoretical Computer Science, volume 16. AMS,Providence, 1 – 41, 1994.

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Patobulintas hibridinis optimizavimo algoritmas kvadratinio paskirstymo uždaviniui

A. Misevicius

Šiame straipsnyje pasi ulytas patobulintas hibridinis euristinis optimizavimo algoritmas geraižinomam, sudetingam kombinatorinio optimizavimo uždaviniui, b utent, kvadratinio paskirsty-mo (KP) uždaviniui. Tai – pagerinta autoriaus ankstesnio hibridinio algoritmo versija. Naujasisalgoritmas pasižymi tuo, jog cia išpletota efektyviu euristiku (atkaitinimo modeliavimo (AM)(angl. simulated annealing) ir tabu paieškos (TP) (angl. tabu search) "hibridizacijos" ideja."Hibridizacija" remiasi vadinamaja iteracine schema: TP algoritmas panaudojamas kaip post-analizes proced ura AM algoritmo gautajam sprendiniui, savo ruožtu, AM algoritmas taikomassprendiniu sekai, gautai sprendiniu diversifikavimo/generavimo keliu. Svarbi pasi ulyto algo-ritmo savybe yra ir ta, kad jame realizuotas vadinamasis "šaltojo pakartotinio starto" principas,kurio paskirtis padeti išvengti galimu paieškos "stagnacijos" situaciju. Naujasis algoritmasišbandytas su KP uždavinio duomenimis iš testiniu pavyzdžiu bibliotekos QAPLIB. Gautieksperimentu rezultatai liudija, jog nagrinetiems KP uždavinio pavyzdžiams si ulomas algorit-mas yra pranašesnis už ankstesnius atkaitinimo modeliavimo ir tabu paieškos algoritmus, taippat už ankstesni autoriaus hibridini algoritma.

!"#$%'&)($+*,-./-0102.435

169–178c© 2004 Technika ISSN 1392-6292

MONOTONE AND CONSERVATIVE DIFFERENCESCHEMES FOR ELLIPTIC EQUATIONS WITHMIXED DERIVATIVES1

I.V. RYBAK

Institute of Mathematics, National Academy of Sciences of Belarus

11 Surganov Str., 220072 Minsk, Belarus

E-mail: 67189;:<#=?>"@A89BCEDFHGI@A817

Received November 13, 2003; revised January 15, 2004

Abstract. In the paper elliptic equations with alternating-sign coefficients at mixed deriva-tives are considered. For such equations new difference schemes of the second order of ap-proximation are developed. The proposed schemes are conservative and monotone. The con-structed algorithms satisfy the grid maximum principle not only for coefficients of constantsigns but also for alternating-sign coefficients at mixed derivatives. The a priori estimates ofstability and convergence in the grid norm C are obtained.

Key words: monotone difference scheme, conservative difference scheme, elliptic equations,mixed derivatives, grid maximum principle

1. Introduction

For the development of difference schemes of the high order of approximation itis important to save properties of both monotonity and conservativeness becausemonotone schemes lead to the well-posed systems of algebraic equations. Iterativemethods converge significantly better in the case of diagonally dominant matrices.

Problems of the development of difference schemes for equations with mixedderivatives were studied in papers [1, 2, 4, 11]. The conservative difference schemesfor elliptic equations with mixed derivatives were considered in [5, p. 286], [6,p. 175], but these schemes do not satisfy the grid maximum principle. For ellip-tic and parabolic equations with mixed derivatives the monotone and conservativedifference schemes were proposed in papers [7, 8, 10], but these schemes can beused only in the case of constant-sign coefficients. If coefficients at mixed deriva-tives changed their sign, then differential equation was rewritten in non-divergent

1 The author thanks Prof. Oleg Iliev and Prof. Raimondas Ciegis for the statement of theproblem, Prof. Piotr Matus and Dr. Mikhail Chuiko for the discussion and useful comments

170 I.V. Rybak

form with first derivatives and monotone schemes were developed by means of theregularization principle [5, p. 183]. But after such a transformation the property ofconservativeness was lost. Such situation is typical in theory of difference schemes.

In the present paper, for elliptic equations with mixed derivatives new mono-tone and conservative difference schemes for both constant-sign and alternating-signcoefficients are proposed. The main idea is based on using the stencil functionalswith absolute values of the coefficients at mixed derivatives. For proposed differenceschemes the a priori estimates of stability and convergence in the grid norm C areobtained. Numerical experiments confirm the theoretical results.

2. Difference scheme

In the rectangle G = 0 ≤ xα ≤ lα, α = 1, 2 with the boundary Γ we considerthe Dirichlet problem for elliptic equations with mixed derivatives

Lu− q(x)u = −f(x), x ∈ G ,

u = µ(x), x ∈ Γ, x = (x1, x2) ,(2.1)

where

Lu =

2∑

α,β=1

Lαβu, Lαβu =∂

∂xα

(

kαβ(x)∂u

∂xβ

)

, q(x) ≥ c0 > 0.

We suppose that the following ellipticity conditions are satisfied

c1

2∑

α=1

ξ2α ≤

2∑

α,β=1

kαβ(x)ξαξβ ≤ c2

2∑

α=1

ξ2α, x ∈ G, (2.2)

where c1, c2 > 0 are positive constants, ξ = (ξ1, ξ2) is an arbitrary nonzero vector.In the rectangle G we consider the uniform grid ωh = ωh ∪ γh:

ωh = x = (x(i1)

1 , x(i2)

2 ) : x(iα)α = iαhα, hαNα = lα, iα = 0, Nα, α = 1, 2,

where ωh is the set of inner grid nodes, γh is the set of boundary grid nodes.Further we will use the following notations of the theory of difference schemes [5]:

v(±1α) = v(x(iα)

α ± hα, x(i3−α)

3−α ), α = 1, 2,

y = y(x(i1)

1 , x(i2)

2 ), yxα=y − y

(−1α)

hα

, yxα=y(+1α)

− y

hα

.

On the grid ωh we approximate differential problem (2.1) by the differencescheme

Λy − dy = −ϕ, x ∈ ωh ,

y = µ(x), x ∈ γh ,

(2.3)

Monotone and conservative difference schemes 171

where

Λy =2

∑

α, β=1

Λαβy, Λααy = (aααyxα)xα

, α = 1, 2 ,

Λαβy =1

4

(

(a−αβyxβ)xα

+ (a−(+1α)

αβ yxβ)xα

+ (a+

αβyxβ)xα

+ (a+(+1α)

αβ yxβ)xα

)

,

a−

αβ = aαβ − |aαβ |, a+

αβ = aαβ + |aαβ |, α 6= β .

Here d ≥ c0, ϕ are some stencil functionals of the coefficient q and the right-handside f respectively. The stencil functionals aαβ can be chosen as follows

aαβ = kαβ iα−

1

2, iβ

= kαβ(xα − 0.5hα, xβ) ,

aαβ =kαβ iα, iβ

+ kαβ iα−1, iβ

2=kαβ + k

(−1α)

αβ

2,

aαβ =2kαβk

(−1α)

αβ

kαβ + k(−1α)

αβ

, α, β = 1, 2 .

A difference scheme is called conservative (divergent), if we have algebraic sumsof unknowns or functions of them only along the boundary after summation of thescheme equations over all grid nodes of the domain [3, p. 280]. If we sum up differ-ence scheme (2.3) over grid nodes of the domain ωh, we obtain algebraic sums offunctions only along the boundary Γ . Hence, the proposed scheme is conservative.

We consider aαβ = kαβ(xα − 0.5hα, xβ) and show that the grid operator Λapproximates the differential operator L with the second order. Let the coefficientskαβ(x) of equation (2.1), all partial derivatives up to the third order inclusively of thecoefficients and up to the fourth order inclusively of the solution u(x) be bounded.By using Taylor expansion of the functions Λαβu in the neighbourhood of the pointx ∈ ωh, we obtain

Λααu− Lααu = O(h21 + h

22) = O(|h|2), α = 1, 2,

Λαβu− Lαβu =hβ

4

∂3u

∂xα∂x2β

(∣

∣

∣

∣

kαβ +hα

2

∂kαβ

∂xα

∣

∣

∣

∣

−

∣

∣

∣

∣

kαβ −

hα

2

∂kαβ

∂xα

∣

∣

∣

∣

)

+O(|h|2), α 6= β .

Using the inequality∣

∣

|a+ b| − |a− b|

∣

∣

≤ 2|b|, we have

|Λαβu− Lαβu| ≤h1h2

4

∣

∣

∣

∣

∂kαβ

∂xα

∣

∣

∣

∣

∣

∣

∣

∣

∣

∂3u

∂xα∂x2β

∣

∣

∣

∣

∣

+O(|h|2) = O(|h|2).

Hence,Λαβu− Lαβu = O(|h|2), α 6= β.

We suppose that the stencil functionals d(x) and ϕ(x) satisfy the usual conditions ofapproximation of the coefficient q(x) and the right-hand side f(x) with the secondorder

172 I.V. Rybak

d(x) − q(x) = O(|h|2), ϕ(x) − f(x) = O(|h|2).

So, difference scheme (2.3) approximates differential problem (2.1) with the secondorder. The stencil of difference scheme (2.3) is presented in Fig. 1.

h1

h2

x

Figure 1. Stencil of difference scheme (2.3).

3. Grid maximum principle

To obtain the a priori estimates of stability in the grid norm C with respect to theright-hand side and the boundary conditions we will use the grid maximum princi-ple [5, p. 258]. Therefore, we have to reduce the difference scheme to the canonicalform

A(x)y(x) =∑

ξ∈S′(x)

B(x, ξ)y(ξ) + F (x), x ∈ ωh, (3.1)

and verify the following sufficient conditions on the coefficients

A(x) > 0, B(x, ξ) ≥ 0, D(x) = A(x) −∑

ξ∈S′(x)

B(x, ξ) > 0, x ∈ ωh. (3.2)

Here A(x), B(x, ξ), F (x) are the known grid functions, S ′(x) = S(x) \ x, S(x)is the stencil of the scheme.

Theorem 1. Let us suppose that conditions (3.2) of the coefficients positivity aresatisfied. Then for the solution of problem (3.1) the following a priori estimate isvalid

‖y‖C ≤ max

∥

∥

∥

∥

F

D

∥

∥

∥

∥

Cγ

,

∥

∥

∥

∥

F

D

∥

∥

∥

∥

C

, (3.3)

where ‖v‖C = maxx∈ωh

|v(x)|, ‖v‖C = maxx∈ωh

|v(x)|, ‖v‖Cγ= max

x∈γh

|v(x)|.

Let us number the nodes of the stencil of difference scheme (2.3) according to Fig. 1and reduce the scheme to canonical form (3.1):

Ay =8

∑

k=1

Bkyk + F, yk = y(xk), xk ∈ S′(x).


If x ∈ ωh, then values of the coefficients are defined by the following formulas

A =a11 + a

(+11)

11

h21

−

|a12| + |a(+11)

12 | + |a21| + |a(+12)

21 |

2h1h2

+a22 + a

(+12)

22

h22

+ d,

B1 =a(+12)

22

h22

−

|a12| + a12 + |a(+11)

12 | − a(+11)

12 + 2|a(+12)

21 |

4h1h2

,

B2 =|a

(+11)

12 | + a(+11)

12 + |a(+12)

21 | + a(+12)

21

4h1h2

≥ 0,

B3 =a(+11)

11

h21

−

2|a(+11)

12 | + |a21| + a21 + |a(+12)

21 | − a(+12)

21

4h1h2

,

B4 =|a

(+11)

12 | − a(+11)

12 + |a21| − a21

4h1h2

≥ 0,

B5 =a22

h22

−

|a12| − a12 + |a(+11)

12 | + a(+11)

12 + 2|a21|

4h1h2

,

B6 =|a12| + a12 + |a21| + a21

4h1h2

≥ 0,

B7 =a11

h21

−

2|a12| + |a21| − a21 + |a(+12)

21 | + a(+12)

21

4h1h2

,

B8 =|a12| − a12 + |a

(+12)

21 | − a(+12)

21

4h1h2

≥ 0,

D = d ≥ c0 > 0, F = ϕ .

For x ∈ γh, the coefficients of the canonical form are given by:

A = 1, B = 0, D = 1, F = µ .

Further we will assume that the following condition is satisfied

maxk1, k2 ≤

h1

h2

≤ mink3, k4, (3.4)

where

k1 =|a12| − a12 + |a

(+11)

12 | + a(+11)

12 + 2|a21|

4a22

,

k2 =|a12| + a12 + |a

(+11)

12 | − a(+11)

12 + 2|a(+12)

21 |

4a(+12)

22

,

k3 =4a11

2|a12| + |a21| − a21 + |a(+12)

21 | + a(+12)

21

,

k4 =4a

(+11)

11

2|a(+11)

12 | + |a21| + a21 + |a(+12)

21 | − a(+12)

21

.

174 I.V. Rybak

Lemma 1. Let coefficients of differential equation (2.1) satisfy the following inequal-ity

kαα ≥ |k(±1α,±1β)

αβ |, α, β = 1, 2. (3.5)

If we choose h1 = h2 = h, then condition (3.4) is always satisfied.

Proof. Let condition (3.5) be satisfied and h1 = h2. In order to prove that in thiscase condition (3.4) is always satisfied, we have to show that

k1 ≤ 1, k2 ≤ 1, k3 ≥ 1, k4 ≥ 1 .

First we prove that k1 ≤ 1, i.e.,

|a12| − a12 + |a(+11)

12 | + a(+11)

12 + 2|a21| ≤ 4a22. (3.6)

Let a12 = 0.5(k(−11)

12 + k12), a21 = 0.5(k(−12)

21 + k21), a22 = 0.5(k(−12)

22 + k22).In this case formula (3.6) can be rewritten in the form∣

∣

k(−11)

12 +k12

∣

∣+∣

∣

k12+k(+11)

12

∣

∣+2∣

∣

k(−12)

21 +k21

∣

∣

−k(−11)

12 +k(+11)

12 ≤ 4(

k(−12)

22 +k22

)

. (3.7)

As condition (3.5) is valid, then k22 ≥ |k21|, k(−12)

22 ≥ |k(−12)

21 | and we have

|k(−12)

21 + k21| ≤ |k(−12)

21 | + |k21| ≤ k(−12)

22 + k22 .

Thus instead of (3.7) we have to prove that

|k(−11)

12 + k12| + |k12 + k(+11)

12 | − k(−11)

12 + k(+11)

12 ≤ 2(k(−12)

22 + k22). (3.8)

1. Let assume that k(−11)

12 + k12 ≥ 0, k12 + k(+11)

12 ≥ 0. Then inequality (3.8)can be rewritten in the form:

k12 + k(+11)

12 ≤ k(−12)

22 + k22.

It is easy to see that this inequality is valid under condition (3.5).


12 + k12 ≥ 0, k12 + k(+11)

12 ≤ 0. In this case from (3.8)

we obtain: k(−12)

22 + k22 ≥ 0. From ellipticity condition (2.2) for ξ = (0, 1) we have0 < c1 ≤ k22 ≤ c2. Hence, the required inequality holds true.


12 + k12 ≤ 0, k12 + k(+11)

12 ≥ 0, then formula (3.8) hasthe form:

−k(−11)

12 + k(+11)

12 ≤ k(−12)

22 + k22.

This inequality is valid under condition (3.5).


12 + k12 ≤ 0, k12 + k(+11)

12 ≤ 0. In this case we rewriteinequality (3.8) in the form:

−k12 − k(−11)

12 ≤ k(−12)

22 + k22.

This inequality is true under condition (3.5).Hence, k1 ≤ 1 if condition (3.5) is satisfied. Analogously we prove that k2 ≤ 1,

k3 ≥ 1, k4 ≥ 1.


Theorem 2. Let us suppose, that for all x ∈ ωh condition (3.4) is satisfied. Thendifference scheme (2.3) is stable with respect to the right-hand side and the boundaryconditions, and for its solution the following a priori estimate is valid

‖y‖C ≤ max‖µ‖Cγ, c

−10 ‖ϕ‖C. (3.9)

Proof. It is easy to see that the coefficients B2k ≥ 0, k = 1, 4 without any limita-tions. The coefficients B2k−1 ≥ 0, k = 1, 4 under condition (3.4):

B1 =1

h1h2

(

a(+12)

22

h1

h2

−

|a12| + a12 + |a(+11)

12 | − a(+11)

12 + 2|a(+12)

21 |

4

)

≥

1

h1h2

(

a(+12)

22 k2 −|a12| + a12 + |a

(+11)

12 | − a(+11)

12 + 2|a(+12)

21 |

4

)

= 0,

B3 =1

h21

(

a(+11)

11 −

h1

h2

2|a(+11)

12 | + |a21| + a21 + |a(+12)

21 | − a(+12)

21

4

)

≥

1

h21

(

a(+11)

11 − k4

2|a(+11)

12 | + |a21| + a21 + |a(+12)

21 | − a(+12)

21

4

)

= 0,

B5 =1

h1h2

(

a22

h1

h2

−

|a12| − a12 + |a(+11)

12 | + a(+11)

12 + 2|a21|

4

)

≥

1

h1h2

(

a22k1 −|a12| − a12 + |a

(+11)

12 | + a(+11)

12 + 2|a21|

4

)

= 0,

B7 =1

h21

(

a11 −h1

h2

2|a12| + |a21| − a21 + |a(+12)

21 | + a(+12)

21

4

)

≥

1

h21

(

a11 − k3

2|a12| + |a21| − a21 + |a(+12)

21 | + a(+12)

21

4

)

= 0.

Coefficient A > 0, if the following condition is true

max

|a21|

2a22

,

|a(+12)

21 |

2a(+12)

22

≤

h1

h2

≤ min

2a11

|a12|,

2a(+11)

11

|a(+11)

12 |

.

This statement follows from the following inequalities

A =1

h21

(

a11 −h1

h2

|a12|

2

)

+1

h21

(

a(+11)

11 −

h1

h2

|a(+11)

12 |

2

)

+1

h1h2

(

a22

h1

h2

−

|a21|

2

)

+1

h1h2

(

a(+12)

22

h1

h2

−

|a(+12)

21 |

2

)

+ d ≥

1

h21

(

a11 −2a11

|a12|

|a12|

2

)

+1

h21

(

a(+11)

11 −

2a(+11)

11

a(+11)

12

|a(+11)

12 |

2

)

+1

h1h2

(

a22

|a21|

2a22

−

|a21|

2

)

+1

h1h2

(

a(+12)

22

|a(+12)

21 |

2a(+12)

22

−

|a(+12)

21 |

2

)

+ d = d > 0.

176 I.V. Rybak

Note, that the above condition is weaker than condition (3.4), i.e., it holds true if con-dition (3.4) is valid. We verify directly that for any grid node x ∈ ωh the coefficientD > 0:

D = A−

8∑

k=1

Bk = d(x) ≥ c0 > 0.

For x ∈ γh, the coefficients of the canonical form are given by: A = 1 > 0, B =0, D = 1 > 0 . Now, all the conditions of Theorem 1 are satisfied. A prioriestimate (3.3) provides the required inequality (3.9).

4. Convergence

Let us consider now the problem of convergence of the proposed difference scheme.Substituting y = z + u into equations (2.3) we get the following problem for theerror of the discrete solution

Λz − dz = −ψ, x ∈ ωh ,

z = 0, x ∈ γh ,

(4.1)

where ψ = Λu − du + ϕ denotes the error of approximation of difference scheme(2.3) corresponding to the exact solution of differential problem (2.1). It was shownabove that the proposed difference scheme approximates the given differential prob-lem with the second order, thus

‖ψ‖C = M(h21 + h

22),

where M > 0 is a positive constant which does not depend on the grid steps h1, h2.

Using Theorem 2 for the solution of problem (4.1), it can be verified that thefollowing theorem takes place.

Theorem 3. Let us suppose that for all x ∈ ωh, condition (3.4) is satisfied. Thenthe solution of difference scheme (2.3) converges to the exact solution of differentialproblem (2.1), and the following a priori estimate

‖y − u‖C ≤

M

c0

(

h21 + h

22

)

is valid.

Remark 1. Results above can be easily extended to p–dimensional (p ≥ 2) ellipticequations with mixed derivatives.

Remark 2. The proposed approach can be also applied for the development of theconservative monotone difference schemes for multidimensional parabolic equationswith mixed derivatives.


5. Numerical results

To solve problem (2.1) by means of difference scheme (2.3) we use the modifiedstrongly implicit method [9]. Therefore, we reduce difference scheme (2.3) to thesystem of algebraic equations

[A]y = C .

Here A is a nine-diagonal matrix. Then we consider matrix [A + P ], which is theproduct of the lower triangular matrix [L] and the upper triangular matrix [U ], anddevelop the iterative process

[A+ P ]yn+1 = C + [P ]yn.

Since [A+ P ] = [L][U ] we obtain the following numerical algorithm

[L][U ]yn+1 = C + [P ]yn.

Matrices [L], [U ] and [P ] are defined in [9].

Numerical experiments were carried out in domain G = [0, 1]×[0, 1]. We choosethe coefficients: k11 = 1, k12 = k21 = cos(π(x1 + x2)), k22 = 1, q = 1 . It iseasy to see that kαβ satisfy ellipticity condition (2.2). The exact solution is given asu = sin(4πx1) sin(4πx2) . By substituting the exact solution into (2.1), we obtainthe boundary conditions and the right-hand side f .

Table 1. The convergence order of difference scheme (2.3).

N ×N 32 × 32 64 × 64 128 × 128 256 × 256 512 × 512

zN 0.043264 0.010734 0.002687 0.000654 0.000167

DN 0.049900 0.010923 0.002690 0.000672 0.000164

pN 2.19 2.02 2.00 2.03 2.04

The results of the numerical experiments are presented in Tab. 1, where

zN = max

x∈ωh

|yh(x) − u(x)|

is the global error of the discrete solution. Since the exact solution is usually un-known, we have computed the solution on the grids ωh, ωh/2, ωh/4, etc. Then theaposteriori error estimate of the solution yh can be obtained by using the Rungeestimator:

DN =

1

3maxx∈ω2h

|yh(x) − y2h(x)| .

Here we take the difference between the values of the solution on the grid with N/2nodes and the solution at the same point on the grid with N nodes.

The second aposteriori estimator pN = log2(DN/2

/DN) estimates the conver-

gence order of the approximation yh.

178 I.V. Rybak

6. Conclusions

In this paper new difference scheme for elliptic equations with mixed derivativesand alternating coefficients is presented. The proposed scheme is conservative, hasthe second order of approximation and satisfies the grid maximum principle. For thedeveloped numerical algorithms the a priori estimates of stability and convergencein the uniform norm are obtained.

The proposed approach to the construction of monotone conservative differenceschemes can be also applied to the development of monotone and conservative nu-merical algorithms for multidimensional parabolic equations with mixed derivatives.

References

[1] T. Arbogast, M.F. Wheeler and I. Yotov. Mixed finite elements for elliptic problems withtensor coefficients as cell-centered finite differences. SIAM J. Numer. Anal., 34(2), 828– 852, 1997.

[2] P. Crumpton, G. Shaw and A. Ware. Discretization and multigrid solution of ellipticequations with mixed derivative terms and strongly discontinuous coefficients. J. Com-put. Phys., 116, 343 – 358, 1995.

[3] S. Godunov and V. Ryabenkii. Difference schemes. Nauka, Moskow, 1973. (in Russian)[4] J. Hyman, J. Morel, M. Shashkov and S. Steinberg. Mimetic finite difference methods

for diffusion equations. Computational Geosciences, 6, 333 – 352, 2002.[5] A. Samarskii. The theory of difference schemes. Marcel Dekker, Inc., New York–Basel,

2001.[6] A. Samarskii and V. Andreev. Difference methods for elliptic equations. Nauka,

Moskow, 1976. (in Russian)[7] A. Samarskii, P. Matus, V. Mazhukin and I. Mozolevski. Monotone difference schemes

for equations with mixed derivatives. Computers and Mathematics with Applications,44, 501 – 510, 2002.

[8] A. Samarskii, V. Mazhukin, P. Matus and G. Shishkin. Monotone difference schemesfor equations with mixed derivatives. Mathematical Modeling, 13(2), 17 – 26, 2001.

[9] G. Schneider and M. Zedan. A modified strongly implicit procedure for the numericalsolution of field problems. Numerical Heat Transfer, 4, 1 – 19, 1981.

[10] G. Shishkin. Grid approximation of the singularly perturbed boundary-value problemfor quasilinear parabolic equations in the case of total degeneracy with respect to spacevariables. In: Yu. A. Kuznetsov(Ed.), Numerical methods and mathematical modeling,Russian Academy of Sciences, Institute of Computational Mathematics, Moskow, 103 –128, 1992.

[11] W. Voligt. Finite-difference schemes for parabolic problems with first and mixed secondderivatives. Z. angew. Math. und Mech., 68(7), 281 – 288, 1988.

Monotoniškos ir konservatyvios baigtiniu skirtumu schemos eliptinio tipo lygtims sumišriomis išvestinemis

I. Rybak

Straipsnyje nagrinejamos eliptinio tipo lygtys su mišriomis išvestinemis. Šioms diferencia-linems lygtims pasiulytos naujos antros eiles baigtiniu skirtumu schemos, kurios yra mono-toniškos ir konservatyvios. Sukonstruoti algoritmai tenkina skaitini maksimumo principa, kaikoeficientai prie mišriuju išvestiniu gali buti bet kokio ženklo. Gauti aprioriniai iverciai mak-simumo normoje. Irodyta baigtiniu skirtumu schemu stabilumas ir konvergavimas.

!"#$%'&)($+*,-#.0/1213.546

179–192c© 2004 Technika ISSN 1392-6292

APPLICATIONS OF INSPECTION GAMES

R. AVENHAUS

Universität der Bundeswehr München

Werner-Heisenberg-Weg 39, 85577 Neubiberg

E-mail: 798:;<7=5>?#@A;BC9DFE#79G5@AHJIK=;5@ALMNOE=:F;P<:;JIRQ:

Received July 27, 2004; revised September 9, 2004

Abstract. An inspection game is a mathematical model of a non-cooperative situation wherean inspector verifies that another party, called inspectee, adheres to legal rules. The inspectorwishes to deter illegal activity on the part of the inspectee and, should illegal activity never-theless take place, detect it with the highest possible probability and as soon as possible. Theinspectee may have some incentive to violate his commitments and violation, if observed, willincur punishment. Therefore if he chooses illegal behaviour, the inspectee will wish to avoiddetection with the highest possible probability.

Three examples of applications are presented. The first one deals with random controls inpublic transportation systems. The second one describes the problem of verification of armscontrol and disarmament in a very general way. The third one deals with inspections over timewhich are important in the context of non-proliferation verification.

Key words: Extensive form game, interim inspection, Nash equilibrium, normal form game,public transportation, verification of arms control and disarmament agreements

1. Introduction

An inspection game is a mathematical model of a situation where an inspection au-thority, called inspector, verifies that another party, called inspectee, adheres to cer-tain legal rules [4]. This legal behaviour may be defined, for example, by an armscontrol treaty, and the inspectee has a potential interest in violating these rules. Typ-ically, the inspector’s resources are limited so that verification can only be partial.

A mathematical analysis should help in designing an optimal inspection scheme,where it must be assumed that an illegal action is executed strategically. This de-fines a game theoretic problem, usually with two players, inspector and inspectee.In some cases, several inspectees are considered as individual players. Game the-ory is not only adequate to describe an inspection situation, it also produces resultswhich may be used in practical applications. But what does that mean? Theoreticiansand practitioners have, as we know, very different views about applications. Insteadof discussing this question in an abstract manner, in this paper three cases will be

180 R. Avenhaus

presented which illustrate three different kinds of applications. For that purpose, wedefine operational, conceptual and structural models.

The first one which deals with random controls of passengers using public trans-portation systems gives a concrete advise what effort the inspector should spend inorder to achieve his objectives. The second one which describes the problem of theverification of arms control and disarmament in a very general way provides insightinto the nature of the inspection problem. The third one which deals with inspec-tions over time which are important in the context of non-proliferation verificationshows how sensibly the best inspection strategies depend on assumptions about theinformation the inspectee gains, or does not gain, in the course of the game. Whereasthese three examples by no means do exhaust the wealth of models developed in thepast forty years - the first serious attempts were made in the early 1960s where gametheoretic studies of arms control inspections were commissioned by the US ACDA[10] – they should at least give an idea of what inspection models can achieve andfurthermore that each inspection problem has its own characteristics which requirenew models and appropriate solution techniques.

Since in this paper the emphasis is put on the modelling aspect and furthermoreon the use of the models for practical applications, proofs are only sketched andresults are not presented in form of theorems; both proofs and theorems can be foundin the references.

2. Operational Model: Passenger Ticket Control

In its edition of July 8th, 1997, the daily Süddeutsche Zeitung reports about the com-plaints by the city treasurer of Munich regarding the passenger ticket control appliedwithin the area of the Munich Transport and Fares Tariff association (MünchnerVerkehrs- und Tarifverbund, MVV). The deployment of inspectors was not worth-while since they make up for only about half of what they cost themselves by charg-ing the extra fares (fines) i.e., the employment of them was not profitable. It is obvi-ous that there must be an optimum high incidence of controlling: if there was onlyone inspector many passengers would go without paying, i.e., this one single inspec-tor would collect a lot of fines which is certainly not in line with the interest of theMVV, although the inspector would pay off. If on the other hand all passengers werechecked, all of them would pay for the fares. This would please the MVV, how-ever, the numerous and expensive ticket inspectors would not take in any fines at all.Where does thus the optimum regulation for the entire MVV system lie?

Since the behaviour of passengers, of whom one can assume strategical con-duct i.e. reflections regarding payment or non-payment (morale aspects should bedisregarded in this incident), must be taken into account when the assessment of theoptimum frequency of controls by the MVV is made, a decision theoretical, preciselya game theoretical analysis of the problem is required.

The ”game” is conducted by the inspector, representing the MVV on one part,the frequency of controls being his strategic variable, and the passenger on the otherside who decides between the alternatives of paying (legal behaviour) or not payingthe fare (illegal behaviour).

Applications of Inspection Games 181

Let f be the normal fare, b the fine and e the costs of controls per passenger. Weassume e < b. Then the payoffs to the two „players“ (inspector, passenger) in thefour possible situations (outcomes) are

(f − e,−f) for control and legal behaviour,

(f,−f) for no control and legal behaviour,

(b − e,−b) for control and illegal behaviour,

(0, 0) for no control and illegal behaviour. (2.1)

We consider the normal form indicated in Table 1 showing a two-person-game be-tween the MVV being represented by the inspector as first player and the passengeras second. In this diagram the pure strategies of the first player (control/no control)are depicted as rows and the second player’s as columns (legal/illegal behaviour); inthe individual squares the payments to the first player resulting from the respectivecombination of strategies are put down on the left bottom and those to the secondplayer on the top right.

Table 1. Normal form of the two person game between the inspector representing the MVVand the passenger. The arrows indicate the preference directions of the two players.

←−

↓

inspector/

passenger legal behaviour illegal behaviour

(q) (1-q)

control -f -b(p) f-e b-e

no control -f 0(1-p) f 0

↑

−→

Taking this formulation of the problem we ignore the costs on the part of theMVV for maintenance of the business since these do not influence the decisions ofboth players immediately and also for the same reason we ignore the ideal or materialgain the passenger has from his trip.

According to John Nash [8], one of the Economics Nobel prize winners of 1994,we understand by a solution of this game a pair of equilibrium strategies implyingthe quality that if one of the two players deviates unilaterally from his equilibriumstrategy he cannot improve his payment. In so doing we ignore the difficult problemof the existence of multiple equilibria since in our cases they do not occur.

Since according to Table 1 the preference directions of the two players, i. e., theincentive to deviate from a chosen strategy go cyclical, there is no equilibrium inpure strategies. The inspector will thus control with probability p, and the passengerwill behave legally with probability q. The expected payments to the two players aregiven in this case by

E1(p, q) = (f − e) p q + (b − e) p (1 − q) + f (1 − p) q ,

E2(p, q) = −f p q − b p (1 − q) − f (1 − p) q . (2.2)

182 R. Avenhaus

If we designate the mixed equilibrium strategies of the two players as p∗

, and q∗

, andthe equilibrium payments as E

∗

i = Ei(p∗

, q∗), i = 1, 2, the equilibrium conditions

according to John Nash are

E∗

1 (p, q) > E1(p, q∗) for all p ∈ [0, 1] ,

E∗

2 (p, q) > E2(p∗

, q) for all q ∈ [0, 1] . (2.3)

In our case, the equilibrium strategies can be determined so that the adversary isindifferent as regards to the choice of his own strategy, see e.g. Morris [7]. As aresult, the equilibrium strategies and – payments are given as follows [1]:

p∗ =

f

b

, E∗

1 = f

(

1 −

e

b

)

, (2.4)

q∗ = 1 −

e

b

, E∗

2 = −f . (2.5)

Thus in the equilibrium the passenger with a positive probability 1 − q∗ behaves

illegally, in the mean, however, he pays the same price he would pay if he alwaysbehaved legally. The reduced price achieved by dodging the fare is compensated bythe obligatory fine.

The mean value of the control expenditure by the MVV per passenger is e p,

while the profit from the fines is b p (1 − q). Thus the difference is

e p− b p (1 − q) =(

e − b (1 − q))

p T 0 fore

b

T 1 − q .

If the passenger chooses his equilibrium strategy q∗ given by (2.5), the following

holds(

e − b (1 − q∗)

)

p = 0 (2.6)

for any control probability p, i.e., the investment of control is just being compensatedby the amount of fines taken in. It must be noted that these considerations only includethe parameters e and b, but not the fare f of the trip.

The optimum control probability p∗ satisfies the condition p

∗

b = f , which canbe understood intuitively: if the passenger behaves legally he has to pay −f, whereashis expected payment in case of illegal behaviour is −b p

∗

. Thus the optimum controlprobability renders the passenger indifferent as regards to the strategy to be chosenby him.

Thus, in conclusion, this game theoretical model gives an advise how frequentlypassengers should be controlled if the fare and the fine are fixed. It should be men-tioned that the actual figures for Munich approximately satisfy (2.4). One may spe-culate why then, according to the City Treasurer’s complaints, the equilibrium con-dition (2.6) is not satisfied. Certainly one reason is that the inspections are nor purelyrandom: passengers who systematically do not buy tickets frequently recognize theinspectors already before they can do their job. Another reason are the considerabledeviations of passengers in frequency, hour of the day and dwelling time from theaverage which is not adequately taken into account by the inspectors.

A stratification of inspection procedures which would be required here has beenanalysed in the context of arms control, see [2]. There it turns out that the inspection


efforts in the different strata have to be the higher, the more profitable illegal ac-tions are, and in turn that the illegal actions are concentrated there as well. It will bepointed out in the last section, however, that one has to be very careful in predictingresults of yet unsolved problems.

Finally, it should be mentioned that there is also not intentional illegal behaviour– e.g., passengers use monthly tickets but forget to take them with them – whichcan be modelled as well [1]; the analysis shows, however that it does not change theresults in a significant way.

3. Conceptual Models: Arms Control and DisarmamentVerification

As a second case, let us consider an international arms control and disarmamentagreement, for example the Treaty for the Non-Proliferation of Nuclear Weapons,or the Chemical Weapons Convention. A State who signs this agreement is obligednot to act illegally in that sense that he does not do anything that is forbidden by theagreement, for example to acquire nuclear or chemical weapons.

Let us assume furthermore that, together with the agreement, a verification sys-tem is established which means that an international authority verifies with the helpof well-defined measures - measurements, on-site inspections and others - that theinspected State adheres to the provisions of the agreement. For the Non-ProliferationTreaty, for example, the International Atomic Energy Agency (IAEA) plays that role.The purpose of the verification is to deter the State from illegal behaviour or, shouldhe behave illegally, to detect this with as high a probability and as quickly as possible.

On the other hand, the inspected State may have some incentive to violate hiscommitments – otherwise the situation is pointless, we will come back to this issue,– and violation, if observed, will incur punishment of the State. Therefore, if hechooses illegal behaviour, the inspected State will wish to avoid detection with thehighest possible probability.

In the following we will describe this conflict situation between the verificationauthority (in short inspector) and the State (in short inspectee) with the help of anon-cooperative two-person game. Let the payoffs to the inspector as the first playerand to the inspectee as the second player be given by

(0, 0) for legal behaviour of the inspectee,

(−a,−b) for detected illegal behaviour of the inspectee,

(−c, d) for undetected illegal behaviour of the inspectee. (3.1)

Note that inspection costs are not taken into account explicitly. We assume

0 < a < c, 0 < b, 0 < d , (3.2)

the first inequality expresses the fact that the highest priority of the inspector is todeter the inspectee from illegal behaviour.

In keeping with common notation, let us call 1 − β be the probability to detectillegal behaviour. Then the expected payoffs to the two players are

184 R. Avenhaus

(0, 0) for legal behaviour of the inspectee, (3.3)

(−a (1 − β) − c β, −b (1 − β) + d β) for illegal behaviour of the inspectee.

Furthermore we assume that the inspector, in a concrete situation, decides eitherto verify or not, and the inspectee, in turn, to behave legally or not. The normal formof this two-by-two-game is given by Table 2.

Table 2. Normal form of the two person game between a State (inspectee) and theverification authority (inspector). False alarms are not possible. The arrows indicatethe preference directions of the two players if (3.4) is fulfilled.

←−

l

inspector/


Verification 0 −b(1− β) + dβ

0 −a(1− β)− cβ

No 0 d

verification 0 −c

↑

−→

As a solution of this game we consider again the Nash equilibrium. Using themethod of incentive directions, we see immediately that legal behaviour is the onlyequilibrium strategy of the inspectee if

0 > −b (1− β) + d β ,

or, equivalently, if

β <

1

1 + db

. (3.4)

Thus, as a result we see that the inspectee will be induced to legal behaviour if thenon-detection probability is smaller than some threshold, which is the lower, thelarger the ratio between gain in case of undetected illegal behaviour and the sanc-tions b in case of detected illegal behaviour is. Otherwise he will behave illegally.Alternatively one may say that the inspectee will behave legally if either the proba-

bility of no detection or the ratiod

b

is small enough.

Now let us consider a more complicated problem: Let us assume that false alarmsmay happen with probability α. Let the payoffs to the two players in case of a falsealarm be −e < 0 and −f < 0. We assume

0 < e < a < c, 0 < f < b, 0 < d .

Then the normal form of the verification game is given by Table 3.We see immediately that legal behaviour is not an equilibrium strategy. However,

for−fα > −b (1− β) + d β ,

or equivalently, for


Table 3. Normal form of the two person game between a State (inspectee) and theverification authority (inspector). False alarms occur with probability α. The arrowsindicate the preference directions of the two players if (3.5) is fulfilled.

←−

↓

inspector/


(q) (1-q)

Verification −fα −b(1− β) + dβ

−eα −α(1− β)− cβ

No 0 d

verification 0 −c

↑

−→

β <

−f

b + d

α +b

b + d

(3.5)

there exists a Nash equilibrium in mixed strategies: The inspectee will act illegallywith probability q

∗ as given by

1

q∗

= 1 +c − a

e

1 − β

α

. (3.6)

Since, as mentioned initially, and contrary to the previous example, where moraleproblems were not dominating, the purpose of the treaty is that the State fulfills theprovisions of the treaty, the question arises if there is any possibility to induce theinspectee to legal behaviour.

Let us assume that α is a strategic variable of the inspector. Of course, β dependson α. For unbiased test procedures we have

α + β < 1 . (3.7)

Let us assume in addition

β = 1 for α = 0 , and β = 0 for α = 1 , (3.8)

and furthermore,dβ

dα

< 0,

d2β

dα2

< 0. (3.9)

Then the problem of choosing an appropriate value of α can be represented graphi-cally as given in Figure 1a.

We see that for α = α0 and β = β0 = β(α0) as defined in the figure theinspectee is indifferent between legal and illegal behaviour.

Now let us change the rules of the game [2]. Instead of the inspector’s two al-ternatives considered so far, namely verifying with a fixed false alarm probability ornot verifying, we now assume that the inspector always verifies, and that his set ofstrategies consists in the possible choices of the false alarm probability. In addition,he will announce his strategy in a credible way. The extensive form of this so-calledinspector leadership game is given by Figure 1b .

186 R. Avenhaus

db

b

+

db

fb

+

-

( )ab

db

b

db

f

+

+

+

-

a

1

1¾®¾a

0a

0b

0

a

÷÷ø

öççè

æ×-

×-

a

a

f

e ( )( ) ÷÷

ø

öççè

æ×+-×-

×--×-

bb

bb

db

ca

1

1

inspector

inspectee

legal illegal

a) b)

Figure 1. a) graphical representation of the value of which makes the inspectee indifferentbetween legal and illegal behaviour, b) extensive form of the leadership game between theverification authority (inspector) and the State (inspectee).

According to the backward induction procedure the inspectee will

behave legally, if − f α > −b (1− β) + d β ,

be indifferent, if − f α > −b (1− β) + d β ,

behave illegally, if − f α > −b (1− β) + d β .

(3.10)

or, equivalently, with α0 as defined in Figure 1a,

behave legally, if α > α0 ,

be indifferent, if α = α0 ,

behave illegally, if α < α0 .

(3.11)

For this best reply of the inspectee, the payoff to the inspector is given as representedgraphically in Figure 2.

a

0

0a

1

e-

a-

C-

a×- e

( ) bb ×--×- ca 1

Figure 2. Payoff to the inspector for the best reply of the inspectee.

One sees immediately that the payoff of the inspector has its maximum at α0.

Now, surprisingly enough it can be shown that α = α0 and


legal behaviour, for α ≥ α0 ,

illegal behaviour, for α < α0

(3.12)

are Nash equilibrium strategies. This means in effect that the inspector chooses α =α0 and that consequently, the inspectee acts legally – even though at that point he isindifferent between legal and illegal behaviour.

In discussions on the usefulness of verification in general, arms control and dis-armament officials, administrators and political scientists have frequently criticized,and still do so, that game theorists or more generally, analysts working quantitatively,always assume that the State might behave illegally even though he has ratified theagreement under consideration. In the beginning we mentioned that without this as-sumption the situation would be pointless. Now we can be more precise: in order toshow that appropriate verification on one hand and legal behaviour of the State onthe other are equilibrium strategies we have to study deviations – quite in the spiritof Nash’s equilibrium concept.

4. Structural Models: Interim Inspections

Finally, as a third case, we consider a single inspected object, for example a nuclearor chemical facility subject to verification in the framework of an international treaty,and a reference period of one time unit (e.g. one calendar year). In order to separatethe timeliness aspect of routine inspection from the overall goal of detecting illegalactivity, we assume that a thorough and unambiguous inspection takes place at theend of the reference period which will detect an illegal activity with certainty once ithas occurred.

In addition there are a number of less intensive and strategically placed „interim“inspections which are intended to reduce the time to detection below the length of thereference period. An interim inspection will detect a preceding or coincident illegalactivity, but with some lower probability. Again in keeping with common notation,we call this probability 1 − β, where β is the probability of an error of the secondkind, or non-detection probability.

Associated with each interim inspection which is not preceded by an illegal ac-tion is a corresponding probability of an error, the false alarm probability α. More-over, again only an unbiased inspection procedure is considered.

We assume that, by agreement, k interim inspections will occur within the refer-ence period. For convenience we label the inspections backwards in time. Also welabel the beginning of the reference time tk+1 and the end t0, so we have

0 = tk+1 < tk < . . . < t1 < t0 = 1. (4.1)

The utilities of the protagonists (inspector, inspectee) are taken to be as follows:

(0, 0) for legal behaviour over the reference time, and no false alarm,

(−le,−lf) for legal behaviour over the reference time, and l false alarms,

l = 1, . . . , k

(−α∆t, d∆t − b) for detection of illegal activity after elapsed time ∆t ≥ 0 ,

188 R. Avenhaus

where0 < e < a, 0 < f < b < d.

Thus the utilities are normalized to zero for legal behaviour without false alarms,and the loss (profit) to the inspector (inspectee) grows proportionally with the timeelapsed to detection of an illegal action. A false alarm is resolved unambiguouslywith time independent costs −e to the inspector and −f to the inspectee, where-upon the game continues. The quantity b is the cost to the inspectee of immediatedetection. Note that, if b > d, the inspectee will behave legally even if there are nointerim inspections at all. Since interim inspections introduce false alarm costs forboth parties, there would be no point in performing them.

The extensive form of the inspection game for one single observable interiminspection is represented graphically in Figure 3. Without going through the analysiswhich uses similar techniques as sketched before, we just present the results [3]:

Figure 3. Extensive form of the two person game between the inspector and the inspectee forone interim inspection.

Let t1 : 0 < t1 < 1 be the set of pure strategies of the inspector, and theprobabilities g2, g1 to start illegal actions at time moments t = 0, t1 be the mixedbehavioral strategies of the inspectee. Let V2 and W2 be the payoffs to the two players(here we use a notation different from that of the second section in order to remainconsistent with the notation in the published literature). Taking into account that theinspectee will behave legally if his payoff is larger than in case he behaves illegally,and vice versa, the equilibrium strategies and payoffs are given as follows.

Under the assumption

b

d

<

1

2 − β

(

1 +f α

d

)

(4.2)


in equilibrium the inspectee acts illegally, with payoffs and strategies given by

V∗

2 = −a A2 − e α B2 ,

W∗

2 = d A2 − f α B2 − b ,

t∗

1 = (1 − β) A2 −f α

d

(

(1 − β) B2 + β

)

,

g∗

2 = A2, g∗

1 = 1,

where A2 and B2 are given by

A2 =1

2 − β

, B2 =1 − β

2 − β

. (4.3)

Under the assumption (we exclude equality being practically not important)

b

d

>

1

2 − β

(

1 +fα

d

)

(4.4)

in equilibrium the inspectee acts legally, with the following payoffs to the two antag-onists:

V∗

2 = −ea, W∗

2 = −fα .

The equilibrium strategy of the inspector is not unique; it is given by what M. Kilgour[6] called the cone of deterrence:

1 −

b

d

6 t∗

1 61

1 − β

(

b

d

−

fα

d

− β

)

. (4.5)

It can be shown that the equilibrium strategy of the inspector in case of illegal be-haviour of the inspectee is an element of the cone of deterrence (4.5). Thus, theinspector is on the safe side if he always uses the former one.

It is also possible to generalize this solution to more than one interim inspectionhowever, the analysis gets rather involved since non-trivial information sets have tobe taken into account and furthermore, since unrealistic solutions may occur wheresome interim inspections may have to be conducted right at the beginning of thereference time [3]. Nevertheless are the realistic solutions for more than one interiminspection of the same structure as that given by (4.3), with more complicated ex-pressions for A and B; whereas it is not easy to find them, it is straightforward toprove their validity via complete induction.

We will not delve into these intricacies. Instead, we consider an inspection prob-lem which differs from the previous one only by the fact that now the interim inspec-tions are unobservable or – formally the same – that prior commitment on the partof the inspectee is assumed. That means that now we consider a simultaneous ratherthan a sequential game.

This game has been analysed by H. Diamond [5] for an arbitrary number ofinterim inspections, however, without taking into account false alarms. The analysisof the game for one interim inspection and the possibility of false alarms is due toSohrweide [9] and again we just present the main results.

190 R. Avenhaus

For one interim inspection an equilibrium strategy for the inspector is to choosehis single interim inspection time t1 on an interval 0 < t1 6 κ < 1 according to thedistribution function

F∗ (t1) = −

1

1− β

ln(

1 −

t1

1 −

f

dα

1−β

)

, (4.6)

where κ is given by

κ =

(

1 −

1

e1−β

) (

1 −

1

d

α

1 − β

)

. (4.7)

Under the assumptionb

d

> 1 − κ (4.8)

the inspectee behaves illegally; he randomizes similarly, however, his distributionfunction Q

∗(t), as given by

Q∗(t) =

1 − κ + ea

α1−β

1 + ea

α1−β

for 0 6 t 6 κ, (4.9)

has an atom at t = 0 :

Q∗(0) = 1 −

κ

1 + ea

α1−β

> 0 .

Thus, both players necessarily play mixed strategies in equilibrium, with payoffs

V∗

2 = −a

[

β (1 − κ) − (1 − κ +e

a

α

1 − β

) ln(

1 −

κ

1 − t + ea

α1−β

)]

,

W∗

2 = d (1 − κ) − f − b . (4.10)

If (4.8) is not fulfilled (we exclude equality being practically not important) the in-spectee behaves legally, with payoffs being the same as in the previous model.

Whereas the inspectee’s payoff W∗

2 as given by (4.10) can be understood easily– it is just his payoff in case he starts his illegal action at time t = κ – this is not soeasy in case of the inspector’s payoff unless we have e = −f which can hardly bejustified.

It turns out, not surprisingly, that the unobservability places the inspectee at adisadvantage: his payoff in case of illegal behaviour is smaller than that for an ob-servable inspection, and the limit for b

dto induce the inspectee to legal behaviour is

lower.

At first sight it is very surprising that for one well specified inspection problemdifferent assumptions about the information the inspectee gains during the courseof the game or does not gain, lead to totally different results: In the first case theinspector plays in equilibrium a pure, in the second case a mixed strategy. This isthe general lesson to be drawn from this very concrete inspection problem: Even ifone has studied so many different problems, one hardly will be able to predict theoutcome of a new or even only modified one.


References

[1] R. Avenhaus. Entscheidungstheoretische Analyse der Fahrgastkontrollen. Der Nahver-kehr 9, Alba Fachverlag Düsseldorf, 27 – 30, 1997.

[2] R. Avenhaus and M.J. Canty. Compliance Quantified: An Introduction to Data Verifica-tion. Cambridge University Press, 1996.

[3] R. Avenhaus and M.J. Canty. Playing for time: A sequential inspection game. TheEuropean Journal for Operational Research, 2004. (in print)

[4] R. Avenhaus, B. von Stengel and S. Zamir. Inspection games. In: R.J. Aumann andS. Hart(Eds.), Handbook of Game Theory, volume 3, North-Holland, Amsterdam, 1947– 1987, 2000.

[5] H. Diamond. Minimax policies for unobservable inspections. Mathematics of Opera-tions Research, 7(1), 139 – 153, 1982.

[6] M. Kilgour. Site selection for on-site inspection in arms control. Arms Control Contemp.Security Policy, 13, 439 – 462, 1992.

[7] P. Morris. Introduction to Game Theory. Springer Verlag, 1994.[8] J. Nash. Noncooperative games. Annals of Mathematics, 54(2), 286 – 295, 1951.[9] K. Sohrweide. Diamond’s inspektionsspiel mit fehlern 1 und 2. Diploma thesis, Univer-

sität der Bundeswehr München, Germany, 2002.[10] United States Arms Control and Disarmament Agency (USAACDA). Applications of

Statistical Methodology to Arms Control and Disarmament. Mathematica, PrincetonNJ, Contract No. ACDA/ST-3, 1963.

192 R. Avenhaus

Kontroles lošimu taikymai

R. Avenhaus

Kontroles lošimas yra matematinis modelis tam tikru nekooperaciniu situaciju, kai inspek-torius kontroliuoja kita puse, skatindamas korektiška elgesi. Inspektorius turi atbaidyti už-draustus veiksmus su kuo galima didesne tikimybe ir kuo greiciau. Tai reiškia, kad už nus-tatytu taisykliu pažeidima mokama tam tikra bauda ir šie pažeidimai aptinkami su maksi-malia tikimybe. Straipsnyje nagrinejami trys šio modelio taikymo pavyzdžiai: atsitiktine vi-suomeninio transporto keleiviu kontrole, ginklu kontroles modelis ir paplitimo ribojimo pertam tikra laika modelis.

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193–200c© 2004 Technika ISSN 1392-6292

THE SECOND BOUNDARY VALUE PROBLEM OFRIEMANN’S TYPE FOR BIANALYTICALFUNCTIONS WITH DISCONTINUOUSCOEFFICIENTS

I. B. BOLOTIN


Przevalskogo 4, 214000 Smolensk, Russia

E-mail: 789:#;=<>?2>A@7B:C?7D=@FEHG2I

Received October 13, 2003; revised February 5, 2004

Abstract. The paper is devoted to the investigation of one of the basic boundary valueproblems of Riemann’s type for bianalytical functions with discontinuous coefficients. In thecourse of work there was made out a constructive method for solution of the problem in a unitcircle. There was also found out that the solution of the problem under consideration consistsin consequent solutions of two Riemann’s boundary value problems for analytical functionsin a unit circle. Besides, the example is constructed.


1. Introduction

Let L = t : |t| = 1, D+ = z : |z| < 1 and D

− = C\D+

⋃

L.Let Gk(t) (k = 1, 2) – given on the contour L functions, satisfying the condi-

tion of Holder everywhere on L, except for a finite number of points, where theyhave simple discontinuity, and Gk(t) 6= 0 on the contour. Also we shall consider,that function G0(t) has derivative which satisfies the condition of Holder, exceptfor a finite number of points, where it may have simple discontinuities. Hereinafter,following N.I. Muskhelishvili (see, for example, [2]), we shall call points of discon-tinuity of the functions G0(t), G

′

0(t) and G1(t) as knots, and remaining points of thecontour L we shall name ordinary. Besides we shall rank all points of discontinuityof the function G0(t) and its derivative as knots of function G1(t).

Further we shall generally use terms and definitions accepted in [3].

JLKM2NPOQNSR5NUTVOXWYWe shall speak, that bianalytical function F

±(z) in domain D±

belongs to the class A2(D±)

⋂

I(2)(L), if it proceeds on the contour L together with

194 I.B. Bolotin

the partial derivatives∂

α+βF

±(z)

∂zα∂z

β(α = 0, 1; β = 0, 1), and so that boundary values

of this function and all specified derivatives satisfy on z the condition of Holdereverywhere, except for, possibly, knots, where the reversion in infinity of integrableorder is possible when α + β < 2.

It is required to find all piecewise bianalytical functions F (z) = F+(z), F−(z),

belonging to the class A2(D±)

⋂

I(2)(L), vanishing on infinity, limited near the

knots of the contour and satisfying in all ordinary points of L the following boundaryconditions:

F+(t) = G0(t)F

−(t) + g0(t), (1.1)

∂F+(t)

∂n+

= −G1(t)∂F

−(t)

∂n−

− g1(t), (1.2)

where∂

∂n+

(

∂

∂n−

)

– derivative on interior (exterior) normal to the contour L,

gk(t) (k = 0, 1) – given on L functions of the class H(1−k)(L), and g0(t) = (t −

c)γcg∗(t), c – any of knots of the function G0(t), γc > 0 - quite defined numbers.

Here, in equality (1.2), factor (−1) at G1(t) and g1(t) is entered for conveniencehereinafter.

We shall name the formulated problem as the second basic boundary value prob-lem of Riemann’s type for bianalytical functions with discontinuous coefficients inthe unit circle or in short the problem R2,2, and appropriate homogeneous problem(g0(t) ≡ g1(t) ≡ 0) shall be named as a problem R

02,2.

Let’s notice, that the problem R2,2, stated by F. D. Gakhov as one of basic bound-ary value problems for bianalytical functions (see, for example, [1], p. 316) in caseof continuous coefficients and smooth closed loops was explicitly investigated in thework of K. M. Rasulov (see [3]).

In the above mentioned statement the problem R2,2 is investigated in the presentwork for the first time.

2. About the Solution of the Problem R2,2

It is known (see [1, 3]), that any vanishing on infinity piecewise bianalytical functionF (z) with line of saltuses L is possible to represent as:

F (z) =

F+(z) = ϕ

+0 (z) + zϕ

+1 (z), z ∈ D

+,

F−(z) = ϕ

−

0 (z) + zϕ−

1 (z), z ∈ D−

,

(2.1)

where ϕ±

k (z) – analytical functions in domain D± (analytical components of piece-

wise bianalytical function), for which the following conditions are fulfilled:

Πϕ−

k ,∞ ≥ 1 + k, k = 0, 1;

here Πϕ−

k ,∞ means the order of the function ϕ−

k (z) in the point z = ∞.Let’s search for the solution of the problem R2,2 as

The Second Basic Boundary Value Problem 195

F (z) = f0(z) + (zz − 1)f1(z). (2.2)

Then the functions fk(z) (k = 0, 1) will be connected with analytical compo-nents of the required bianalytical function F (z) by the formulas:

ϕ0(z) = f0(z) − f1(z), ϕ1(z) = zf1(z). (2.3)

As known (see [1] p. 304)

∂

∂n±

= ±i

(

t′

∂

∂t

− t′

∂

∂t

)

, (2.4)

then taking into account (2.2) and the fact that the equality t =1

t

is fulfilled on L,

the boundary conditions (1.1) and (1.2) can be copied accordingly in the aspect:

f+0 (t) = G0(t)f

+0 (t) + g0(t), (2.5)

f+

1 (t) = G1(t)f−

1 (t) +1

2

(

−t

df+

0 (t)

dt

+ tG1(t)df

−

0 (t)

dt

+ g1(t)

)

. (2.6)

The equalities (2.5) and (2.6) represent boundary conditions of usual Riemann’sproblems for analytical functions with discontinuous coefficients in the unit circle(see [1] or [2]).

Thus, as a matter of fact, solution of the initial problem R2,2 is reduced to se-quential solution of two auxiliary problems of Riemann (2.5) and (2.6) in classes ofpiecewise analytical functions with the line of saltuses L. But as in the problem R2,2

we search the solutions, limited close to the knots of the contour and vanishing oninfinity, there arises the necessity in a choice of defined classes of analytical func-tions at the solution of auxiliary problems (2.5) and (2.6). Therefore, at first we shallfind out, in what classes it is necessary to search for solutions of boundary valueproblems (2.5) and (2.6).

From equalities (2.3) we can see that the function f−

0 (z) on the infinity shouldhave zero not below than the first order, and f

−

1 (z) – zero not below than the thirdorder.

Let’s study the behaviour of function F (z) near the knots of the contour L. Let c

be any of knots, then cc = 1 and |c| = 1.We have the following serieses of inequalities:

|F (z)| = |f0(z) + (zz − 1)f1(z)| ≤ |f0(z)| + |f1(z)||zz − 1|

= |f0(z)| + |f1(z)||(z − c + c)(z − c + c) − 1|

≤ |f0(z)| + 2|f1(z)||z − c| + |f1(z)||z − c|2; (2.7)

|F (z)| = |f0(z) + (zz − 1)f1(z)| ≥ |f0(z)| − |f1(z)||zz − 1|

= |f0(z)| − |f1(z)||(z − c + c)(z − c + c) − 1|

≥ |f0(z)| − 2|f1(z)||z − c| − |f1(z)||z − c|2. (2.8)

Thus, for the function F (z) would be limited close to the knots of the contour L,it is necessary and enough that the function f0(z) would be limited, and the functionf1(z) supposed the evaluation:

196 I.B. Bolotin

|f1(z)| ≤const

|z − c|α

, 0 ≤ α < 1. (2.9)

Really, if the function f0(z) is limited close to c and function f1(z) supposesthe evaluation (2.9), from inequalities (2.7) it follows, that the required bianalyticalfunction F (z) will be limited in a neighbourhood of the knot c.

Back, if the function F (z) of the class A2(D±)

⋂

I(2)(L) is limited close to c,

from inequalities (2.8) it follows, that the function f1(z) has to suppose the evalu-ation (2.9) (otherwise all solutions of the problem R2,2 will not be found), so thefunction f0(z) has to be limited in a neighbourhood of the knot c.

Therefore, it is required to search the solution of the problem (2.5) in the classof functions, vanishing on infinity and limited near the knots, and solution of theproblem (2.6) is required to search in the class of functions, having zero of the thirdorder on infinity and infinity of the integrable order near the knots of the contour L.

Let’s solve the boundary value problem of Riemann (2.5) using the method of-fered by F.D. Gakhov (see, for example, [1], p. 448).

Let index of the problem (2.5) be equal κ0 in the specified class.Then, if κ0 ≥ 0, a common solution of the problem (2.5) is set by the formula

(see [1, 2]):

f0(z) = X0(z)

1

2πi

∫

L

g0(τ)

X+

0 (τ)

dτ

τ − z

+ Pκ0−1(z)

, (2.10)

where X0(z) – canonical function of the problem (2.5), Pκ0−1(z) – the polynomialof a degree not higher then κ0 − 1 with arbitrary complex coefficients.

In the case when κ0 < 0, the solution of the problem (2.5) also will be expressedby the formula (2.10) with only one modification, that Pκ0−1(z) ≡ 0, at observanceof |κ0| conditions of solvability of the aspect:

∫

L

g0(τ)

X+

0 (τ)τ

k−1dτ = 0, k = 1, . . . , |κ0|.

Let’s define numbers γc specified in the statement of the problem R2,2. Let c1, c2, . . . ,

cm be knots of the function G0(t).Below we shall consider that

γck>

1

2π

(arg G(ck − 0) − argG(ck + 0)), k = 2, . . . , m

γc1>

1

2π

(argG(c1 − 0) − argG(c1 + 0) − 2πκ0). (2.11)

Further, on the found function f0(z) with the help of differentiation and in viewof the formulas Sokhotzky-Plemelj (see [4], p. 333, [1, 2]), we shall find out bound-

ary valuesdf

±

0 (t)

dt

of the functiondf0(z)

dz

.

Note 1. We shall notice, that if the knot c is not singular or c – singular, butln |G0(c − 0)| − ln |G0(c + 0)| = 0 from the conditions (2.11), it follows that the


functionsdf

±

0 (t)

dt

satisfy the condition of Holder everywhere on L except for, pos-

sibly, knots, where they may have a singularity of the integrable order (knots of the

first type). Otherwise functionsdf

±

0 (t)

dt

will have a singularity of the first order near

the knots (knots of the second type).

Further we shall solve the boundary value problem of Riemann (2.6).Let the index of the problem (2.6) be equal κ1 in the specified class.As it is known (see [1, 2]), if κ1 ≥ 3, a common solution of the problem (2.6) is

set by the formula:

f1(z) = X1(z)

1

2πi

∫

L

Q1(τ)

X+1 (τ)

dτ

τ − z

+ Pκ1−3(z)

, (2.12)

where X1(z) – canonical function of the problem (2.6), Pκ1−3(z) – the polynomialof a degree not higher then κ1 − 3 with arbitrary complex coefficients,

Q1(t) =1

2

(

−t

df+

0 (t)

dt

+ tG1(t)df

−

0 (t)

dt

+ g1(t)

)

.

If κ1 ≤ 2, the solution of the problem (2.6) also will be expressed by the formula(2.12) with only one modification that Pκ1−3(z) ≡ 0, at observance of −κ1 + 2conditions of solvability of the aspect:

∫

L

Q1(τ)

X+1 (τ)

τk−1

dτ = 0, k = 1, . . . ,−κ1 + 2.

Note 2. Generally speaking, absolute term Q1(t) of the problem (2.6) satisfiesthe condition of Holder everywhere on L except for, possibly, knots c1, c2, . . . , cm,where it may have singularity of the first order (knots of the second type), and re-maining knots, where it may have an integrable singularity. And, if the knot of thesecond type of the problem (2.5) is the singular knot of the problem (2.6), then theproblem R2,2 will be insoluble in the class A2(D

±)⋂

I(2)(L).

Further on the found functions f0(z) and f1(z), using the formulas (2.3), werestore analytical components of the required piecewise bianalytical function, andthen the piecewise bianalytical function F (z) itself under the formula (2.1).

Thus, the following basic outcome is fair.

Theorem 1. Let L = t : |t| = 1, D+ = z : |z| < 1 and D

− = C\D+

⋃

L.Then the solution of the problem R2,2 is reduced to the sequential solution of twoscalar boundary value problems of Riemann (2.5) and (2.6) with discontinuity coef-ficients in classes of analytical functions in the unit circle, and that the solution of theproblem (2.5) is searched in the class of functions vanishing on infinity and limitedin the knots of the contour; and the solution of the problem (2.6) is searched in theclass of functions, having on infinity zero of the third order and infinity of the inte-grable order in the knots of the contour L. The problem R2,2 is solvable if and only if

198 I.B. Bolotin

the problems (2.5) and (2.6) in the specified classes of functions are simultaneouslysolvable and knots of the second type are not singular for the coefficient G1(t) of theproblem (2.6).

Example 1. Let L = t : |t| = 1, D+ = z : |z| < 1 and D

− = C\D+

⋃

L.It is required to find all piecewise bianalytical functions F (z) = F

+(z), F−(z)belonging to the class A2(D

±)⋂

I(2)(L) vanishing on infinity, limited near the

knots of the contour and satisfying in all ordinary points L the following boundaryconditions:

F+(t) = G0(t)F

−(t) + (t − 1)3

2 (t + 1)3

2 , (2.13)

∂F+(t)

∂n+

= −t4 ∂F

−(t)

∂n−

− (3t2(t2 − 1)

1

2 + 2t2), (2.14)

Here

G0(t) =

1, t ∈ L1 = t : t = eis

, 0 ≤ s ≤ π,

−1, t ∈ L2 = t : t = eis

, π ≤ s ≤ 2π,

G1(t) = t4, g0(t) = (t − 1)

3

2 (t + 1)3

2 and g1(t) = (3t2(t2 − 1)

1

2 + t2).

Using equalities (2.2) – (2.4), the boundary condition (2.13) will take the follow-ing aspect:

f+

0 (t) = G0(t)f−

0 (t) + (t − 1)3

2 (t + 1)3

2 , (2.15)

Knots of the problem (2.15) are the points t = 1 and t = −1, in which functionG0(t) has simple discontinuity.

Let’s calculate the index of the problem (2.15). Let’s choose as the initial pointt = 1. We have,

G0(1 + 0) = 1 = ei0

, θ1 = 0,

the change of argument of the function G0(t) on the arc L1 will be equal

∆θ1 = [argG0(t)]L1= 0.

ThereforeG0(−1− 0) = 1 = e

i0.

Let G0(−1 + 0) = −1 = eiθ2 . Let’s choose a value θ2 so that the inequality is

fulfilled0 ≤ 0− θ2 < 2π,

That is θ2 = −π

The change of argument of the function G0(t) on the arc L2 will be equal to zero.So G0(1 − 0) = −1 = e

−iπ.Let’s define the whole number κ0, satisfying the following condition:

0 ≤ −π − 2πκ0 < 2π.

Thus, the index of the problem (2.15) will be equal to -1.The common solution of the problem will look like:


f+

0 (z) = (z2− 1)

3

2 ,

f−

0 (z) ≡ 0,

at observance of one condition of solvability:∫

L

τ(τ2− 1)dτ = 0,

which is obviously fulfilled.The boundary values of the derivative of the solution of the problem (2.15) will

be set by the formulas:

df+0 (t)

dt

= 3t(t2 − 1)1

2 , (2.16)

df−

0 (t)

dt

≡ 0. (2.17)

With the account (2.16) – (2.17), the boundary condition (2.14) will take theaspect:

f+

1 (t) = t4f−

1 (t) + t2. (2.18)

Generally speaking, the coefficient of the problem (2.18) is the continuous functionon L, but following the statement of the problem (2.18) the knots will be representedby the points t = 1 and t = −1.

Let’s calculate the index of the problem (2.18). Let’s choose as the initial pointt = 1. We have,

G1(1 + 0) = 1 = ei0

, θ1 = 0,

The change of argument of function G1(t) on the arc L1 will be equal

∆θ1 = [argG1(t)]L1= 4π.

ThereforeG1(−1 − 0) = 1 = e

i4π.

Let G1(−1 + 0) = −1 = eiθ2 . Let’s choose a value θ2 so that the inequality is

fulfilled−2π < 4π − θ2 ≤ 0,

That is θ2 = 4π

The change of argument of function G1(t) on the arc L2 will be equal 4π. SoG1(1 − 0) = 1 = e

i8π.Let’s define whole number κ1, satisfying the following condition:

−2π < 8π − 2πκ1 ≤ 0.

Thus, the index of the problem (2.18) will be equal to 4.Hence, the common solution of the problem (2.18) will look like:

f+1 (z) = z

2 + a1z + a0,

f−

1 (z) =a1

z3

+a0

z4,

200 I.B. Bolotin

where a0 and a1 – arbitrary complex coefficients.On the found functions f0(z) and f1(z), using (2.3), we restore analytical com-

ponents of the required piecewise bianalytical function F (z):

ϕ+

0 (z) = (z2− 1)

3

2− z

2 + a1z + a0, (2.19)

ϕ−

0 (z) = −

a1

z3−

a0

z4, (2.20)

ϕ+

1 (z) = z3 + a1z

2 + a0z, (2.21)

ϕ−

1 (z) =a1

z2

+a0

z3. (2.22)

Thus, common solution of the problem (2.13) – (2.14) is represented by the for-mula:

F (z) =

F+(z) = ϕ

+0 (z) + zϕ

+1 (z), z ∈ D

+,

F−(z) = ϕ

−

0 (z) + zϕ−

1 (z), z ∈ D−

,

where the functions ϕ±

0 (z) and ϕ±

1 (z) are defined by the formulas (2.19) – (2.22).

References

[1] F.D. Gakhov. Boundary value problems. Nauka, Moscow, 1977. (in Russian)[2] N.I. Muskhelishvili. Singular integral equations. Nauka, Moscow, 1968. (in Russian)[3] K.M. Rasulov. Boundary value problems for polyanalytical functions and some of their

applications. Smolensk State Pedagogical University, Smolensk, 1998. (in Russian)[4] N.P. Vekua. Systems of singular integral equations and some boundary value problems.

Nauka, Moscow, 1970. (in Russian)

Antrasis kraštinis uždavinys Rimano tipo bianalizinems funkcijoms su trukiais koefi-cientais

I.B. Bolotin

Darbe sprendžiamas antrasis kraštinis uždavinys Rimano tipo bianalizinems funkcijoms sutrukiais koeficientais. Parodoma, kad sprendžiamas uždavinys suvedamas i sprendima dviejuRimano uždaviniu analizinems funkcijoms su trukiais koeficientais. Randamos analiziniufunkciju klases, kuriose gali buti sprendinys. Pateikiamas pavyzdys, iliustruojantis nagrine-jamo uždavinio sprendimo algoritma.

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201–208c© 2004 Technika ISSN 1392-6292

DIFFUSION OF POPULATION UNDER THEINFLUENCE INDUSTRIALIZATION IN ATWIN-CITY ENVIRONMENT

J. DHAR1 and H. SINGH2

1Department of Applied Mathematics, Beant College of Engineering andTechnology

Gurdaspur – 143521, Punjab, India

E-mail: 728:9;:<=>?:@;A9BB%C6DEB%CFHG2Department of Mathematics, Baba Banda Singh Bahadur Engineering College

Fatehgarh Sahib - 140407, Punjab, India

Received February 12, 2004; revised August 6, 2004

Abstract. A mathematical model of a living population in a twin-city is proposed. Here pop-ulations are migrating from one place to another for their resource and settlement under theinfluence of industrialization. The long term effect of industrialization on the movement ofhuman population is considered in two adjoining cities. It is shown that the steady state distri-bution of population is positive, continuous, monotonic and the system is stable under certainset of conditions. Further, numerical solution of the steady state distributions of populationand industrialization are shown by taking particular values of the parameters.

Key words: Diffusion of population, industrialization, twin-city, steady state distribution,stability

1. Introduction

Modelling is very useful in understanding the behaviour of any environmental sys-tem. The models, in fact, represent the system in an abstract form, and providenecessary information about the system. It is a sequential and iterative processwhich helps in conceptualization, synthesis, simulation and analysis of the synthesis[1, 13, 14, 15, 16].

Every population is characterized with such characteristics as dispersion, fluctua-tion, sex ratio, birth rate and death rate [12, 14]. Population growth in a particular re-gion is directly related to the continuous changes taking place in that environment asthe environment is never static and keeps on changing from time to time due to seve-ral reasons of which some are natural while others are man-made [5, 6, 8, 9, 10, 11].

202 J. Dhar, H. Singh

One of the man-made reasons responsible for population migration from one placeto another, is industrialization [2], which has positive effects in terms of employmentand resources, while some major negative effects in the form of air and water pollu-tion. Moreover, the effluents from industries contain many chemicals that are toxic toliving organisms. Industrial effluents may also contain some radioactive substanceswhich cause many deadly diseases [17].

An interesting problem, in a twin city the continuous movement of populationtakes place due to the influence of different level of industrialization. In this paperwe therefore propose a mathematical model to understand the long time effect ofindustrialization in two adjoining cities.


We consider a linear environment 0 ≤ x ≤ L2, consisting of two adjoining cities0 ≤ x ≤ L1 and L1 ≤ x ≤ L2 with L1 as the interface between the two cities.Here the regions are divided either by a river, highway or any other geographical ortopographical condition. Let Ii(x, t) and Ni(x, t) be the respective industrializationand population densities at the location x at time t in i-th region, where i = 1, 2 (seeFig. 1).

Population density N1(x, t) Population density N2(x, t)

Industrialization density I1(x, t) Industrialization density I2(x, t)

Figure 1. Twin city environment.

It is assumed that Ii(x, t) grows logistically in both regions, with growth rate ai

and carrying capacity Ci. Further, we assume that in i-th city the population growsin absence of industrialization in a general logistical manner with growth rate ri, car-rying capacity Ki, growth factor βi and in presence of industrialization, the growthrate of population varies with interaction rate αi (i.e. which includes all positive andnegative effects of industrialization on population). Here, D1i and D2i are the dif-fusion coefficients of industrialization and population respectively in the i-th region.It is also assumed that, when the industrialization reaches its highest level (i.e atthe carrying capacity), then the population in that region also reaches its carryingcapacity.

Keeping in view all these assumptions, we get the following mathematical model:

∂Ii

∂t

== aiIi

(

1 −

Ii

Ci

)

+ D1i

∂2Ii

∂x2

,

∂Ni

∂t

= riNi

(

1 −

(

Ni

Ki

)βi

)

+ αiNi

(

1 −

Ii

Ci

)

+ D2i

∂2Ni

∂x2

,

(2.1)

Diffusion of population under the influence industrialization 203

where ai, Ci, ri, Ki, αi, D1i, D2i > 0, i = 1, 2; 0 < x < L2.

We are interested only in coexistence of the population and industrialization atthe equilibrium (N∗

i , I∗

i ). Thus we want to find N∗

i > 0, I∗

i > 0, such that

aiI∗

i

(

1−

I∗

i

Ci

)

= 0 ,

riN∗

i

(

1 −

(

N∗

i

Ki

)βi

)

+ αiN∗

i

(

1 −

I∗

i

Ci

)

= 0 ,

which implies N∗

i = Ki and I∗

i = Ci. Therefore, the nonzero equilibrium point isgiven by the carrying capacities of the industrialization and population, respectivelyin that habitat.

We also assume the continuity and flux matching conditions at the interface x =L1. The continuity conditions at the interface x = L1 for this system are

I1(L1, t) = I2(L1, t), N1(L1, t) = N2(L1, t) (2.2)

and the continuous flux matching conditions at the interface s = L1 for Ii(x, t) andNi(x, t) are written as

D11

∂I1(L1, t)

∂x

= D12

∂I2(L1, t)

∂x

, (2.3)

D21

∂N1(L1, t)

∂x

= D22

∂N2(L1, t)

∂x

.

The model is studied under no-flux boundary conditions i.e.

∂I1(0, t)

∂x

= 0,

∂I2(L2, t)

∂x

= 0 , (2.4)

∂N1(0, t)

∂x

= 0,

∂N2(L2, t)

∂x

= 0 .

Finally, the model is completed by assuming some positive initial distribution

I1(x, 0) = f1(x) > 0, N1(x, 0) = g1(x) > 0, 0 < x < L1 , (2.5)

I2(x, 0) = f2(x) > 0, N2(x, 0) = g2(x) > 0, L1 < x < L2 .

We first study the existence and stability behaviour of system (2.1) in homoge-neous habitat, the effect of patchiness will be investigated later.

3. Model in a Homogeneous Habitat

The corresponding model of (2.1) in a single homogeneous habitat without diffusion,i.e.

ai = a, ri = r, Ki = K, Ci = C, αi = α, βi = β, i = 1, 2


becomes

dI

dt

= aI

(

1 −

I

C

)

,

dN

dt

= rN

(

1 −

(

N

K

)β)

+ αN

(

1 −

I

C

)

.

(3.1)

Here the positive equilibrium is (K, C), which is locally asymptotically stable if

(

αK

C

)2

6 4arβ . (3.2)

This can be easily verified by using Lyapunov’s direct method and taking the follow-ing positive definite function:

V (t) =1

2

(

(I − C)2 + (N − K)2)

.

Example 1. By taking the following values of the parameters

a = 0.05, r = 0.06, α = 0.005, β = 2, C = 1000, K = 80, 000,

we see that condition (3.2) holds true.

4. Steady State Problem in Twin City Environment

We denote the steady state of the industrial density by ui(x) and the populationdensity by vi(x) in the i-th city, for i = 1, 2.

Then the steady state problem becomes:

D1i

d2ui

dx2

+ aiui

(

1 −

ui

Ci

)

= 0 ,

D2i

d2vi

dx2

+ rivi

(

1 −

(

vi

Ki

)βi

)

+ αivi

(

1 −

ui

Ci

)

= 0 .

(4.1)

The continuity and flux matching conditions at the interface x = L1 are given by

D11

du1

dx

(L1) = D12

du2

dx

(L1), D21

dv1

dx

(L1) = D22

dv2

dx

(L1) ,

u1(L1) = u2(L1), v1(L1) = v2(L1) , (4.2)

and the no-flux boundary conditions are

du1

dx

(0) = 0,

du2

dx

(L2) = 0, (4.3)

dv1

dx

(0) = 0,

dv2

dx

(L2) = 0 .

By using a similar analysis as in [3, 4, 7], we can prove the following theorem.


Theorem 1. The steady state distribution ui and vi are positive, continuous andmonotonic in the whole region.

Now we will find the stability conditions of the system by using Lyapunov’sdirect method.

Theorem 2. The steady state system (4.1) for βi = 1, i = 1, 2 is locally asymptoti-cally stable if

ui >Ci

αi

, vi >Ki

2, (4.4)

(

αiKi

Ci

)2

6 4airi . (4.5)

Proof. Linearizing the steady state system (4.1) by using

eIi(x, t) = ui(x) + ni(x, t), Ni(x, t) = vi(x) + mi(x, t)

we get a system of linear equations

∂ni

∂t

= aini

(

1 −

2ui

Ci

)

+ D1i

∂2ni

∂x2

,

∂mi

∂t

= rimi

(

1 −

2vi

Ki

)

+ αi

[

mi

(

1 −

ui

Ci

)

− ni

vi

Ci

]

+ D2i

∂2mi

∂x2

.

We use the following positive definite function

V (t) =

2∑

i=1

∫ Li

Li−1

1

2

(

n2i + m

2i

)

dx.

By using the boundary and flux matching conditions at the interface L1, we obtain

2∑

i=1

∫ Li

Li−1

D1ini

∂2ni

∂x2

dx = −

2∑

i=1

∫ Li

Li−1

D1i

(

∂ni

∂x

)2

dx ,

2∑

i=1

∫ Li

Li−1

D2imi

∂2mi

∂x2

dx = −

2∑

i=1

∫ Li

Li−1

D2i

(

∂mi

∂x

)2

dx .

Hence, the system will be asymptotically stable if(

1 −

2ui

Ci

)

6 0, ri

(

1 −

2vi

Ki

)

+ αi

(

1 −

ui

Ci

)

6 0 , (4.6)

[

αi

vi

Ci

]2

6 4ai

(

1−

2ui

Ci

)[

ri

(

1 −

2vi

Ki

)

+ αi

(

1−

ui

Ci

)]

. (4.7)

It can be easily verified that, (4.6) is automatically satisfied if inequalities (4.4)are true. Moreover from (4.7) by using a simple concept that if f(x) 6 g(x) thenmax f(x) 6 min g(x), we get (4.5). Hence the theorem is proved.


5. Discussion and Numerical Simulation

There are several twin cities in India, for example Secundrabad–Hyderabad, Chandi-garh–Mohali, Delhi–Gaziabad. First pair of cities are separated by highways, secondpair by inter state border and third pair by river. We can also find many examplesround the globe.

Moreover in our study the industrialization means not only small and big in-dustries, it also includes all man-made projects, e.g. marketing complex, housingcomplex. Hence it is very difficult to get complete or concrete data of population andindustrialization at different time and space in both the cities.

0 5 10 15 20 0

4000

8000

12000

16000

20000

24000

28000

32000

PATCH-II PATCH-I

Ste

ady

Sta

te D

istr

ibut

ions

Population with high dispersal rates Population with low dispersal rates Industrialization

Figure 2. Steady state distributions

For better understanding, in this section we study, the steady state distributionsof the population and industrialization in two adjoining regions (see Fig. 2) with fluxmatching conditions at interface and no-flux boundary conditions as stated in (4.1) –(4.3). A particular set of parameters is shown in Table 1, the region parameters arethe following:

β = 1.2, L1 = 10, L2 = 20 .

It can be easily verified that the stability conditions of Theorem 2 are satisfied forthese values of parameters. We see in Fig. 2, that both population and industrializa-tion distributions are continuous and monotonic from one end to the other end of thehabitat. Moreover, if the dispersal rates D21 and D22 are very high, then the steadystate distribution of the population is almost linear and when dispersal rates are verylow then the population distribution is almost at the level of carrying capacities ofthe respective cities except near to the interface of cities where abrupt changes takeplace (see Fig. 2).


Table 1. Values of the parameters for Fig. 2.

Parameters Patch 1 Fig. 2 Patch 2 Fig. 2

Growth rate of Industrialization a1 0.02 a2 0.03Growth rate of Population r1 0.04 r2 0.03Carrying capacity of Population K1 15000 K2 30000Carrying capacity of Industrialization C1 4000 C2 8000Interaction rate α1 0.005 α2 0.003Migration coefficient of Industrialization D11 0.5 D12 0.5For high dispersal rate of population D21 0.8 D22 0.7For low dispersal rate of population D21 0.015 D22 0.02

References

[1] L.J.S. Allen. Persistence and extinction in Lotka-Volterra reaction-diffusion equations.Math. Biosci., 65, 1 – 12, 1983.

[2] J. Dhar. Modelling and analysis: the effect of industrialization on diffusive forest re-source biomass in closed habitat. African Diaspora Journal of Math., 2(1), 142 – 159,2003.

[3] J. Dhar. A prey-predator model with diffusion and a supplementary resource for the preyin a two-patch environment. Mathematical Modelling and Analysis, 9(1), 9 – 25, 2004.

[4] J. Dhar and J.B. Shukla. A single species model with diffusion and harvesting in atwo-patch habitat. Mathematical Analysis and Application, 79 – 92, 2000.

[5] J. Dhar and H. Singh. Steady state distribution and stability behavior of a single speciespopulation with diffusion in n-patch habitat. Far East J. Appl. Math., 11(2), 103 – 119,2003.

[6] H.I. Freedman and T. Krisztin. Global stability in model of population dynamics withdiffusion. I. Patchy environment. In: Proc. Royal Soci. of Edinburgh, volume 122A, 69– 84, 1992.

[7] H.I. Freedman, J.B. Shukla and Y. Takeuchi. Population diffusion in a two-patch envi-ronment. Math. Biosci., 95, 111 – 123, 1989.

[8] H.I. Freedman and J. Wu. Steady state analysis in a model for population diffusion in amulti-patch environment. Nonl. Analy., Theor., Meths. & Appl., 18, 517 – 542, 1992.

[9] K. Gopalsamy. Competition, dispersion and co-existence. Math. BioSci., 33, 25 – 33,1977.

[10] K.P. Hadeler, V. Heiden and F. Rothe. Nonhomogeneous spatial distributions of popula-tions. J. Math. Biol., 1, 165 – 176, 1974.

[11] A. Hastings. Global stability in Lotka-Volterra systems with diffusion. J. Math. Biol., 6,163 – 168, 1978.

[12] M. Kot. Elements of Mathematical Ecology. University Press, Cambridge, UK, 2001.[13] S.A. Levin. Population model and community structure in heterogeneous environments.

In: Mathematical Ecology, Springer, New York, 295 – 321, 1986.[14] A. Okubo. Diffusion and ecological problem: Mathematical models. In: Lecture Notes.

Springer-Verlag, 1980.[15] C.V. Pao. Coexistence and stability of a competition diffusion system in population

dynamics. J. Math. Anal. Appl., 83, 54 – 76, 1981.[16] F. Rothe. Global solution of reaction-diffusion systems. In: Lecture Notes in Mathemat-

ics, volume 1072. Springer-Verlag, 1984.[17] P.D. Sharma. Ecology and Environment. Rastogi Publications, Meerut, India, 2001.

Seventh Edition


Populiacijos difuzija miestu-dvyniu aplinkoje, esant industralizacijos poveikiui

J. Dhar, H. Singh

Straipsnyje pasi ulytas populiacijos dinamikos miestuose-dvyniuose matematinis modelis. Da-roma preilaida, kad populiacija migruoja iš vienos vietos i kita industralizacijos poveikyje.Tiriamas ilgalaikis industralizacijos poveikis žmoniu judejimui dviejuose gretimai esanciuosemiestuose. Irodyta, kad esant išpildytoms tam tikroms salygoms, nusistovejes režimas yratolydus, monotoniškas ir stabilus. Taip pat pateiktas skaitiniais metodais gautas stacionarusispasiskirstymas, esant pasirinktam parametru rinkiniui.

!"#$%'&)($+*,-#.0/1213.546

209–222c© 2004 Technika ISSN 1392-6292

A REVIEW OF NUMERICAL ASYMPTOTICAVERAGING FOR WEAKLY NONLINEARHYPERBOLIC WAVES 1

A. KRYLOVAS and R. CIEGIS

Vilnius Gediminas Technical University

Sauletekio al. 11, LT-10223 Vilnius, Lithuania

E-mail: 728:9;%<=;#>:?2@2ACBEDFG2HIDKJGReceived June 8, 2004; revised July 5, 2004

Abstract. We present an overview of averaging method for solving weakly nonlinear hyper-bolic systems. An asymptotic solution is constructed, which is uniformly valid in the "large"domain of variables t + |x| ∼ O(ε

−1). Using this method we obtain the averaged system,

which disintegrates into independent equations for the nonresonant systems. A scheme fortheoretical justification of such algorithms is given and examples are presented. The averagedsystems with periodic solutions are investigated for the following problems of mathematicalphysics: shallow water waves, gas dynamics and elastic waves. In the resonant case the ave-raged systems must be solved numerically. They are approximated by the finite differenceschemes and the results of numerical experiments are presented.

Key words: small parameter method, perturbations, hyperbolic systems, averaging, reso-nance, finite difference schemes, numerical solution, gas dynamics, shallow water, elasticwaves

1. Introduction

We consider a hyperbolic system of weakly nonlinear differential equations with asmall positive parameter ε:

Ut +A(U)Ux = εB(t, x, εt, εx, U, Ux, Uxx, Uxxx) , (1.1)

whereU(t, x; ε) = (u1, u2, . . . , un)

T, A(U) = ‖aij(U)‖n×n .

Let define a constant solution U0, which satisfies the equation

1 This work was supported by by the Lithuanian State Science and Studies Foundation (V-27)within the framework of the Eureka project OPTPAPER E!-2623, E-2002.02.27

210 A. Krylovas, R. Ciegis

B(t, x, εt, εx, U0, 0, 0, 0) = 0.

We assume that all coefficients in (1.1) are sufficiently smooth functions. Ourgoal is to find a small amplitude solution

U(t, x; ε) = U0 + εU1(t, x; ε).

The system (1.1) must be solved in the "large" domain:

(t, x) ∈[

0,τ0

ε

]

×

[

−ξ0

ε

,

ξ0

ε

]

−→

ε→0

[0,+∞) ×R.

For small ε the problem of solving (1.1) numerically is a very difficult task. Asymp-totic methods are used for the analysis of such problems. Often (but not always) theasymptotic solution satisfies some simple equations. If a problem for asymptotic so-lution is still complicated, then a combination of numerical and asymptotic methodscan be used (see [1]).

In this article we present an overview of new asymptotic methods for solutionof problems with a small parameter. Recent developments in theoretical analysis, aswell as numerical algorithms are discussed. We present numerical algorithms forsolving the averaged systems which are obtained applying the asymptotic avera-ging method for the system (1.1) with periodic initial conditions. Three examplesof applied problems are investigated, including resonant interaction of shallow waterwaves, one dimensional waves of gas dynamics and resonances in elastic waves. Thetheoretical aspects of the asymptotic analysis of these examples where considered inour papers [16, 17, 18, 19]. In this article we focus our attention on the investigationof finite difference schemes for solving the averaged systems of equations.

Note, that analogous integro-differential systems were also investigated in [21,22, 23]. However numerical algorithms were not considered in these papers and nocomputational examples were given.

2. The Method of Averaging

Let assume that the problem is hyperbolic in neighborhood ofU0, thus we can rewritethe system using the well know Riemann invariants

Λ = diagλ1, λ2, . . . , λn = RA(U0)R−1,

V (t, x; ε) = RU1(t, x; ε) = (v1, v2, . . . , vn) ,

then we get:

Vt + ΛVx = εF (t, x, εt, εx, V, Vx, Vxx, Vxxx) + o(ε) . (2.1)

If the parameter ε = 0, then system (2.1) disintegrates into independent equa-tions, which describe linear waves vj(x − λj t). If ε 6= 0 and t + |x| ∼ ε

−1, thenthe exact solution of (2.1) is not close to these simple waves. For example, the initialvalue problem

A Review of Numerical Asymptotic Methods 211

vt + vx = εvvx ,

v(0, x) = sin(x)

describes a nonlinear wave, which can be obtained from the implicit relation

v = sin(x− t− εtv) .

For εt = O(1) the solution v(t, x; ε) can not be approximated by a simple wavesin(x − t) and thus it is a nontrivial task to construct an asymptotic approximation,which is uniformly valid in the region t+ |x| = O(ε−1).

The key idea of all asymptotic methods for solving problem (2.1) (or (1.1)) isto introduce new slow variables, e.g., τ = εt, ξ = εx, and to define explicitly thedependence on fast variables (principle of multiple scales, see [8, 24]). For examplein [28] the solution of the system (2.1) is obtained in the following form

uj = ψj(η, ζj), η = ε1+a

t, ζj = εa(x− λjt+ ε

1−aϕ(t, x)) .

Substituting these expressions into (1.1), using the Taylor expansion with respect toε, and collecting equal powers of εk, we get equations for unknown functionsψj . TheBurgers and Korteweg – de Vries equations are examples of such problems. We willshow, that these asymptotics are not uniformly valid in the region t+ |x| = O(ε−1)for the systems with periodic initial conditions in the case of resonant interaction ofwaves, therefore some modifications of such algorithms should be proposed.

2.1. Formulation of the integro-differential system

Our method of asymptotic integration is based on principles of multiple scales andaveraging. We introduce slow variables τ = εt, ξ = εx and fast characteristic vari-ables yj = x− λjt, j = 1, 2, . . . , n. Our goal is to construct the asymptotic solutionin the following form

vj(t, x; ε) = Vj(τ, ξ, yj) + o(1), j = 1, 2, . . . , n, ε→ 0 .

The basic idea of our method is the special averaging along characteristics:

Mj [g(τ, ξ, t, x, y1, y2, . . . , yn)] ≡ limT→+∞

1

T

∫ T

0

g

(

τ, ξ, s, yj + λjs, (2.2)

yj + (λj − λ1)s, . . . , yj + (λj − λn)s)

ds .

We also will use the following notation

< g >j (τ, ξ, yj) = Mj [g(τ, ξ, t, x, y1, y2, . . . , yn)] .

The asymptotic solution satisfies the averaged system:

∂Vj

∂τ

+ λj

∂Vj

∂ξ

= Mj [fj(t, x, τ, ξ, V1, V2, . . . , Vn, . . .)] (2.3)

with periodic initial conditions


Vj(0, ξ, yj) = voj(ξ, yj) = v0j(ξ, yj + 2π), j = 1, 2, . . . , n .

Our method also can be used for the non-periodic solutions, e.g., for almost pe-riodic functions or for functions vj(τ, ξ, yj), which satisfy conditions

limyj→±∞

v0(τ, ξ, yj) = 0 .

In the 2π-periodic case with integer coefficients λ1, λ2, . . ., λn the operator (2.2) canbe written in the simpler form:

< g >j=1

2π

∫ 2π

0

g(τ, ξ, s, yj + λjs, yj + (λj − λ1)s, . . . , yj + (λj − λn)s) ds.

After averaging each function depends only on one fast characteristic indepen-dent variable yj . The new feature of this method is that the averaging operatoris applied for functions, which themselves are solutions of the obtained averagedequations. Thus we get integro-differential problems (see [12], where our averagingscheme is compared with the other averaging methods). This idea was presented in[27] and developed in papers of the first author of this article [9, 10, 11] (see also[2, 5, 21, 22, 23] and a survey of mathematical results in [7]). The aspects of mathe-matical substantiation of our method were considered in [13, 14, 15].

The method of [21] is very close to the method of averaging along characteristicsfrom [9, 27] but only quadratic nonlinearities were considered in [21]. A mathematicsubstantiation of the asymptotic method and especially the construction of higherorder terms in the asymptotic series leads to the problem of small denominators:

δjl = l1(λj − λ1) + l2(λj − λ2) + · · · + ln(λj − λn) ,

where l = (l1, l2, . . . , ln) is a vector with integer components. In [27] all com-binations λj−λi

λj−λk

were rational numbers and therefore δjl were equal to zero or

|δjl| > const. In a general case

min||l||=L, δjl 6=0

|δjl| = o(1), L→ ∞ ,

therefore numbers λ1, λ2, . . ., λn, satisfying the following condition

min‖l‖=L, δjl 6=0

|δjl| >c

Lr ,

where c and r are some positive constants, were considered in [9] (see also [6]). Theproperties of

min‖l‖=L, δjl 6=0

|δjl|

were studied in [10], here small perturbations of numbers λj = λj0 + λj1(ε) wereinvestigated. In [21] the conditions of resonant interaction of waves were given onlyfor quadratic nonlinearities, more general relations of resonance δjl = 0 were pro-posed in [11].


2.2. Numerical algorithms

The integro-differential system (2.3), must be solved numerically in the compactdomain of variables:

(τ, ξ, y1, y2, . . . , yn) ∈ [0, τ0] × [−ξ0, ξ0] × [0, 2π]n .

It is important to note that the averaged system is solved only once and the obtainedsolution later can be used for all ε.

We always try to split the operators Mj into a sum of two operators

Mj(V1, . . . , Vn) = LjVj +Nj(V1, . . . , Vn)

and LjVj is included into the differential part of (2.3). For many applications we getsystems of nonlinear differential equations such that efficient numerical methods al-ready exist for solving this type of equations. In some cases even numerical softwareis available and this fact provides a possibility to tackle real-life problems at smallprogramming cost. The remaining integral part of the integro-differential system isapproximated explicitly. A fixed-point iteration method can be used to improve thestability of the obtained numerical algorithm.

3. Theoretical Justification of the Method

In this section we describe the main steps of the theoretical analysis. First we showthat the averaging operator Mj [g(τ, ξ, t, x)] takes out secular terms, which were ob-tained after integration of the equation

∂u

∂t

+ λj

∂u

∂x

= εg(τ, ξ, t, x)

along the characteristic x− λjt = const. Thus the following equality

limε→0

ε

∫ t

0

(

g −M [g])

(t = s, x = x− λjt+ λjs) ds = 0 (3.1)

should be valid uniformly in |x| + t 6c0

ε

.

Let us denote

F (ε) = maxj = 1, 2, . . . , nτ + ξ ∈ [−c0, c0]t+ |x| ∈ [0, c0/ε]

ε

∫ t

0

(fj −M [fj ]) ds. (3.2)

Here fj are functions from (2.1). Let us assume that in (1.1) the right-hand sidevector B = B(U) =

(

b1(U), . . . , bn(U))

and all functions bj(U) are continuouslydifferentiable functions. If the averaged system satisfies condition (3.1), i.e.

F (ε) = o(1), ε→ 0,


then a solution of system (1.1) is approximated uniformly in the region |x|+ t 6c0

ε

by a solution of the averaged system, i.e.:

uj(t, x; ε) = vj(τ, ξ, yj) +O

(

F (ε))

.

Secondly, we should prove that averaged system (2.1) has a solution, which sa-tisfies condition (3.1). In case of periodical initial conditions it is sufficient to notethe following property of operatorMj .

Let us denote by C12π([−c0, c0] × [0, 2π]) a class of 2π – periodical functions

u(τ, y), which have a continuous derivative with respect to y. Let assume that

fj(v1, v2, . . . , vn) ∈ C1(Rn), vj(τ, yj) ∈ C

12π([−c0, c0] × [0, 2π]) .

Then we obtain, that

Mj [fj(v1, . . . , vn)] = gj(τ, yj) ∈ C12π([−c0, c0] × [0, 2π]) .

We see that all averaged functions preserve properties of continuity and periodicity,thus we can prove the existence and uniqueness of the solution of averaged system(2.1) by using the standard Picard method.

The proposed averaging method can be used not only in case of periodical func-tions. This scheme can be applied to construct almost periodical asymptotical ap-proximations or to consider functions decreasing at infinity

limyj→±∞

v0(τ, ξ, yj) = 0 .

The other generalization of the method is obtained for a case when the averagingoperator is nonuniform with respect to the arguments of functions:

vj(τ, ξ, yj) ≈ v+

0j(τ, ξ, yj), yj → +∞,

vj(τ, ξ, yj) ≈ v−

0j(τ, ξ, yj), yj → −∞.

If functions v+

0 , v−0 are periodical or almost periodical with respect to yj or de-creasing at infinity, then averaging operator can be applied in each region (t, x) ∈

λit < x < λi+1t. Therefore we can consider not only the initial value problem, butalso to formulate initial – boundary value problem in t = 0, x > 0 and x = 0, t > 0(see [13, 14, 15]).

4. Shallow Water Equations

4.1. Averaged system

In this section we consider the system of shallow water equations:

A Review of Numerical Asymptotic Methods 215(

Z

W

)

t

+

(

0 11 0

)(

Z

W

)

x

(4.1)

= −ε

(

1

3Wxxx + (H(x)W )x + (ZW )x

WWx

)

.

Here Z denotes the water surface level, W denotes the horizontal velocity of thefluid, L

∗is some typical horizontal size, H

∗is some typical vertical size, ε =

(

H∗

L∗

)2

<< 1, H = 1 + εh(x) is the bottom equation.

The asymptotic solution of (4.1) satisfies the averaged system of two equations:

∂V+

∂τ

+3

2V

+∂V

+

∂y+

+1

6

∂3V

+

∂y+3

= −1

2

∂

∂y+〈H(x)V −〉

+,

∂V−

∂τ

−3

2V

−

∂V−

∂y−

−1

6

∂3V

−

∂y−

3=

1

2

∂

∂y−

〈H(x)V +〉−

.

(4.2)

Here we use notation

W = v+ − v

−

, Z = v+ + v

−

, y± = x∓ t ,

and V ± are approximations of v±:

v±(t, x; ε) = V

±(τ, ξ, y±) + o(1) .

In the nonresonant case the expressions on the right hand side of (4.2) are equalto zero and we get two independent Korteweg de Vries equations. We consider theresonant interaction of 2π-periodic waves:

V+τ +

3

2V

+V

+

y+ +1

6V

+

y+y+y+

= −1

4π

∂

∂y+

2π∫

0

H(y+ + s)V −(τ, y+ + 2s) ds,

V−

τ −3

2V

−

V−

y−−

1

6V

−

y−y−y−

=1

4π

∂

∂y−

2π∫

0

H(y− − s)V +(τ, y− − 2s) ds.

(4.3)

4.2. Finite difference scheme

We define the space ωh and time ωτ meshes and assume that the space mesh size hand time mesh size τ are uniform. We denote by vn

j = v(tn, yj) a discrete functiondefined on ωh × ωτ . The following common notations of difference derivatives areused in our paper (see, e.g., [25]):

vτ =v

n+1 − vn

τ

, vy =vj − vj−1

h

,

vy =vj+1 − vj

h

, v

y=vj+1 − vj−1

2h.


The finite difference approximation of system (4.3) is defined as follows (see also[4]):

Pτ = −1

6

(

Pn+1 + P

n

2

)

yy

y

−3

4

(

(

Pn+1)2

+ Pn+1

Pn + (Pn)2

3

)

y

−F+

(

Mn+1

,Mn, j + 1

)

− F+

(

Mn+1

,Mn, j − 1

)

4h,

Mτ =1

6

(

Mn+1 +M

n

2

)

yy

y

+3

4

(

(

Mn+1)2

+Mn+1

Mn + (Mn)

2

3

)

y

+F−

(

Pn+1

, Pn, j + 1

)

− F−

(

Pn+1

, Pn, j − 1

)

4h,

where the integrals are approximated as follows:

F+

(

Mn+1

,Mn, j

)

=1

2π

N∑

i=1

H(yj − ih)M

n+1

j−2i +Mnj−2i

2h ,

F−

(

Pn+1

, Pn, j

)

=1

2π

N∑

i=1

H(yj + ih)P

n+1

j+2i + Pnj+2i

2h .

Here P and M approximate V +, V

−, respectively.

The approximation error of this finite finite difference scheme is estimated asO

(

τ2 + h

2)

. Numerical methods for solving the Korteweg-de Vries equation areinvestigated in [3, 26]. A special formula for averaging in time is used in order tosatisfy some conservation properties, which are valid for the solution of the differen-tial problem.

4.3. Linear dispersion problem

In this section we consider a linear problem

Zt + (HW )x = −ε

3Wxxx,

Wt + Zx = 0.(4.4)

First we will prove that system (4.4) defines an ill-posed problem. Let consider thecase H = 1. After simple computations we get the equation for W :

Wtt −Wxx =ε

3Wxxxx. (4.5)

Considering the k-th Fourier mode we get that the solution of (4.5) is unstable fork

2ε ≥ 3. In order to define a stable solution we use the following regularized problem

Zt + (HW )x = −ε

3Wxxx −

ε2

20Wxxxxx,

Wt + Zx = 0.

(4.6)


We note that the averaged system (2.3) also gives a nontrivial regularization ofthis ill-posed problem.

Extensive computational results are presented in [16, 18].

5. One Dimensional Gas Dynamics Equations


Let ρ denotes the gas density, u is the velocity and θ is the temperature. We introducea vector U = (ρ, u, θ)T and matrixes

A =

u ρ 0

Rθ

ρ

u R

0Rθ

cv

u

cvρ

, B =1

ρ

0

γ

∂2u

∂x2

κ

cv

∂2θ

∂x2

+γ

cv

(

∂u

∂x

)2

, (5.1)

where cv is the specific heat at constant volume,R is the gas constant for a politropicideal gas:

p = Rρθ ,

κ and γ are small viscosity and heat conduction coefficients (i.e. ∼ O(ε) as ε → 0).Then the gas dynamics problem can be formulated as system (1.1).

In this case the averaged system (2.3) is described by the linear Burgers equationscoupled through integral terms:

∂V1

∂τ

− f111V1

∂V1

∂y1

− f11

∂2V1

∂y12

=

⟨

f123V2

∂V3

∂y3

+ f132V3

∂V2

∂y2

⟩

1

,

∂V2

∂τ

− f22

∂2V2

∂y22

= 0 ,

∂V3

∂τ

− f333V3

∂V3

∂y3

− f33

∂2V3

∂y32

=

⟨

f321V2

∂V1

∂y1

+ +f312V1

∂V2

∂y2

⟩

3

.

(5.2)

The explicit expressions for coefficients in (5.2) are presented in [17].


The averaged system (5.2) is approximated by the following finite difference scheme:

V1,τ = f11Vn+11,yy + 1

2f111 (V n+1

1 )2

y+ f123 S1(V2, V3) + f132 S1(V3, V2) ,

V2,τ = f22Vn+1

2,yy ,

V3,τ = f33Vn+13,yy + 1

2f333 (V n+1

3 )2

y+ f312 S2(V1, V2) + f321 S2(V2, V1) ,

where the integrals are approximated as follows:


Si(Vj , Vk) =1

4π

N∑

l=1

(

Vk

(

yi + ((λi − λk)l + 1)h)

− Vk

(

yi + ((λi − λk)l − 1)h)

)

Vj

(

yi + ((λi − λj)l − 1)h)

.

The approximation error is given by O(τ + h2).

5.3. Numerical experiments

In this section we present results of numerical experiments. The following coeffi-cients

cv = 1, R = 1, ν = 1, κ = 1

are used in all tests. Initial conditions are selected as

v01(x) = cosx, v02(x) = sin 2x, v03(x) = cosx.

Figure 1 shows the solution of system (5.1) and the asymptotic solution at t =1

ε

for

two different values of the small parameter ε. We present graphics of the density andvelocity functions.

0 1 2 3 4 5 60.8

0.85

0.9

0.95

1

1.05

1.1

1.15 ρρ

asympt

UU

asympt

0 1 2 3 4 5 60.9

0.95

1

1.05

1.1ρρ

asympt

UU

asympt

ε = 0.1 ε = 0.05

Figure 1. Asymptotic and exact solutions of Euler problem (5.1) for t =1

ε

.

6. Elastic Waves Equations


We consider a problem of wave propagation in two dimensional elastic materialsand assume that displacements do not depend on the y coordinate. Restricting ourattention to the axial displacements along x and y directions, we have equations[20]:


ρutt = σx ,

ρvtt = τx ,

where ρ is the density of material, u and v denote the displacements along two direc-tions, σ and τ ate longitudinal normal and shear stresses along the x-axes. The linearwave equation can be obtained from this equations if we use assumptions

σ = (λ+ 2µ)ux, τ = µvx, (6.1)

where λ and µ are the Lamé coefficients. The equations (6.1) are obtained by usingthe simple approximation for the full energy of the system

F ≈ F0 + F11u2x + F22v

2x,

then σ = ∂F∂ux

and τ = ∂F∂vx

(see [20]). In order to get high order approximations weuse more terms in a Taylor series of F (see [24]):

σ = (λ+ 2µ)ux + 4ρ(a1u2x + a2uxvx + a3v

2x),

τ = µvx + 4ρ(b1u2x + b2uxvx + abv

2x).

LetP = ux, Q = vx, R = ut, S = vt, U = (P,Q,R, S)T

.

Then we get the system (1.1) with

A(U) = −

0 0 1 00 0 0 1

λ+ 2µ

ρ

0 0 0

0µ

ρ

0 0

, B = 4∂

∂x

00

a1P2 + a2PQ+ a3Q

2

b1P2 + b2PQ+ b3Q

2

.

In this case the averaged system is given by the system of four equations:

P±

τ −2a1

c2p

P±

P±

y±= −

2a3

c2s

My±

[

∂

∂x

(S+S−)

]

∓a2

cscp

My±

[

∂

∂x

(P∓(S+ − S−))

]

,

S±

τ −2b3c2s

S±

S±

z±= −

2b1c2p

Mz±

[

∂

∂x

(P+P

−)

]

∓b2

cscp

Mz±

[

∂

∂x

(S∓(P+ − P−))

]

.

(6.2)

Here we use notation


y± = x∓ cpt, z

± = x∓ cst, cp =

√

λ+ 2µ

ρ

, cs =

√

µ

ρ

,

P± = R± cpP, S

± = S ± csQ.

In the nonresonant case we get four independent nonlinear Burgers equations.


The integrals on the right-hand side of (6.2) are approximated by the trapezoidalrule, the derivatives of functions are computed using the central difference approxi-mations.

The upwind method is used to approximate the Burgers equation

Un+1

j = Unj + µ

τ

h

(

F (Unj , U

nj+1) − F (Un

j−1, Unj ))

,

here F is the numerical flux function

F (v, w) =

0.5w2 if 0 6 w 6 v or (v < w and v + w > 0) ,

0.5v2 else.

Thus we get the explicit approximation for the system of integro-differential equa-tions. We have used implicit approximations in two previous examples. Now ourgoal is to show that explicit schemes also can be used to solve averaged equations, ifsuch approximations are efficient for solving the differential part of the system (i.e.the Burgers equations for this example).

6.3. Numerical experiments

The results of numerical experiments are presented in Fig. 2. Here we present only

one wave p+ and its asymptotic approximation P+ for t =1

ε

, ε = 0.01.

We see that the averaged system approximates uniformly the differential problem

till time moments t = O(1

ε

)

and the effect of resonant interaction of waves is also

identified correctly.

7. Conclusions

For weakly nonlinear hyperbolic systems with internal resonances the analysis canbe done using the combination of numerical and asymptotic methods. The proposedmethod for constructing asymptotic solution of weakly nonlinear hyperbolic systemcan be used in nonresonant and in resonant cases. This solution is uniformly validin large domain 0 6 t 6 O(ε−1). The averaging of system (2.1) reduces it to theintegro-differential system of averaged equations (2.2). The averaged problem gives


0 1 2 3 4 5 6

-0.8

-0.4

0

0.4

0.8p

+

P+

Figure 2. Asymptotic and exact solutions of P+ wave for ε = 0.01.

a system of integro-differential equations. However, in the nonresonant case the so-lution can be obtained as independent nonlinear waves. In the resonant case the so-lution is a superposition of waves, which satisfy the averaged system of nonlinearintegro-differential equations. Such systems are solved numerically in the compactdomain of variables (τ, x) ∈ [0, τ0] × [0, 2π]).

References

[1] N. S. Bakhvalov, G. P. Panasenko and A. L. Shtaras. The averaging method for par-tial differential equations (homogenization) and its applications. In: Partial DifferentialEquations V, Springer, New York, 211 – 239, 1999.

[2] S. C. Chikwendu and J. Kevorkian. A perturbation method for hyperbolic equations withsmall nonlinearities. SIAM J. Appl. Math., 22, 235 – 258, 1972.

[3] B. Fornberg and T. Driscoll. A fast spectral algorithm for nonlinear wave equations withlinear dispersion. J. of Comp. Phys., 155, 456 – 467, 1999.

[4] D. Furihata. Finite Difference Schemes that Inherit Energy Conservation or DissipationProperty. Number 1212. 1998.

[5] J. K. Hunter and J. B. Keller. Weakly nonlinear high frequency waves. Comm. PureAppl. Math., 36, 547 – 569, 1983.

[6] L. A. Kalyakin. Asymptotic integration of hyperbolic equations with weakly nonlinearperturbation. Differencialnye uravnenija, 20(2), 351 – 353, 1984.

[7] L. A. Kalyakin. Integrability equations as asymptotic limits of nonlinear systems. Us-pechi Matematicheskich Nauk, 44(1), 5 – 34, 1989.

[8] J. Kevorkian and J. D. Cole. Multiple Scale and Singular Pertubation Methods. Springer– Verlag, Berlin, New-York, 1996.

[9] A. V. Krylov. About the asymptotic interaction of first order hyperbolic systems. Lith.Math. J., 23(4), 12 –17, 1983.

[10] A. V. Krylov. Asymptotic integration of weakly nonlinear partial differential systems.Zhurnal Vytchisl. Matem. i Matemat. Fiziki, 26(1), 72 – 79, 1986. (in Russian)

[11] A. V. Krylov. The method of research of weakly nonlinear interaction one dimensionalwaves. Prikladnaja Matem. i Mech., 51(4), 933 – 940, 1987. (in Russian)


[12] A. V. Krylov. The internal averaging of first order partial differential systems. Matem-atitcheskije Zametki, 46(6), 112 – 113, 1989. (in Russian)

[13] A. V. Krylov. The substantiation of the method of the internal averaging along charac-teristics in weakly nonlinear systems. I. Lith. Math. J., 29(4), 721 – 732, 1989.

[14] A. V. Krylov. The substantiation of the method of the internal averaging along charac-teristics in weakly nonlinear systems. II. Lith. Math. J., 30(1), 87 – 100, 1990.

[15] A. V. Krylov. The averaging of weakly nonlinear hyperbolic systems with non-uniformintegral average. Ukrainian Math. J., 43(5), 611 – 618, 1991.

[16] A. Krylovas and R. Ciegis. Asymptotic analysis of weakly nonlinear systems. In:A. A. Samarskii R. Ciegis and M. Sapagovas(Eds.), Finite Difference Schemes Theoryand Applications, Proceedings of the Conference FDS2000, September 1 - 4, Palanga,Institute of Mathematics and Informatics, Vilnius, 142 – 151, 2000.

[17] A. Krylovas and R. Ciegis. Asymptotical analysis of one dimensional gas dynamicsequations. Mathematical Modelling and Analysis, 6(1), 103 – 112, 2001.

[18] A. Krylovas and R. Ciegis. Asymptotical approximation of hyperbolic weakly nonlinearsystems. J. of Nonlinear Math. Phys., 8(4), 458 – 470, 2001.

[19] A. Krylovas and R. Ciegis. On the interaction of elastic waves. Journal of Civil Engi-neering and Management, 9(3), 218 – 224, 2003.

[20] L. D. Landau and E. M. Lifshitz. Theory of Elasticity: Course of Theoretical Physics.Butterworth V Heinemann, Oxford, 1986.

[21] A. Majda and R. Rosales. Resonantly interacting weakly nonlinear hyperbolic waves. 1.a single space variable. Studies in Appl. Math., 71(2), 149 – 179, 1984.

[22] V. P. Maslov. Asimptoticheskie metody reshenija psevdodifferencialnych uravnenii.Nauka, Moskva, 1987. (in Russian)

[23] V. P. Maslov and P. P. Mosolov. Uravnenija odnomernogo barotropnogo gaza. Nauka,Moskva, 1990. (in Russian)

[24] A. H. Nayfeh. Pertubation Methods. Jon Willay & Sons Inc., New-York, 2000.[25] A.A. Samarskii. The theory of difference schemes. Marcel Dekker, Inc., New-York,

Basel, 2001.[26] J. M. Sanz-Serna. Symplectic integrators for hamiltonian problems. Acta Numerica, 1,

243 – 286, 1992.[27] A. L. Štaras. The asymptotic integration of weakly nonlinear partial derivatives equa-

tions. Doklady Akademii Nauk SSSR, 237(3), 525 – 528, 1977.[28] T. Taniuti. Reductive perturbation method and far fields of wave equations. Suppl. Progr.

Theor. Phys., 55, 1 – 35, 1974.

Silpnai netiesiniu hiperboliniu sistemu skaitinio asimptotinio vidurkinimo apžvalga

A. Krylovas, R. Ciegis

Darbe nagrinejamas silpnai netiesiniu hiperboliniu sistemu ilguju bangu asimptotinis spren-dinys. Si ulomas jo konstravimo metodas, pagristas vidurkinimu bei dvieju masteliu principu.Užrašytos skirtumines schemos suvidurkintu lygciu sistemoms spresti. Ištirti trys periodiniuasimptotiniu sprendiniu pavyzdžiai: sekliuju vandenu modelis, duju dinamikos lygtys bei tam-priuju bangu saveika.

!"#$%'&)($+*,-#.0/1213.546

223–228c© 2004 Technika ISSN 1392-6292

ABOUT THE SOLUTION IN CLOSED FORM OFGENERALIZED MARKUSHEVICH BOUNDARYVALUE PROBLEM IN THE CLASS OFANALYTICAL FUNCTIONS

K. M. RASULOV

214000, Przevalskogo 4, Smolensk, Russia


E-mail: 7289:;=<>987@?69BA5CDEF59HGI?KJ2<

Received October 13, 2003; revised June 28, 2004

Abstract. The paper is devoted to the investigation of the problem of obtaining piecewiseanalytical functions F (z) =

˘

F+(z), F

−

(z)¯

with the jump line L, vanishing on the infinityand satisfying on L the boundary condition

F+

[(α(t)] = G(t) F−

(t) + b(t)F−(t) + g(t), t ∈ L,

where α(t) is the preserving orientation homeomorphism of L onto itself and G(t), b(t), g(t)

are given on L functions of Holder class and G(t) 6= 0 on L.The algorithm for the solution of this problem was obtained and particular cases, when it

is solvable in closed form are determined.


1. The Formulation of the Problem

Let T+ be a bounded simply connected region on the plane of the complex vari-

able z = x + iy, bounded by the simple closed Liapunov’s contour L, and T− =

C\ (T+∪ L). For determination we shall suppose, that the point z = 0 belongs to

T+. Let us denote by α(t) the function, mapping the contour L onto itself with the

preservation of the rule and having the derivative, satisfying the Holder conditionH(L). We shall use notations from [8].

Let us consider the following problem. It is required to find all piecewise analy-tical functions F (z) = F

+(z), F−(z) from the class A(T±) ∩ H(L), vanishingon infinity and satisfying on L the following boundary condition

F+ [α(t)] = G(t) F

−(t) + b(t) F−(t) + g(t), t ∈ L, (1.1)

224 K. M. Rasulov

where G(t), b(t), g(t) are given on L functions of the class H(L), and G(t) 6= 0 onL.

It should be noted that the problem in form (1.1) in case α(t) ≡ t firstly wasformulated in 1946 by A.I. Markushevitch [5]. So we shall call the problem formu-lated above as the Markushevitch boundary value problem or, in short, the problemM , and the corresponding homogeneous problem (g(t) ≡ 0) as the problem M

0.If α(t) 6= t we shall call this problem generalized Markushevitch boundary valueproblem for analytical functions, or, in short, the GM problem.

During the last 50 years many original works have been devoted to the problem(1.1) (see, for example [2, 4, 6, 7, 9, 11] and the bibliography there). Even in the firstworks [1, 10], devoted to the investigation of the problem GM it was established,that if the condition

G(t) 6= 0, t ∈ L, (∗)

is fulfilled, it is the Noeter problem.In the author’s work [7] the constructive algorithm for solution of the problem

M was obtained in the general case, i.e. if only one of the conditions (*) is fulfilled.In this article we shall obtain the constructive algorithm for solution of the problemGM and show the cases, when the problem GM can be solved in a closed form (inquadratures).

2. Solution of the Markushevitch Problem in a Closed Form forRational Coefficients

Let the region T+ be the unity circle, i.e. T + = z : |z| < 1. Then, as it was proved

in [7], if κ = Ind G(t) ≥ 0 the problem M is equivalent to the following integralequation of Fredholm type:

F−(t) +

∫

L

K(t, τ)F (τ)dτ = Q(t) + X−(t)Pκ−1(t), (2.1)

where

K(t, τ) =τ

2X

−(t)

2πi

[

b(τ)

X+(τ)

−

b(t)

X+(t)

]

1

τ − t

+b(t)

τX+(t)

, (2.2)

Q(t) = −

g(t)

2α(t)+

X−(t)

2πi

∫

L

g(τ)

X+(τ)

dτ

τ − t

, (2.3)

X+(t), X

−(t) are canonical functions of the Riemann boundary value problem withthe coefficient G(t) and Pκ−1(z) is the polynomial of the degree not higher thanκ − 1 with arbitrary complex coefficients.

If κ < 0, by the following conditions

∫

L

b(τ)F−(t) + g(t)

X+(t)

tk−1

dτ = 0, k = 1, . . . ,−κ, (2.4)

About the Solution in Closed Form of Generalized Markushevich Problem 225

the problem M is also equivalent to the integral equation of the form (2.1), wherewe put Pκ−1(z) ≡ 0.

Now it is easy to notice, that if the coefficients G(t) and b(t) are rational func-tions, then the kernel K(t, τ), determined by the formula (2.2), will be degenerate(see e.g. [8], p.181), i.e.

K(t, τ) =

N∑

j=1

rj(t)qj(τ), (2.5)

where rj(t), qj(t) are determined rational functions of there arguments. Hence, inthis case integral equation (2.1) has the solution in closed form (see for example,[3], p.37). But then the boundary value problem M also has the solution in closedform.

Thus, the following statement is valid.

Theorem 1. If L = t : |t| = 1 and the coefficients G(t) and b(t) are rationalfunctions, then the problem M is equivalent to the integral equation of Fredholmtype (2.1) with degenerate kernel, and, consequently, it can be solved in a closedform (in quadratures).

Remark 1. The statement of the theorem 1 can be also obtained from the fact, thenin the case of rational coefficients and unity circle T

+ = z : |z| < 1 the solutionof boundary value problem M , as it is known (see, for example, [4], p.223), is equi-valent to the solution of the two-dimensional vector-matrix Riemann boundary valueproblem, where coefficient is non-singular matrix with rational elements. The latterproblem also can be solved in a closed form (see, for example, [11], p.40).

3. Solution of the Markushevitch Boundary Value Problem inRational Images of the Unity Circle

Let for finite simple connected region T+, bounded by the simple closed Liapunov’s

contour, the rational function exists

z = ω(ζ), ζ = ξ + iv, (3.1)

mapping the unity circle K1 = ζ : |ζ| < 1 conformally on this region. So we shellcall the region T

+ the rational image of the unity circle.The following statement is valid.

Theorem 2. If T+ is the rational image of the unity circle and the coefficients G(t),

b(t) are rational functions, then the problem M can be solved in closed form.

Proof. Introducing the following notations:

f±(ζ) = F

±(z) = F±(ω(ζ)),

G1(ζ) = G(ω(ζ)), b1(ζ) = b(ω(ζ)), g1(ζ) = g(ω(ζ)),

we can rewrite the boundary condition of M in this way

226 K. M. Rasulov

f+(τ) = G1(τ)f−(τ) + b1(τ)f−(τ) + g1(τ), |τ | = 1, (3.2)

The equality (3.2) is the boundary condition of the usual Markushevitch boundaryvalue problem with the rational coefficients in the class of the analytical functions,permitting the poles in the points ζ = ∞ and ζ = ai, i = 1, 2, . . . , m, whereai ∈ ζ : |ζ| > 1. Reasoning further as in the work [7], we can prove, that theproblem (3.2) is equivalent to the Fredholm’s integral equation of the form (2.1)with the degenerate kernel (i.e. of the form (2.5)). Therefore, the problem (3.2) canbe solved in closed form, which means that the problem M can also be solved inquadratures.

4. Solution of the Generalized Markushevitch Boundary ValueProblem

As it is known (see, for example, [2], p. 153), the regions T+ and T

− can be confor-mally mapped on two mutually supplementary for the full plane regions T

+1 and T

−

2

with the common boundary L1, so, that the analytical functions F+(z) and F

−(z)will be transferred to the functions F

+

1 (z) and F−

2 (z) defined in T+

1 and T−

2 corre-spondingly, and on L1 the boundary condition of the following form will be fulfilled

F+

1 (ζ) = G1(ζ) F−

1 (ζ) + b1(ζ) F−

1 (ζ) + g1(ζ), ζ ∈ L1, (4.1)

where

ζ = ω−(t) = ω

+ [α(t)] , G1(ζ) = G [(σ(ζ)] , b1(ζ) = b [σ(ζ)] , σ

[

ω−(t)

]

≡ t.

Here the functions ω±(z), mapping the regions T

± onto T±

1 conformally, canbe uniquely determined as the solution of the following Riemann boundary valueproblem (see, for example [2], p.154)

ω+ [α(t)] = ω

−(t), ω−(z) = z +

c1

z

+c2

z2

+ . . . , z → ∞. (4.2)

It is important to notice, that the indexes of the problems (1.1) and (4.1) are equal,i.e. IndL1

G1(ζ) = IndLG(t). Therefore, the generalized boundary value problem(1.1) is equivalent to the usual boundary value problem (4.1) both in the sense ofsolvability, and the number of linear independent solutions of the corresponding ho-mogeneous problems.

Solving the problem (4.1) by the method, proposed in [7], we can obtain thefunctions F

+

1 (z) and F−

1 (z). By the formula

F+(z) = F

+

1

[

ω+(z)

]

, F−(z) = F

−

1

[

ω−(z)

]

we can obtain the solutions of the problem (1.1). We conclude that the followingstatement is valid.

Theorem 3. The generalized Markushevitch boundary value problem (1.1) permitsthe solutions in closed form, when we can solve in closed form the Riemann boundaryvalue problem (4.2) and the usual Markushevitch boundary value problem (4.1).

About the Solution in Closed Form of Generalized Markushevich Problem 227

Example 1. Let T+z : |z| < 1 and L = t : |t| = 1. It is required to find all

piecewise analytical functions F (z) = F+(z), F−(z), belonging to the class

A(T±)∩H(L), vanishing on the infinity and satisfying on L the following boundarycondition:

F+(−t) = t

2F

−(t) +1

t

F−(t) +

2

t2. (4.3)

Solution. Here α(t) = −t, b(t) =1

t

, g(t) =2

t2

. The functions ω+(z) = −z,

ω−(z) = z are solutions of the Riemann boundary value problem (4.2). Conse-

quently, in this case the boundary value problem (4.1) will have the form

F+

1 (t) = t2F

−

1 (t) +1

t

F−

1 (t) +2

t2, |t| = 1.

It should be noted, that here

κ = Ind

(

t2)

= 2, X+(z) = 1, X

−(z) =1

z2, Pκ−1(z) = C0 + C1z,

where C0 and C1 are arbitrary complex constants. Hence, taking into considerationformulas (2.1) and (2.2), we get in this case:

K(t, τ) ≡ 0, Q(t) =2

t4,

F−

1 (z) =C1

z

+C0

z2−

2

z4,

F+1 (z) = (C0 + C1) + (C1 + C0)z − 2z

3.

Then the following functions will be the solution of the problem (4.3):

F−

1 (z) =C1

z

+C0

z2−

2

z4,

F+(z) = (C0 + C1) − (C1 + C0)z + 2z

3.

References

[1] B.V. Bojarski. About the generalized Gilbert boundary problem. In: Proc. Ac. Sc. GSSR,volume 25, 385 – 390, 1960. (in Russian)

[2] F.D. Gachow. Boundary value problems. Nauka, Moscow, 1977. (in Russian)[3] M.P. Krasnow. Integral equations. Nauka, Moscow, 1975. (in Russian)[4] G.S. Litvinchuk. Boundary value problems and singular integral equations with the

displacement. Nauka, Moscow, 1977. (in Russian)[5] A.I. Markushevitch. About the boundary value problem of the theory of analytical func-

tions. In: Proc. of Moskow State University, volume 1, 20 – 30, 1946.[6] L.G. Michailow. The new class of singular integral equations and its applications to the

differential equations with singular coefficients. Dushanbe, 1963. (in Russian)[7] K.M. Rasulov. About the method of solution of the Markushevitch boundary value prob-

lem in the class of analytical functions. In: Investigations on boundary value problemsof complex analysis and differential equations, number 3, 98 – 108, 2001. (in Russian)

228 K. M. Rasulov

[8] K.M. Rasulow. Boundary value problems for polyanalytycal functions and some of theirapplications. Smolensk state pedagogical institute, Smolensk, 1998. (in Russian)

[9] I.Ch. Sabitow. About the boundary value problem of linear displacement. In: Math.proc., volume 64(106)(2), 262 – 274, 1964. (in Russian)

[10] N.P. Vekua. About the boundary value problem of the theory functions of complexvariable. In: Proc. Ac. Sc. USSR, volume 86, 457 – 460, 1952. (in Russian)

[11] N.P. Vekua. Systems of singular integral equations and some boundary value problems.Nauka, Moscow, 1970. (in Russian)

Apie apibendrintojo Markuševiciaus uždavinio sprendima analiziniu funkciju klaseje

K.M. Rasulov

Darbe pateikiamas algoritmas Markuševiciaus uždavinio, kai ieškomos dalimis analizinesfunkcijos F (z) = F +

(z), F−

(z) nykstancioje begalybeje, savo šuoliu linijoje L tenki-nancios salyga

F=[α(t)] = G(t)F (t) + b(t)F−(t) + g(t), t ∈ L,

kur G(t), b(t), g(t) – apibrežtos konture L funkcijos Golderio klases, o α(t) – homemor-fizmas konturo i save. Atvejui α(t) ≡ t uždavini suformulavo A.I. Markuševicius 1946 m.Irodyta, kad uždavinio sprendimas suvedamas i integralines antrosios rušies Fredholmo tipolygties sprendima. Pateikiamas pavyzdys, iliustruojantis gautus teorinius rezultatus.

!"#$%'&)($+*,-#.0/1213.546

229–242c© 2004 Technika ISSN 1392-6292

A PIEZOELECTRIC CONTACT PROBLEM WITHSLIP DEPENDENT COEFFICIENT OF FRICTION

M. SOFONEA1 and EL-H. ESSOUFI21Laboratoire de Mathématiques et Physique pour les Systèmes, Université dePerpignan

52 Avenue de Paul Alduy, 66 860 Perpignan, France

E-mail: 78298:;<=>:5?@5ACB;ED2BGFH92D2 Groupe d’Analyse Non Linéaire, Faculté des Sciences et Techniques, UniversitéMoulay Ismail

52000 Boutalamine, Errachidia, Maroc

E-mail: ;778>9?E=I5<J2KGFH:;EJ

Received March 16, 2004; revised July 3, 2004

Abstract. We consider a mathematical model which describes the static frictional contactbetween a piezoelectric body and an obstacle. The constitutive relation of the material is as-sumed to be electroelastic and involves a nonlinear elasticity operator. The contact is modelledwith a version of Coulomb’s law of dry friction in which the coefficient of friction dependson the slip. We derive a variational formulation for the model which is in form of a coupledsystem involving as unknowns the displacement field and the electric potential. Then we pro-vide the existence of a weak solution to the model and, under a smallness assumption, weprovide its uniqueness. The proof is based on a result obtained in [14] in the study of ellipticquasi-variational inequalities.

Key words: piezoelectric material, electroelasticity, static frictional contact, Coulomb’s law,slip dependent coefficient of friction, quasivariational inequality, weak solution

1. Introduction

The piezoelectric effect was discovered in 1880 by Jacques and Pierre Curie; it con-sists on the apparition of electric charges on the surfaces of some crystals after theirdeformation. The reverse effect was outlined in 1881; it consists on the generationof stress and strain in crystals under the action of electric field on the boundary.A deformable material which undergoes piezoelectric effects is called a piezoelec-tric material. An elastic material with piezoelectric effect is called an electroelasticmaterial and the discipline dealing with the study of electroelastic materials is thetheory of electroelasticity. Their bases were underlined by Voigt [24] who provided

230 M. Sofonea, El-H. Essoufi

the first mathematical model of a linear elastic material which takes into account theinteraction between mechanical and electrical properties.

General models for elastic materials with piezoelectric effects can be found in[10, 11, 12, 22, 23] and, more recently, in [1, 21]. Currently, there is a consider-able interest in frictional contact problems involving piezoelectric materials, see forinstance [2, 9] and the references therein. Indeed, situations which involve contactphenomena abound in industry and everyday life. The contact of the braking padswith the wheel, the tire with the road and the piston with skirt are just three simpleexamples. Because of the importance of contact processes a considerable effort hasbeen made in their modelling and the engineering literature concerning this topicis extensive. However, there are very few mathematical results concerning contactproblems involving piezoelectric materials and therefore there is a need to extendthe results on models for contact with deformable bodies to models for contact withdeformable bodies which include coupling between mechanical and electrical prop-erties.

The aim of this paper is to provide such an extension. Indeed, we consider herea model for the process of frictional contact between an electrolastic body, which isacted upon by forces and electric charges, and a foundation. The process is static,the contact is frictional and it is modeled with a version of Coulomb’s law of dryfriction in which the coefficient of friction depends on the slip. Such kind of depen-dence was pointed out in [18] in the study of the stick-slip phenomenon and wasconsidered in various papers, see for instance [16, 19]. Frictional contact boundaryvalue problems with elastic materials and slip dependent friction were considered in[3, 6] in the static case and in [4] in the quasistatic case. Here we extend the frictionalmodel in [3] to the case of nonlinear electroelastic materials. Taking into account thepiezoelectric behavior of the body consists the main trait of novelty of the model. Wederive a variational formulation of the model then we prove its weak solvability and,under an additional assumption, its unique solvability. As in [3], the proof of theseresults are based on an abstract theorem on quasivariational inequalities derived in[14]; however, keeping in mind the coupling of the electrical and mechanical effects,we apply this result in a different setting and with a different choice of operatorsand functionals. An important continuation of this paper consists in the numericalanalysis of the model, including numerical simulations, and will be presented in aforthcoming work.

The paper is structured as follows. In Section 2 we state the model of the equilib-rium process of the elastic piezoelectric body in frictional contact with a foundation.In Section 3 we introduce some preliminary material, list assumptions on the prob-lem data and state our main existence and uniqueness result, Theorem 1. The proofof the theorem is presented in Section 5; it is based on an abstract existence anduniqueness result that we recall in Section 4.

2. Problem Statement

We consider the following physical setting. An elastic piezoelectric body occupies abounded domain Ω ⊂ IRd

, d = 2, 3 with a smooth boundary ∂Ω = Γ . The body issubmitted to the action of body forces of density f 0 and volume electric charges of

A Piezoelectric Contact Problem with Slip Dependent Coefficient 231

density q0. It is also submitted to mechanical and electric constraints on the boundary.To describe them, we consider a partition of Γ into three measurable parts Γ1, Γ2,Γ3, on one hand, and on two measurable parts Γa and Γb, on the other hand, suchthat measΓ1 > 0 and measΓa > 0. We assume that the body is clamped on Γ1

and surfaces tractions of density f 2 act on Γ2. On Γ3 the body is in frictional contactwith an obstacle, the so-called foundation. We model the contact with a version ofCoulomb’s law of dry friction, already used in [3] and [6], in which the normal stressis prescribed and the coefficient of friction depends on the slip. We also assume thatthe electrical potential vanishes on Γa and a surface electric charge of density q2is prescribed on Γb. We denote by Sd the space of second order symmetric tensorson R

d or, equivalently, the space of symmetric matrices of order d. Also, below ν

represents the unit outward normal on Γ while “ ·” and ‖ ·‖ denote the inner productand the Euclidean norm on Rd and Sd, respectively.

With the assumption above, the problem of equilibrium of the electroelastic bodyin frictional contact with a foundation is the following.

Problem P . Find a displacement field u : Ω → Rd, a stress field σ : Ω → Sd, anelectric potential ϕ : Ω → R and an electric displacement field D : Ω → Rd suchthat

σ = Fε(u) − ETE(ϕ) in Ω, (2.1)

D = Eε(u) + βE(ϕ) in Ω, (2.2)

Div σ + f0 = 0 in Ω, (2.3)

div D = q0 in Ω, (2.4)

u = 0 on Γ1, (2.5)

σν = f 2 on Γ2, (2.6)

− σν = S on Γ3, (2.7)

‖στ‖ ≤ µ(‖uτ‖)|S|, on Γ3,

στ = −µ(‖uτ‖)|S|uτ

‖uτ‖, if uτ 6= 0

(2.8)

ϕ = 0 on Γa, (2.9)

D · ν = q2 on Γb. (2.10)

In (2.1) – (2.10) and below, in order to simplify the notation, we do not indicateexplicitly the dependence of various functions on the spatial variable x ∈ Ω ∪ Γ .Equations (2.1) and (2.2) represent the electroelastic constitutive law of the mate-rial in which F is a given nonlinear function, ε(u) denotes the small strain tensor,E(ϕ) = −∇ϕ is the electric field, E represents the third order piezoelectric tensor,ET is its transposite and β denotes the electric permitivitty tensor. Details of thelinear version of the constitutive relations (2.1) and (2.2) can be find in [1, 2]. Equa-tions (2.3) and (2.4) represent the equilibrium equations for the stress and electric-displacement fields, respectively, (2.5) and (2.6) are the displacement and tractionboundary conditions, respectively, and (2.9), (2.10) represent the electric boundaryconditions.

We now provide some comments on the frictional contact conditions (2.7) and(2.8), which are our main interest. Condition (2.7) states that the normal stress σν


is prescribed on Γ3 since S denotes a given function. Condition (2.8) represents theassociated friction law in which στ is the tangential stress, uτ denotes the tangentialdisplacement and µ is the coefficient of friction. This law should be seen either as amathematical model suitable for proportional loadings or as a first approximation ofa more realistic model, based on a friction law involving the time derivative of uτ

(see for instance [4, 13]). Note that in (2.8) the coefficient of friction depends on theslip ‖uτ‖ which leads to a nonstandard frictional contact problem.

3. Variational Formulations and Main Result

In this section we list the assumptions on the data, derive a variational formulation forthe contact problem (2.1) – (2.10) and state our main existence and uniqueness result,Theorem 1. To this end we need to introduce notation and preliminary material.

We recall that the inner products and the corresponding norms on Rd and Sd aregiven by

u · v = uivi , ‖v‖ = (v · v)1

2 ∀u,v ∈ Rd,

σ · τ = σijτij , ‖τ‖ = (τ · τ )1

2 ∀σ, τ ∈ Sd.

Here and everywhere in this paper i, j, k, l run from 1 to d, summation over repeatedindices is implied and the index that follows a comma represents the partial derivative

with respect to the corresponding component of the spatial variable, e.g. ui,j =∂ui

∂xj

.

Everywhere below we use the classical notation for Lp and Sobolev spaces asso-ciated to Ω and Γ . Moreover, we use the notation L2(Ω)d, H1(Ω)d and H and H1

for the following spaces:

L2(Ω)d = v = (vi) | vi ∈ L

2(Ω) , H1(Ω)d = v = (vi) | vi ∈ H

1(Ω) ,

H = τ = (τij) | τij = τji ∈ L2(Ω) , H1 = τ ∈ H | τij,j ∈ L

2(Ω) .

The spaces L2(Ω)d, H1(Ω)d, H and H1, are real Hilbert spaces endowed with thecanonical inner products given by

(u,v)L2(Ω)d =

∫

Ω

u · v dx, (u,v)H1(Ω)d =

∫

Ω

u · v dx+

∫

Ω

ε(u) · ε(v) dx,

(σ, τ )H

=

∫

Ω

σ · τ dx, (σ, τ )H1

= (σ, τ )H

+ (Div σ,Div τ )L2(Ω)d

and the associated norms ‖ · ‖L2(Ω)d , ‖ · ‖H1(Ω)d , ‖ · ‖H

and ‖ · ‖H1

, respectively.Here ε : H1 → H and Div : H1 → H are the deformation and divergence operators,respectively, that is

ε(v) = (εij(v)), εij(v) =1

2(vi,j + vj,i) ∀v ∈ H

1(Ω)d,

Div τ = (τij,j) ∀τ ∈ H1.

For every element v ∈ H1(Ω)d we also write v for the trace of v on Γ and

we denote by vν and vτ the normal and tangential components of v on Γ given byvν = v · ν, vτ = v − vνν.


Let us now consider the closed subspace of H1(Ω)d defined by

V = v ∈ H1(Ω)d | v = 0 on Γ1 .

Since meas (Γ1) > 0, the following Korn’s inequality holds:

‖ε(v)‖H

≥ cK ‖v‖H1(Ω)d ∀v ∈ V, (3.1)

where cK > 0 is a constant which depends only on Ω and Γ1. A proof of Korn’sinequality can be found in, for instance, [15] p. 79. Over the space V we considerthe inner product given by

(u,v)V = (ε(u), ε(v))H

(3.2)

and let ‖ · ‖V be the associated norm. It follows from Korn’s inequality (3.1) that‖ · ‖H1(Ω)d and ‖ · ‖V are equivalent norms on V . Therefore (V, ‖ · ‖V ) is a realHilbert space. Moreover, by the Sobolev trace theorem, (3.1) and (3.2), there existsa constant c0 depending only on the domain Ω, Γ1 and Γ3 such that

‖v‖L2(Γ3)d ≤ c0‖v‖V ∀v ∈ V. (3.3)

We also introduce the spaces

W = ψ ∈ H1(Ω) | ψ = 0 on Γa ,

W = D = (Di) | Di ∈ L2(Ω), div D ∈ L

2(Ω) ,

where div D = (Di,i). The spaces W and W are real Hilbert spaces with the innerproducts

(ϕ, ψ)W = (ϕ, ψ)H1(Ω), (D,E)W

= (D,E)L2(Ω)d + (div D, divE)L2(Ω).

The associated norms will be denoted by ‖ · ‖W and ‖ · ‖W

, respectively. Notice alsothat, since meas (Γa) > 0, the following Friedrichs-Poincaré inequality holds:

‖∇ψ‖L2(Ω)d ≥ cF ‖ψ‖W ∀ψ ∈ W, (3.4)

where cF > 0 is a constant which depends only on Ω and Γa.In the study of the contact problem (2.1) – (2.10) we assume that

(a) F : Ω × Sd → Sd.

(b) There existsMF> 0 such that

‖F(x, ξ1) −F(x, ξ2)‖ ≤MF‖ξ1 − ξ2‖ ∀ ξ1, ξ2 ∈ Sd

, a.e. x ∈ Ω.

(c) There existsmF> 0 such that

(F(x, ξ1)) −F(x, ξ2)) · (ξ1 − ξ2) ≥ mF‖ξ1 − ξ2‖

2

∀ ξ1, ξ2 ∈ Sd, a.e. x ∈ Ω.

(d) The mapping x 7→ F(x, ξ) is Lebesgue measurable on Ω for any ξ ∈ Sd.

(e) The mapping x 7→ F(x,0) belongs to H.(3.5)


(a) E = (eijk) : Ω × Sd → Rd.

(b) eijk = eikj ∈ L∞(Ω).

(3.6)

(a) β = (βij) : Ω × Rd → Rd.

(b) βij = βji ∈ L∞(Ω).

(c) There existsmβ > 0 such that βij(x)EiEj ≥ mβ‖E‖2 ∀E ∈ Rd,

a.e. x ∈ Ω.

(3.7)

f 0 ∈ L2(Ω)d

, f2 ∈ L2(Γ3)

d (3.8)

q0 ∈ L2(Ω), q2 ∈ L

2(Γb), (3.9)

S ∈ L∞(Γ3) and ‖S‖L∞(Γ3) > 0· (3.10)

(a) µ : Γ3 × IR → IR+.

(b) There exist cµ1 ≥ 0 and cµ2 ≥ 0 such thatµ(x, r) ≤ c

µ1 |r| + c

µ2 ∀ r ∈ IR+, a.e. x ∈ Γ3.

(c) The mapping x 7→ µ(x, r) is Lebesgue measurable on Γ3 for any r ∈ IR.

(d) The mapping r 7→ µ(x, r) is continuous on IR+, a.e. x ∈ Γ3.

(3.11)

There exists Lµ > 0 such that

(µ(x, r2) − µ(x, r1)) · (r1 − r2) ≤ Lµ |r1 − r2|2 ∀ r1, r2 ∈ IR, a.e. x ∈ Γ3.

(3.12)We make in what follows some comments on the assumptions (3.5) – (3.12). As

stated in Section 2, below we suppress the dependence of various functions on thespatial variable x ∈ Ω ∪ Γ .

First, we note that the condition (3.5) is satisfied in the case of the linear elasticconstitutive law σ = Fε(u) in which

Fξ = (fijklξkl), (3.13)

provided that fijkl ∈ L∞(Ω) and there exists α > 0 such that

fijkl(x)ξkξl ≥ α‖ξ‖2 ∀ ξ ∈ Sd, a.e. x ∈ Ω.

To provide examples of nonlinear constitutive laws which satisfy (3.5), for everytensor ξ ∈ Sd we denote by tr ξ the trace of ξ and let ξ

D be the deviatoric part of ξ

given by

tr ξ = ξii, ξD = ξ −

1

d

(tr ξ)Id,


where Id ∈ Sd represents the identity tensor. Let K denotes a nonempty closed con-vex set in Sd and let PK represents the projection mapping on K. We also considera forth order symmetric and positively defined tensor E : S

d → Sd and take

F(ξ) = Eξ +1

λ

(ξ − PKξ) ∀ξ ∈ Sd, (3.14)

where λ is a strictly positive constant. Using the properties of the projection mappingit is straightforward to see that the elasticity operator F defined by (3.14) satisfiescondition (3.5). Constitutive laws of the form σ = Fε(u)) with F given by (3.14)have been considered by many authors, see. e.g. [8], [17] p. 97 and [20] p. 68. Mostof them have defined the convex K by the relationship K = ξ ∈ Sd | G(ξ) ≤ k

where G : Sd → IR is a convex continuous function such that G(0) = 0 and k > 0.A second example of nonlinear elastic equations is provided by nonlinear Hencky

materials (see [25] for details). For a Hencky material, the stress-strain relation isgiven by

σ = K0(tr ε(u)) Id + ψ(‖εD(u)‖2) εD(u),

so that the elasticity operator is

F(ξ) = K0(tr ξ) Id + ψ(‖ξD‖2) ξD ∀ ξ ∈ S

d. (3.15)

Here, K0 > 0 is a material coefficient, the function ψ is assumed to be piecewisecontinuously differentiable, and there exist positive constants c1, c2, d1 and d2, suchthat for s ≥ 0

ψ(s) ≤ d1, −c1 ≤ ψ′(s) ≤ 0, c2 ≤ ψ(s) + 2ψ′(s) s ≤ d2.

Under these assumption it can be shown that the elasticity operator F defined in(3.15) satisfies condition (3.5).

Next, as it is shown in (3.6) and (3.7), we see that the piezoelectric operator E aswell as the electric permitivitty operator β are assumed to be linear and, moreover, βis symmetric and positive definite. Recall also that the transposite tensor ET is givenby ET = (eT

ijk) where eTijk = ekij , and the following equality holds:

Eσ · v = σ · E∗

v ∀σ ∈ Sd, v ∈ R

d. (3.16)

We also remark that (3.8) represent regularity assumptions on the densities ofvolume forces and surface tractions while (3.9) represent regularity assumptions onthe densities of volume and surface electric charges. Condition ‖S‖L∞(Γ3) > 0 in(3.10) is imposed here in order to obtain a genuine frictional contact problem. Indeed,if S = 0 a.e. on Γ3 then by (2.7) and (2.8) it follows that the Cauchy stress vector σν

vanishes on Γ3 and therefore problem (2.1) – (2.10) becomes a purely displacement-traction problem for electroelastic materials.

Finally, we observe that the assumptions (3.11) on the coefficient of friction µare pretty general. Clearly, these assumptions are satisfied if µ is a bounded functionwhich is continuously differentiable with respect to the second variable, as it wasconsidered in [6]. We also remark that assumptions (3.11) and (3.12) are satisfied ifµ does not depend on the second argument and is a positive function which belongs


to L∞(Γ3). This is the case when the coefficient of friction does not depend on theslip. Frictional contact problems involving this last assumption on the coefficient offriction were studied in [5, 17] in the case of purely elastic materials. Notice also thatassumption (3.12) is satisfied if µ(x, ·) : IR → IR+ is an increasing function, a.e.x ∈ Γ3.

We now turn to the variational formulation of Problem P and, to this end, weintroduce further notation. Let h : V × V −→ IR be the functional

h(u,v) =

∫

Γ3

µ(‖uτ‖) |S| ‖vτ‖da, ∀u, v ∈ V (3.17)

and, using Riesz’s representation theorem, consider the elements f ∈ V and q ∈ W

given by

(f ,v)V =

∫

Ω

f 0 · v dx+

∫

Γ2

f2 · v da+

∫

Γ3

S vν da ∀v ∈ V, (3.18)

(q, ψ)W =

∫

Ω

q0ψ dx+

∫

Γb

q2ψ da ∀ψ ∈W. (3.19)

Keeping in mind assumptions (3.8) – (3.11) it follows that the integrals in (3.17) –(3.19) are well-defined.

Using integration by parts, it is straightforward to see that if (u,σ, ϕ,D) aresufficiently regular functions which satisfy (2.3) – (2.10) then

(σ, ε(v) − ε(u))H

+ h(u,v) − h(u,u) ≥ (f ,v − u)V ∀v ∈ V, (3.20)

(D, ψ)L2(Ω)d = (q, ψ)W ∀ψ ∈W. (3.21)

We plug (2.1) in (3.20), (2.2) in (3.21) and use the notation E = −∇ϕ to obtainthe following variational formulation of Problem P , in the terms of displacementfield and electric potential.

Problem PV . Find a displacement field u ∈ V and an electric potential ϕ ∈ W

such that

(Fε(u), ε(v) − ε(u))H

+ (ET∇ϕ,v − u)L2(Ω)d (3.22)

+h(u,v) − h(u,u) ≥ (f ,v − u)V ∀v ∈ V,

(β∇ϕ,∇ψ)L2(Ω)d − (Eε(u),∇ψ)L2(Ω)d = (q, ψ)W ∀ψ ∈W. (3.23)

Our main existence and uniqueness result which we establish in Section 5 is thefollowing.

Theorem 1. Assume that (3.5)–(3.10) hold. Then :1) Under the assumption (3.11), Problem PV has at least one solution.2) Under the assumptions (3.11) and (3.12), there exists L0, which depends only onΩ, Γ1, Γ3, Γa, F , β, S, such that if Lµ < L0 then Problem PV has unique solution(u, ϕ) which depends Lipschitz continuously on f ∈ V and q ∈W .


A “quadriplet" of functions (u, σ, ϕ, D) which satisfy (2.1), (2.2), (3.22) and(3.23) is called a weak solution of the piezoelectric contact problem P . We concludeby Theorem 1 that, under the assumptions (3.5) – (3.11), the piezoelectric contactproblem (2.2) – (2.10) has at least a weak solution (u, σ, ϕ, D) such that u ∈ V ,ϕ ∈ W . Moreover, it is easy to see that in this case σ ∈ H1 and D ∈ W . Thesolution is unique and depends Lipschitz continuously on the data f 0, f2, q0 andq2 if (3.12) holds with a sufficiently small constant Lµ. In particular, this case arisewhen the coefficient of friction is a given positive bounded function which does notdepend on the slip.

4. An Abstract Existence and Uniqueness Result

To prove Theorem 1 we shall use an abstract existence and uniqueness result on el-liptic quasivariational inequalities that we recall in what follows, for the convenienceof the reader.

Everywhere in this section X will represent a real Hilbert space endowed withthe inner product (·, ·)X and the associated norm ‖·‖X . We denote by “ ′′ the weakconvergence on X . Let A : X −→ X be a non linear operator, j : X ×X −→ IRand f ∈ X . With these data we consider the following quasivariational inequality:find x ∈ X such that

(Ax, y − x)X + j(x, y) − j(x, x) ≥ (f, y − x)X ∀ y ∈ X (4.1)

In order to solve (4.1) we assume that A is strongly monotone and Lipschitzcontinuous, i.e.

(a) There exists m > 0 such that(Ax1 −Ax2, x1 − x2)X ≥ m‖x1 − x2‖

2X ∀x1, x2 ∈ X.

(b) There existsM > 0 such that‖Ax1 −Ax2‖X ≤M‖x1 − x2‖X ∀x1, x2 ∈ X.

(4.2)

The functional j : X ×X → IR satisfies

j(η, ·) : X → IR is a convex functional onX, for all η ∈ X. (4.3)

Keeping in mind (4.3) it is well known that there exists the directional derivative ofj with respect to the second argument given by

j′

2(η, x; y) = limλ↓0

1

λ

[

j(η, x+ λy) − j(η, x)]

∀η, x, y ∈ X. (4.4)

We formulate in what follows some conditions on j and we recall that below m

represents the positive constant defined in (4.2).


For every sequence xn ⊂ X with ‖xn‖X → ∞

and every sequence tn ⊂ [0, 1] one has

lim infn→∞

[ 1

‖xn‖2X

j′(tnxn, xn;−xn)

]

< m.

(4.5)

For every sequence xn ⊂ X with ‖xn‖X → ∞

and every bounded sequence ηn ⊂ X one has

lim infn→∞

[ 1

‖xn‖2X

j′(ηn, xn;−xn)

]

< m.

(4.6)

For every sequences xn ⊂ X and ηn ⊂ X such thatxn x ∈ X, ηn η ∈ X and for every y ∈ X one haslim sup

n→∞

[

j(ηn, y) − j(ηn, xn)]

≤ j(η, y) − j(η, x).(4.7)

There exists α < m such thatj(x, y) − j(x, x) + j(y, x) − j(y, y) ≤ α ‖x− y‖2

X ∀x, y ∈ X.(4.8)

In the study of the quasivariational inequality (4.1) we have the following result.

Theorem 2. Let conditions (4.2) – (4.3) hold. Then :1) Under the assumptions (4.5) – (4.7) there exists at least one element x ∈ X whichsolves (4.1).2) Under the assumptions (4.5) – (4.8), problem (4.1) has unique solution x = xf

which depends Lipschitz continuously on f with the Lipschitz constant (m− α)−1.

Theorem 2 has been obtained in [14] and therefore we do not provide here thedetails of the proof. We just specify that the proof was obtained in several steps andit is based on standard arguments of elliptic variational inequalities and topologicaldegree theory.

5. Proof of Theorem 1

The proof of Theorem 1 will be carried out in several steps. To present it we considerthe product space X = V ×W together with the inner product

(x, y)X = (u,v)V + (ϕ, ψ)W ∀x = (u, ψ), y = (v, ψ) ∈ X (5.1)

and the associated norm ‖ · ‖X . Everywhere below we assume that (3.5) – (3.11)hold.

We introduce the operator A : X → X defined by

(Ax, y) = (Fε(u), ε(v))H

+ (β∇ϕ,∇ψ)L2(Ω)d + (ET∇ϕ, ε(v))H

(5.2)

− (Eε(u),∇ψ)L2(Ω)d ∀x = (u, ψ), y = (v, ψ) ∈ X

and we extend the functional h defined by (3.17) to a functional j defined onX×X ,that is

j(x, y) = h(u,v) ∀x = (u, ψ), y = (v, ψ) ∈ X. (5.3)

Finally, we consider the element f ∈ X given by

f = (f , q) ∈ X. (5.4)

We start with the following equivalence result.


Lemma 1. The couple x = (u, ϕ) is a solution to Problem PV if and only if

(Ax, y − x)X + j(x, y) − j(x, x) ≥ (f, y − x)X ∀ y ∈ X. (5.5)

Proof. Let x = (u, ϕ) ∈ X be a solution to Problem PV and let y = (v, ψ) ∈ Y .We use the test function ψ − ϕ in (3.21), add the corresponding inequality to (3.20)and use (5.1) – (5.4) to obtain (5.5). Conversely, let x = (u, ϕ) ∈ X be a solutionto the quasivariational inequality (5.5). We take y = (v, ϕ) in (5.5) where v is anarbitrary element of V and obtain (3.22); then we take successively y = (v, ϕ+ ψ)and y = (v, ϕ − ψ) in (5.5), where ψ is an arbitrary element of W ; as a result weobtain (3.23), which concludes the proof.

Notice that the quasivariational inequality (5.5) derived in Lemma1 is of the form(4.1). Therefore, in order to apply the abstract result provided by Theorem 2, we startwith the study of the the properties of the operator A given by (5.2).

Lemma 2. The operator A : X → X is strongly monotone and Lipschitz continu-ous.

Proof. Consider two elements x1 = (u1, ϕ1), x2 = (u2, ϕ2) ∈ X . Using (5.2) wehave

(Ax1 −Ax2, x1 − x2)X = (Fε(u1) −Fε(u2), ε(u1) − ε(u2))H

+ (β∇ϕ1 − β∇ϕ2,∇ϕ1 −∇ϕ2)L2(Ω)d + (ET∇ϕ1

− ET∇ϕ2, ε(u1) − ε(u2))H − (Eε(u1) − Eε(u1),∇ϕ1 −∇ϕ2)L2(Ω)d

and, since it follows by (3.16) that (ET∇ϕ, ε(u))H

= (Eε(u),∇ϕ)L2(Ω)d for allx = (u, ϕ) ∈ X , we find

(Ax1 −Ax2, x1 − x2)X =

(Fε(u1) −Fε(u2), ε(u1) − ε(u2))H + (β∇ϕ1 − β∇ϕ2,∇ϕ1 −∇ϕ2)L2(Ω)d .

We use now (3.5), (3.7) and Friedrichs-Poincaré inequality (3.4) to see that thereexists c1 > 0 which depends only on F , β, Ω and Γa such that

(Ax1 −Ax2, x1 − x2)X ≥ c1(‖u1 − u2‖2V + ‖ϕ1 − ϕ2‖

2W )

and, keeping in mind (5.1), we obtain

(Ax1 −Ax2, x1 − x2)X ≥ c1 ‖x1 − x2‖2X . (5.6)

In the same way, using (3.5) – (3.7), after some algebra it follows that there existsc2 > 0 which depends only on F , β and E such that

(Ax1 −Ax2, y)X ≥c2(‖u1 − u2‖V ‖v‖V + ‖ϕ1 − ϕ2‖W ‖v‖V

+ ‖u1 − u2‖V ‖ψ‖W + ‖ϕ1 − ϕ2‖W ‖ψ‖W )


for all y = (v, ψ) ∈ V . We use (5.1) and the previous inequality to obtain

(Ax1 −Ax2, y)X ≤ 4c2 ‖x1 − x2‖V ‖y‖V ∀y ∈ X

and, taking y = Ax1 −Ax2 ∈ X , we find

‖Ax1 −Ax2‖X ≤ 4c2 ‖x1 − x2‖V . (5.7)

Lemma 2 is now a consequence of inequalities (5.6) and (5.7).

Next we investigate the properties of the functional j given by (5.3), (3.17). Wefirst remark that j satisfies condition (4.3). Moreover, we have the following results.

Lemma 3. The functional j satisfies conditions (4.5), (4.6) and (4.7).

Proof. Let η = (w, ξ), x = (u, ϕ) ∈ X and let λ ∈]0, 1]. Using (5.3) and (3.17) itresults that

j(η, x− λx) − j(η, x) = −λ

∫

Γ3

µ(‖wτ‖) |S| ‖uτ‖ da

and, keeping in mind (4.4), we deduce that

j′

2(η, x;−x) ≤ 0 ∀η, x ∈ X. (5.8)

We conclude by (5.8) that the functional j satisfies conditions (4.5) and (4.6).Let now consider two sequences xn = (un, ϕn) ⊂ X and ηn =

(wn, ξn) ⊂ X such that xn x = (u, ϕ) ∈ X , ηn η = (w, ξ) ∈ X . Usingthe compactness property of the trace map it follows that un → u and wn → w inL

2(Γ3)d, which imply that

‖unτ‖ → ‖uτ‖ in L2(Γ3), (5.9)

‖wnτ‖ → ‖wτ‖ in L2(Γ3). (5.10)

Moreover, (3.12), (5.10) and Kranoselski’s theorem (see for instance [7]) yield

µ(‖wnτ‖) → µ(‖wτ‖) in L2(Γ3). (5.11)

Therefore, we use the definition of j, (5.9) and (5.11) to deduce that

j(ηn, y) → j(η, y) ∀y ∈ X and j(ηn, xn) → j(η, x), as n→ ∞.

We conclude that the functional j satisfies the condition (4.7).

Lemma 4. If (3.12) holds, then the functional j satisfies the inequality

j(x, y) − j(x, x) + j(y, x) − j(y, y) ≤ c20Lµ‖S‖L∞(Γ3)‖x− y‖2

X ∀x, y ∈ X.

(5.12)


Proof. Let x = (u, ϕ), y = (v, ψ) ∈ V . Using (5.3), (3.17) and (3.12) it followsthat

j(x, y) − j(x, x) + j(y, x) − j(y, y)

=

∫

Γ3

|S|(

µ(‖uτ‖) − µ(‖vτ‖))

(‖vτ‖ − ‖uτ‖) da

≤ Lµ‖S‖L∞(Γ3)

∫

Γ3

∣

∣

∣‖vτ‖ − ‖uτ‖

∣

∣

∣

2

da ≤ Lµ‖S‖L∞(Γ3)

∫

Γ3

‖u− v‖2da.

Using now (3.3) and (5.1) in the previous inequality we deduce (5.12).

We have now all the ingredients to prove the Theorem.

Proof. [Proof of Theorem 1.]1) Assume that (3.5) – (3.11) hold. Then, Lemmas 2 and 3 allow us to use the

abstract results provided by the first part of Theorem 2; we obtain that the quasivari-ational inequality (5.5) has at least a solution x = (u, ϕ) ∈ X and, using Lemma1,we deduce that (u, ϕ) is a solution to Problem PV , which satisfies u ∈ V , ϕ ∈W .

2) Assume that (3.5) – (3.12) hold and let L0 =c1

c20‖S‖L∞(Γ3)

where c1 and c0

are defined by (5.6) and (3.3), respectively. Clearly L0 depends only on Ω, Γ1, Γ3,Γa, F , β and S. Let now assume that Lµ < L0. Then, there exists α ∈ R suchthat c20Lµ‖S‖L∞(Γ3) < α < c1. Using (5.12) and (5.6) we obtain that the functionalj satisfies condition (4.8). Therefore, by the second part of Theorem 2, Lemma 1and (5.4), we obtain that problem PV has a unique solution which depends Lipschitzcontinuously on f ∈ V and q ∈W , which concludes the proof.

References

[1] R.C. Batra and J.S. Yang. Saint-Venant’s principle in linear piezoelectricity. Journal ofElasticity, 38, 209 – 218, 1995.

[2] P. Bisenga, F. Lebon and F. Maceri. The unilateral frictional contact of a piezoelectricbody with a rigid support. In: Contact Mechanics, Kluwer, Dordrecht, 347 – 354, 2002.

[3] C. Ciulcu, D. Motreanu and M. Sofonea. Analysis of an elastic contact problem withslip dependent coefficient of friction. Mathematical Inequalities & Applications, 4, 465– 479, 2001.

[4] C. Corneschi, T.-V. Hoarau-Mantel and M. Sofonea. A quasistatic contact problem withslip dependent coefficient of friction for elastic materials. Journal of Applied Analysis,8, 59 – 80, 2002.

[5] G. Duvaut and J.-L. Lions. Inequalities in Mechanics and Physics. Springer-Verlag,Berlin, 1976.

[6] I.R. Ionescu and J.-C. Paumier. On the contact problem with slip displacement dependentfriction in elastostatics. Int. J. Engng. Sci., 34, 471 – 491, 1996.

[7] O. Kavian. Introduction à la théorie des points critiques et Applications aux équationselliptiques. Springer-Verlag, Paris, Berlin, 1993.

[8] F. Léné. Sur les matériaux élastiques à énergie de déformation non quadratique. Journalde Mécanique, 13, 499 – 534, 1974.


[9] F. Maceri and P. Bisegna. The unilateral frictionless contact of a piezoelectric body witha rigid support. Math. Comp. Modelling, 28, 19 – 28, 1998.

[10] R. D. Mindlin. Polarisation gradient in elastic dielectrics. Int. J. Solids Structures, 4,637 – 663, 1968.

[11] R. D. Mindlin. Continuum and lattice theories of influence of electromechanical cou-pling on capacitance of thin dielectric films. Int. J. Solids Structures, 4, 1197 – 1213,1969.

[12] R. D. Mindlin. Elasticity, piezoelasticity and crystal lattice dynamics. J. of Elasticity, 4,217 – 280, 1972.

[13] D. Motreanu and M. Sofonea. Evolutionary variational inequalities arising in quasistaticfrictional contact problems for elastic materials. Abstract and Applied Analysis, 4, 255– 279, 1999.

[14] D. Motreanu and M. Sofonea. Quasivariational inequalities and applications in frictionalcontact problems with normal compliance. Adv. Math. Sci. Appl., 10, 103 – 118, 2000.

[15] J. Necas and I. Hlavácek. Mathematical Theory of Elastic and Elastico-Plastic Bodies:An Introduction. Elsevier Scientific Publishing Company, Amsterdam, Oxford, NewYork, 1981.

[16] J.T. Oden and J.A.C. Martins. Models and computational methods for dynamic frictionphenomena. Computer Methods in Applied Mechanics and Engineering, 52, 527 – 634,1985.

[17] P. D. Panagiotopoulos. Inequality Problems in Mechanics and Applications. Birkhauser,Basel, 1985.

[18] E. Rabinowicz. Friction and Wear of Materials. Wiley, New York, 1965.[19] S.C. Scholz. The Mechanics of Earthquakes and Faulting. Cambridge Univ. Press,

Cambridge, 1990.[20] R. Temam. Problèmes mathématiques en plasticité, Méthodes mathématiques de

l’informatique. Gauthiers-Villars, Paris, 1983.[21] B. Tengiz and G. Tengiz. Some dynamic problems of the theory of electroelasticity.

Memoirs on Differential Equations and Mathematical Physics, 10, 1 – 53, 1997.[22] R. A. Toupin. The elastic dielectrics. J. Rat. Mech. Analysis, 5, 849 – 915, 1956.[23] R. A. Toupin. A dynamical theory of elastic dielectrics. Int. J. Engrg. Sci., 1, 101 – 126,

1963.[24] W. Voigt. Lehrbuch der Kristall-Physik. Teubner, Leipzig, 1910.[25] E. Zeidler. Nonlinear Functional Analysis and its Applications. IV: Applications to

Mathematical Physics. Springer-Verlag, New York, 1988.

Pjezoelektriko salycio su priklausomu nuo slydimo trinties koeficiento uždavinys

M. Sofonea, El-H. Essoufi

Mes nagrinejame matematini modeli, kuris aprašo salyti tarp pjezoelektriko ir kli uties. Lai-koma, kad medžiaga yra elektroelastine ir nusakoma netiesiniu elastingumo operatoriumi.Salytis modeliuojamas remiamtis sausos trinties Coulomb’o desniu, kuriame trinties koefi-cientas priklauso nuo slydimo. Mes gavome variacini modelio formulavima lygciu sistemosformoje, kurios nežinomaisiais yra perkeltasis laukas ir elektrinis potencialas. Irodomas spren-dinio silpnaja prasme egzistavimas ir su nedidelemis prielaidomis vienatis. Irodymas paremtasrezultatais gautais [14] darbe, kuriame tiriamos elipsines kvazivariacines nelygybes.

!"#$%'&)($+*,-#.0/1213.546

243–252c© 2004 Technika ISSN 1392-6292

STABILITY OF THREE-LEVEL DIFFERENCESCHEMES WITH RESPECT TO THERIGHT-HAND SIDE1

E.L. ZYUZINA

Department of Applied Mathematics and Informatics, Belarussian State University

F. Scoriny av. 4, 220050 Minsk, Belarus

E-mail: 728297:<;=>@?29?BADC8

Received January 11, 2004; revised June 1, 2004

Abstract. In this paper we investigate three-level difference schemes on non-uniform gridsin time. The a priori estimates of stability with respect to the initial data and the right-handside are obtained. New schemes of the raised order of approximation for wave equations areconstructed and investigated.

Key words: Three-level difference scheme, non-uniform grid, stability

1. Introduction

The main results on the theory of the stability of operator-difference schemes havebeen obtained using grids uniform in time [6, 8, 9]. Necessary and sufficient condi-tions of stability were already obtained in the sense of the initial data and the right-hand side in finite-dimensional Hilbert spaces.

For three-level difference schemes on non-uniform grids there are a few partic-ular results. In [1, 2] difference schemes of the first order of approximation wereconsidered for the case τn+1 ≥ τn. In [5] a priori estimate of uniform stability withrespect to initial data was received under special condition on operators and timegrid. The condition on time steps leaded us to the grid satisfying the geometricalprogression law τn+1 = qτn, q = const > 0. In the paper [7] basic canonical formshave been first introduced for three-level difference schemes on non-uniform in timegrids and important theorems concerning the stability with respect to initial data havebeen formulated.

In the work [4] for three-level difference schemes the a priori estimate of absolutestability of solution was obtained with respect to the initial data without assuming

1 The author wish to thank Prof. P. Matus for the statement of the problem and useful com-ments

244 E.L. Zyuzina

the Lipschitz–continuity of operators on a time variable. In [4] the special case ofgrid τn+1 ≥ τn was discussed and the a priori estimate of stability of three-leveloperator-difference schemes with respect to the initial data and the right-hand sidewas received. However the technique introduced in [4] doesn’t allow us to carry outthe investigation for the inverse relations of time steps.

Investigation of stability with respect to the right-hand side of three-level dif-ference schemes on non-uniform in time grids causes certain difficulties. In presentwork new a priori estimates of stability with respect to the initial data and the right-hand side are received with the use of specific technique, which consists in separateinvestigation of two cases τn+1 ≥ τn and τn+1 ≤ τn and it is represented in proofsof Theorems 1, 2. The stability of new computational methods on non-uniform gridswith respect to the initial data and the right-hand side is investigated on the basisof general a priori estimates obtained for three-level operator-difference schemes.Difference schemes of the second order of local approximation are constructed andinvestigated on non-uniform grids in time on standard stencils for hyperbolic equa-tions. Computational experiments for introduced schemes confirm the theoretical re-sults received.

2. Statement of the Problem

Let us note some features of the investigation of difference schemes on non-uniformin time grids. If in the initial differential problem the coefficients are constant, ap-proximation on a non-uniform grid leads us to operator-difference schemes depen-dent on grid node tn. If we require that these operators be Lipschitz–continuous, itwould lead us to an unnatural condition of the quasi-uniformity of a time grid. Thesecond problem is connected with a reduction of the order of local approximationwhen we go from a uniform grid to a non-uniform one.

The main problem we solve is to build new stable three-level difference schemesof the raised order of local approximation on the non-uniform grid using the standardstencils. We consider a three-level operator-difference scheme

Dytt + By

t+ Ay = ϕ,

y0 = u0, y1 = u1

(2.1)

on a non-uniform in time grid

ˆωτ = tn = tn−1 + τn, n ∈ 1, 2, · · · , N, t0 = 0, tN = T = ωτ ∪0, T . (2.2)

Here y = yn = y(tn) ∈ H is the sought function; u0, u1, ϕ(tn) ∈ H are given; H isthe finite-dimensional Hilbert space; D, B, A are linear operators acting in H ;

ytt =yt − yt

τ∗

, yt =yn+1 − yn

τn+1

, yt =yn − yn−1

τn

, y

t=

yn+1 − yn−1

τn + τn+1

,

y = yn+1, y = yn−1, τ = τn, τ+ = τn+1, τ∗ = 0.5(τn + τn+1),

y(σ1,σ2) = σ1y + (1 − σ1 − σ2)y + σ2y, y

(0.5) = 0.5(y + y).

Stability of Three-Level Difference Schemes 245

For any arbitrary functions u, v ∈ H the Cauchy–Schwartz inequality and theε-inequality hold true:

|(u, v)| ≤ ‖u‖‖v‖ ≤ ε‖u‖2 +1

4ε

‖v‖2, ε > 0 . (2.3)

For the self-adjoint and nonnegative operator A we define a semi-norm of the gridfunction u:

‖u‖2A = (Au, u), A = A

∗ ≥ 0.

To obtain a priori estimates of stability with respect to the right-hand side we’lluse stability conditions with respect to the initial data [4].

3. Auxiliary Results

Let us formulate some auxiliary results separately for cases when τn+1 ≥ τn andτn+1 ≤ τn.

Theorem 1. Let operators of scheme (2.1) satisfy the following conditions

D(t) = D∗(t) > 0, A = A

∗

> 0, (3.1)

R = D −τnτn+1

4A > 0, (3.2)

B ≥τn+1 − τn

4A, τn+1 ≥ τn. (3.3)

Let R, A be constant operators. Then the solution of problem (2.1) is stable withrespect to the initial data and the right-hand side and the following estimate is valid:

‖yt,n‖R + ‖y(0.5)n ‖A ≤

√2

(

‖yt,0‖R + ‖y(0.5)0 ‖A +

n∑

k=1

τk+1‖ϕk‖R−1

)

. (3.4)

Proof. To prove the theorem we scalar multiply scheme (2.1) by 2τ∗

y

t:

2τ∗

(

Dytt, y

t

)

+ 2τ∗

(

By

t, y

t

)

+ 2τ∗

(

Ay, y

t

)

= 2τ∗(ϕ, y

t).

Using the proof of Theorem 4.1 [4] and the following representation

2τ∗(ϕ, y

t) = τn+1(ϕ, yt) + τn(ϕ, yt),

we receive the energy inequality:

‖yt‖2R + ‖y(0.5)

n ‖2A −

(

‖yt‖2R + ‖y

(0.5)n−1 ‖

2A

)

+ τ∗

τ+ − τ

2‖ytt‖

2R

+ 2τ∗‖y

t‖2

B−

τ+−τ

4A

= τn+1(ϕ, yt) + τn(ϕ, yt) . (3.5)

In conditions of the theorem

246 E.L. Zyuzina

‖yt‖2R + ‖y(0.5)

n ‖2A − (‖yt‖

2R + ‖y

(0.5)n−1 ‖

2A) ≤ τn+1(ϕ, yt) + τn(ϕ, yt).

Let us introduce a notation

G = ‖yt‖2R + ‖y(0.5)

n ‖2A. (3.6)

Then taking into account that τn+1 ≥ τn and using the Cauchy–Schwartz inequality(2.3), we have

τGt ≤ τ+(ϕ, yt) + τ(ϕ, yt) ≤ τ+‖ϕ‖R−1‖yt‖R + τ‖ϕ‖R−1‖yt‖R

≤ τ+‖ϕ‖R−1(‖yt‖R + ‖yt‖R) ≤ τ+‖ϕ‖R−1(G1/2 + G1/2).

(3.7)

Using the following identity [3]:

(

G1/2

)

t=

G1/2 − G

1/2

τn

=G − G

τn(G1/2 + G1/2)

=Gt

G1/2 + G

1/2(3.8)

from (3.7) we get the estimate

τn

(

G1/2

)

t≤ τn+1‖ϕ‖R−1 ,

or

(

‖yt,n‖2R + ‖y(0.5)

n ‖2A

)1/2

≤(

‖yt,n‖2R + ‖y

(0.5)n−1 ‖

2A

)1/2

+ τn+1‖ϕn‖R−1 .

And then we have

(

‖yt,n‖2R + ‖y(0.5)

n ‖2A

)1/2

≤(

‖yt,0‖2R + ‖y

(0.5)0 ‖2

A

)1/2

+

n∑

k=1

τk+1‖ϕk‖R−1 .

(3.9)Now using the evident relations:

|a| + |b| ≤√

2(a2 + b2),

√

a2 + b

2 ≤ |a| + |b|, (3.10)

the statement of the theorem follows from inequality (3.9).

In order to formulate the theorem about stability for the inverse relations of timesteps we rewrite the three–level operator–difference scheme (2.1) in the followingform:

D

yn+1 − 2yn + yn−1

τnτn+1

+

(

B −τn+1 − τn

τn+1τn

D

)

yn+1 − yn−1

2τ∗

+ Ayn = ϕn,

y0 = u0, y1 = u1 .

(3.11)

Theorem 2. Let operators D(t), B(t), A satisfy the following conditions:

D(t) = D∗(t) > 0, B(t) > 0, A = A

∗

> 0, (3.12)


R = D −τnτn+1

4A > 0, (3.13)

and operators A, R are constant. Let

τn+1 ≤ τn . (3.14)

Then operator-difference scheme (2.1) is stable with respect to the initial data andthe right-hand side and the a priori estimate is true:

‖yt,n‖R + ‖y(0.5)n ‖A ≤

τ1

√2

τn+1

‖yt(0)‖R + ‖y(0.5)0 ‖A +

n∑

k=1

τk‖ϕk‖R−1

.

(3.15)

Proof. Let us scalarly multiply scheme (3.11) by τnτn+1(yn+1 − yn−1) and userepresentations for scalar products given in Theorem 4.1 [4]:

(

D((yn+1 − yn) − (yn − yn−1)), (yn+1 − yn) + (yn − yn−1))

+

((

τnτn+1

2τ∗

B −τn+1 − τn

2τ∗

D

)

(yn+1 − yn−1), yn+1 − yn−1

)

+ τnτn+1(Ayn, yn+1 − yn−1) = τnτn+1(ϕ, yn+1 − yn−1).

Taking into consideration conditions of the theorem (3.12), (3.13) and the followingidentity

τnτn+1(ϕ, yn+1 − yn−1) = τnτn+1(ϕ, τn+1yt) + τnτn+1(ϕ, τnyt),

we get the energy inequality

τ2n+1‖yt‖

2R + τnτn+1‖y

(0.5)n ‖2

A − (τ2n‖yt‖

2R + τnτn+1‖y

(0.5)

n−1 ‖2A)

≤ τnτn+1(ϕ, τn+1yt) + τnτn+1(ϕ, τnyt).(3.16)

Since τ2n+1 ≤ τnτn+1 ≤ τ

2n (see (3.14)), then expression (3.16) takes the

following form:

τ2n+1(‖yt‖

2R + ‖y(0.5)

n ‖2A)− τ

2n(‖yt‖

2R + ‖y

(0.5)

n−1 ‖2A) ≤ τ

2n((ϕ, τn+1yt) + (ϕ, τnyt)).

Using notation (3.6) and the Cauchy–Schwartz inequality, we rewrite the last rela-tion:

τn(τ2n+1Gn)t ≤ τ

2n(τn+1(ϕ, yt) + τn(ϕ, yt))

≤ τ2n‖ϕ‖R−1(τn+1G

1/2n + τnG

1/2

n−1). (3.17)

Let us note that the following identity similar to (3.8) is valid:

(

τn+1G1/2n

)

t=

τn+1G1/2n − τnG

1/2

n−1

τn

=τ

2n+1Gn − τ

2nGn−1

τn(τn+1G1/2n + τnG

1/2

n−1)

=(τ2

n+1Gn)t

τn+1G1/2n + τnG

1/2

n−1

. (3.18)

248 E.L. Zyuzina

Taking into account identity (3.18) we receive from (3.17) the following estimate:

τn(τ2n+1Gn)t ≤ τ

2n‖ϕ‖R−1

or

τn+1(‖yt,n‖2R + ‖y(0.5)

n ‖2A)1/2 (3.19)

≤ τn(‖yt,n−1‖2R + ‖y

(0.5)n−1 ‖

2A)1/2 + τ

2n‖ϕ‖R−1 .

Recursively we get

τn+1(‖yt,n‖2R + ‖y(0.5)

n ‖2A)1/2 (3.20)

≤ τ1

(‖yt(0)‖2R + ‖y

(0.5)

0 ‖2A)1/2 +

n∑

k=1

τk‖ϕk‖R−1

.

Now using relations (3.10) from the last inequality one can easy get a priori estimate(3.15).

Remark 1. Let us note that stability estimates (3.4), (3.15) are received without usingthe Gronuoll lemma, which is usually applied for investigation of stability with re-spect to the right-hand side, and the estimates don’t include the constant e

cT , whichbecomes large with the growth of T . If the series

n∑

k=1

τk+1‖ϕk‖R−1 ,

n∑

k=1

τk‖ϕk‖R−1

converge when n → ∞, then estimates (3.4), (3.15) express the global stability ofthree-level difference scheme (2.1).

4. Stability with Respect to the Right-Hand Side on ArbitraryGrids

Let us combine the results obtained and formulate general theorem about uniformstability of three-level operator-difference schemes. We assume an arbitrary timegrid, where principle of mesh refinement changes k times [4]:

Figure 1. The non-uniform time grid.


Theorem 3. We assume that operators of difference scheme (2.1) D(t), B(t), A sat-isfy the following conditions:

D(t) = D∗(t) > 0, A = A

∗

> 0, (4.1)

R = D −τnτn+1

4A > 0, B ≥ max

τn+1 − τn

4A, 0

, (4.2)

and A, R are constant operators. We also assume that time steps are interrelated as

τmj

τmj+1

≤ cmj+1≤ c0, j = 0, 1, . . . , k,

τm0= τ1, τmk+1

= τN0,

(4.3)

where k is the finite number of changes of mesh refinement principle.Then the solution of problem (2.1) is stable with respect to the initial data and

the right-hand side, and for arbitrary τn the following a priori estimate holds true(an absolute stability):

(

‖yt,n‖2R + ‖y(0.5)

n ‖2A

)1/2

≤ ck0

(

‖yt,0‖2R + ‖y

(0.5)0 ‖2

A

)1/2

+N0−1∑

s=1

maxτs, τs+1‖ϕs‖R−1 .

(4.4)

Proof. Let t ∈ [tmk, tN0

] and time steps become finer to the end of the interval.Then according to Theorem 2 (see (3.20)) the following estimate is valid:

‖yn+1‖1 ≤τmk

τN0

‖ymk‖1 +

N0−1∑

s=mk

τs‖ϕs‖R−1

,

n = mk, mk + 1, . . . , N0 − 1,

(4.5)

where

‖yn+1‖1 =(

‖yt,n‖2R + ‖y(0.5)

n ‖2A

)1/2

.

Let in the moment tmkthe principle of mesh refinement changes, i. e., the time

steps become related as τ+ > τ . Then according to the Theorem 1 (see (3.9))

‖yn+1‖1 ≤ ‖ymk−1‖1 +

mk−1∑

s=mk−1

τs+1‖ϕs‖R−1 ,

n = mk−1, mk−1 + 1, · · · , mk − 1.

Substituting the last inequality to (4.5), we get the estimate

‖yn+1‖1 ≤τmk

τN0

‖ymk−1‖1 +

N0−1∑

s=mk−1

maxτs, τs+1‖ϕs‖R−1

,

n = mk−1, . . . , N0 − 1.

(4.6)

Recursively continuing (4.6) and taking into account steps interrelations (4.3), wereceive the required estimate of stability.

250 E.L. Zyuzina

5. Examples

5.1. Wave equation

In the domain Q = Ω ∪ [0, T ], Ω = 0 ≤ x ≤ l it is necessary to find thesolution of the first boundary value problem for the one dimensional wave equation:

∂2u

∂t2

=∂

2u

∂x2

+ f(x, t), 0 < x < l, t > 0, (5.1)

u(0, t) = u(l, t) = 0, u(x, 0) = u0(x),∂u

∂t

(x, 0) = u0(x). (5.2)

On the uniform space grid ωh = xi = ih, i = 0, N, hN = l and the non-uniform time grid (2.2) we replace the problem (5.1), (5.2) by the class of differenceschemes with weights

ytt + Ay(σ1,σ2) = ϕ, y0 = u0, y1 = u1, (5.3)

wherey(σ1,σ2) = σ1y + (1 − σ1 − σ2)y + σ2y,

(Av)i = −vxx,i = (vi+1 − 2vi + vi−1)/h2, i = 1, N − 1, v0 = vN = 0,

ϕ(t) = (ϕ1(t), ϕ2(t), . . . , ϕN−1(t))T, ϕi(t) = f(xi, t);

the operator A : H → H , where the linear space H = Ωh consists of a set of vectorsv = (v1(t), v2(t), . . . , vN−1(t))

T ; a scalar product and a norm in H are assigned asusual

(y, v) =

N−1∑

i=1

hyivi, ‖y‖ =√

(y, y).

A reduction of scheme (5.3) to the canonical form was done and the validity ofall requirements of Theorem 3 were proved in [4]. Therefore the following theoremis true.

Theorem 4. Assume that

σ1 ≥ σ2 + max

τn − τn+1

2(τn+1 + τn), 0

, σ1 + σ2 =1

2. (5.4)

Then, the difference scheme with weights (5.3) is stable in the sense of the initial dataand the right-hand side and the following a priori estimate holds true

(

‖yt,n‖2 + ‖y(0.5)

n ‖2A

)1/2

≤ ck0

(

‖yt,0‖2 + ‖y

(0.5)0 ‖2

A

)1/2

+

N0−1∑

s=1

maxτs, τs+1‖ϕs‖.

The parameters σj are defined by taking into account the second order accuracyapproximation condition:

σ1τ+ − σ2τ =τ+ − τ

3

and the stability requirement (5.4). Thus we get, that (see, [4]):

σ1 =2τ+ + τ

6(τ+ + τ), σ2 =

τ+ + 2τ

6(τ+ + τ). (5.5)


5.2. Numerical experiment

The accuracy of the proposed scheme (5.3), (5.5) was examined in numeroustests. The obtained results were compared with results obtained by using classi-cal schemes for numerical solution of differential problem (5.1), (5.2) on uniformand non-uniform grids in time. Numerical experiment was undertaken in the domainΩ = [0, 1], t ∈ [0, 1] . The exact solution is given as

u(x, t) = sin(

πx

l

)

sin

(

πt

l

)

+ t2(x2 − lx), l = 1 .

In order to check the second order of approximation and convergence of differencescheme (5.3), (5.5), the initial boundary-value problem was solved on the sequenceof grids: ωh ×ωτ , ωh/2 ×ωτ/2, ωh/4 ×ωτ/4, . . . . The non-uniform time grid ωτ

was built using the random-number generator. The remaining grids were obtained bydividing each time interval into two equal parts. Thus the number of points whereτj 6= τj+1 remains constant for all experiments. In this sense the applied time gridsbecome close to the uniform grid.

The absolute error of solution y is given by

zN = max

(x,t)∈ωhτ

|y(x, t) − u(x, t)|, ωhτ = ωh × ωτ .

Since in real problems the exact solution is usually unknown, let us introduce theaposteriori error estimate of the solution yhτ , which can be obtained using the Rungeestimator:

DN =

1

3max

(x,t)∈ω2(hτ)

|yhτ (x, t) − y2(hτ)(x, t)|.

The second aposteriori estimate

pN = log2(D

N/2/D

N)

gives the convergence order of the approximation yhτ . The results of the experimentsfor the time level t=1 are presented in Tab. 1.

Table 1. Convergence analysis for a test problem.

Nx×Nt 20 × 20 40 × 40 80 × 80 160 × 160 320 × 320 640 × 640

zN 0.015670 0.004022 0.001017 0.000255 0.000064 0.000016

DN

0.014482 0.003883 0.001001 0.000254 0.000064 0.000016

pN

1.89914 1.95513 1.97902 1.98868 2.0

252 E.L. Zyuzina

References

[1] A. Bokov. Energetic inequalities for three–level difference schemes on grids changing intime. Godishn. Vissh. Uchebni Zabed. Prilozh. Mat., 12(2), 87 – 96, 1976.

[2] E. Diakonov and A. Bokov. Energetic inequalities for difference schemes on grids chang-ing in time. Dokl. Bolg. AN., 28(2), 157 – 160, 1975.

[3] B. Jovanovich, S. Lemeshevsky and P. Matus. On the stability of differential–operatorequations and operator–difference schemes as t→∞. Comput. Meth. Appl. Math., 2(2),153 – 170, 2002.

[4] P. Matus and E. Zyuzina. Three–level difference schemes on nonuniform in time grids.Comput. Meth. Appl. Math., 1(3), 265 – 284, 2001.

[5] V. Mazhukin, P. Matus and I. Mozolevski. About stability of three–level differenceschemes on non-uniform time grids. Dokl. NAN Belarusi, 44(6), 23 – 25, 2000.

[6] A. Samarskii. The Theory of Difference Schemes. Marcel Dekker Inc., New York-Basel,2001.

[7] A. Samarskii, P. Vabishchevich, E. Makarevich and P. Matus. Stability of three–leveldifference schemes on non-uniform in time grids. Dokl. Russ. Acad. Nauk, 376(6), 738 –741, 2001.

[8] A. Samarskii, P. Vabishchevich and P. Matus. Difference Schemes with Operator Factors.Kluwer Academic Publishers, Boston/Dordrecht/London, 2002.

[9] A. A. Samarskii and A. V. Goolin. Stability of Difference Schemes. Nauka, Moskow,1973. (in Russian)

Trisluoksniu baigtiniu skirtumu schemu stabilumas dešiniosios puses atžvilgiu

E. Zyuzina

Straipsnyje nagrinejamos trisluoksnes baigtiniu skirtumu schemos su netolygiu laikinu žin-gsniu. Gauti aprioriniai stabilumo iverciai pradiniu duomenu ir dešiniosios puses atžvilgiu.Pasiulytos naujos aukštesnes aproksimacijos eiles baigtiniu skirtumu schemos vienmatei ban-gos lygciai. Pateikti skaitinio eksperimento rezultatai.

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xs(A) = 0.5x, x

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∂cs(A)

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∂2cs(A)

∂x2, x1 < x < x2(t)

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B

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∂cs(A)

∂x= 0, x = x1

∂cl(j)

∂x= 0, x = x3, j = A, B.

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∂cs(j)

∂x

∣

∣

∣

∣

∣

x2 (t)−0

− Dl(j)

∂cl(j)

∂x

∣

∣

∣

∣

∣

x2 (t)+0

= −υph(cs(j) − cl(j)),

j = A, B, x = x2(t),

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xsAC = 4γA x

(A)x

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(

∆HmeltAC

R TmeltAC

TmeltAC − T

T

)

,

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(B)x

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(

∆HmeltBC

R TmeltBC

TmeltBC − T

T

)

.

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é9îÈéT

Å ÇfOUé\OÈé9×*PéVî6&OºÆîºéR

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meltAC

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meltAC

T

meltBC

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xsAC

Ox

sBC

S¾îºÒ× +UÎÏ /îÈéÕÇÈÆðÇVOÈÅ,OÈÆROºé9wÅ OºÒ +UÎ ÍH/TÙ

x(A) + k x

(B) =η

1 − x(A) − x

(B),

+²Î ì2/

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k =γB

γA

exp

(

∆HmeltBC

R TmeltBC

TmeltBC − T

T

)

exp

(

∆HmeltAC

R TmeltAC

TmeltAC − T

T

), η =1

4 γA exp

(

∆HmeltAC

R TmeltAC

TmeltAC − T

T

).

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F(c(A), c

(B)) = 0,

cs(j) = f(c(A)

, c(B)

, T ) j = A, B, x = x2(t)

+²ÎIÑ/

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υph

3Å² éÅ fO)UéÕéVÔ ÓTÒÒî6Å &OºéÄÇÈÊÇOºéV× OUéÕÅ OºéVî)S¹ÓVéÅ Çk]N`é91

d d d d d dx1 x2(t) x3 0 1 2

JV= B J9= B

?802! L> & Dþ FIE3JVý B ý J9E JTýFKE

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Å OÈÒ3O)UéÇ)PÓVé"ÒSÓTÒ×QPÆNOº&OºÅ ÒGOØê½îÈÅ OðØ é9Ç

(x, t)îºé9GÇ9Ù

t = t, x = ϕ(x, t) =

x1 + x ls, 0 ≤ x ≤ 1,

x2 + (x − 1) ll, 1 ≤ x ≤ 2,

lsll

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x = 1

IUé XÒ½êé9îÈÅ NXAé9Ë3Æ&OºÅ ÒGÇÅ fOUé ©TÓ¦I¯¹§Õ·ª¼½©V¤(0 ≤ x < 1)

O6OÉé-O)Ué S¾ÒØ Ø Ò½ÔÅ NXS¾Òîº× a Íc²Ù

ls∂c

s(A)

∂t

− ϕt

∂cs(A)

∂x

=∂

∂x

(

Ds(A)

ls

∂cs(A)

∂x

)

,

(aA + aC)cs(A) + (aB + aC)cs(B) = ρs,

∂cs(A)

∂x

∣

∣

∣

∣

∣

x=0

= 0 .

+Uñ ÍH/

IUé"ÇÈé9ÓTÒG¥é9Ë3ÆOÈÅ ÒÅ +Uñ ÍH/ÔòÇÒGðROºÅ é9*S¾îºÒ×ÓVÒÅOºÅ ÒG +UÎ ÍH/ ÔîºÅ,OVOÈé9Å QOUéS¾Òîº×

xs(A) + x

s(B) =1

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sÅ ÇÔUéé9ÇÈÅOÊwÒSBOUéÇÈÒØ Å 1

aA

aBO

aC

îÈéÄ&OºÒ×Å Ó¢ÔéVÅ XUO6ÇòÒSÓTÒ×QPÒé9OA

B

OC

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(1 < x ≤ 2)Oîºé XÅ êé9¬Ç9Ù

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ll∂c

l(j)

∂t

− ϕt

∂cl(j)

∂x

=∂

∂x

(

Dl(j)

ll

∂cl(j)

∂x

)

, j = A, B ,

∂c(j)

∂x

∣

∣

∣

∣

∣

x=2

= 0, j = A, B .

+UñÎ /

ª¼½©T¤® È¼½©¯°®U¯¹´O¯°®¤ ¾µT¼GÀ6¤ é9Ë3ÆOÈÅ ÒÇòîÈé XÅ êé9wOx = 1

Ù

Ds(A)

ls

∂cs(A)

∂x

∣

∣

∣

∣

∣

x=1−0

−D

l(A)

ll

∂cl(A)

∂x

∣

∣

∣

∣

∣

x=1+0

= −υph(cs(A) − cl(A)) ,

Ds(B)

ls

∂cs(B)

∂x

∣

∣

∣

∣

∣

x=1−0

−D

l(B)

ll

∂cl(B)

∂x

∣

∣

∣

∣

∣

x=1+0

= −υph(cs(B) − cl(B)) ,

F(c(A), c

(B)) = 0, cs(j) = f(c(A)

, c(B)

, T ), j = A, B .

+Uñ ñ2/

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υph.

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ú 5 ú@.?@¾ö> @0>G÷>=? ù >õKù > >

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ÉÒ½Ô'S¾ÆÓ_OºÅ ÒGÇcs(A)

cs(B)

υph

c(A)

c(B)

OîºéÓVÒ×QPÆROºé9 &O!O)Ué*XîºÅ Òé9Ç9_ÔeUÅ Ø é

ϕ

lsllD

sD

lOîºé'ÓVØ ÓVÆØ OÈéK &O OUé.ÓTéVOºéVîAÒSeO)UéXîºÅ ÓVéVØ Ø Ç9

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υph

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h

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2D(s/l)

υph

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−N1

OÈÒN2

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0

IUé!]Å,OÈé Åé9îÈé9ÓTé ÇºÓUé9×¥éÕÓ9Owð#éÕÔîºÅO)OÈé9.GÇ OÙ2 M= F ýFIE þ F¸=6JVý B ýC B E B J¥=KEF þGBBþ F þ E þB ( J F¸J B û FIýC.JCJVý

= B 9E þ ýýC B J þ JVE3= B6þ ýF ÕB B B

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ls

hi− 1

2

2(c

s(A)

i − cs(A)

i ) + lshi+ 1

2

2(c

s(A)

i − cs(A)

i )

− τ

[

ϕt

2

∣

∣

∣

∣

xi−

1

2

(cs(A)

i − cs(A)

i−1 ) +ϕt

2

∣

∣

∣

∣

xi+

1

2

(cs(A)

i+1 − cs(A)

i )

]

+Uñ Ì /

= τ

[

Ds(A)

ls

cs(A)

i+1 − cs(A)

i

hi+ 1

2

−D

s(A)

ls

cs(A)

0 − cs(A)

i−1

hi− 1

2

]

, −N1 ≤ i < 0 ,

(aA + aC)cs(A)

i + (aB + aC)cs(B)

i = ρs, −N1 ≤ i < 0 ,

+¹ñ Ï2/

lsh−

1

2

2(c

s(j)0 − c

s(j)0 ) + ll

h 1

2

2(c

l(j)0 − c

l(j)0 )

− τ

[

ϕt

2

∣

∣

∣

∣

x−

1

2

(cs(j)0 − c

s(j)−1 ) +

ϕt

2

∣

∣

∣

∣

x 1

2

(cl(j)1 − c

l(j)0 )

]

+Uñ ì2/

= τ

[

−(cs(j)0 − c

l(j)0 )υph +

Dl(j)

ll

cl(j)1 − c

l(j)0

h 1

2

−D

s(j)

ls

cs(j)0 − c

s(j)−1

h−

1

2

]

, j = A, B ,

F(c(A)

0 , c(B)

0 ) = 0, cs(j)

0 = f(c(A)

0 , c(B)

0 , T ), j = A, B,

+¹ñÑ /

llhi− 1

2

2(c

l(j)

i − cl(j)

i ) + llhi+ 1

2

2(c

l(j)

i − cl(j)

i )

− τ

[

ϕt

2

∣

∣

∣

∣

xi−

1

2

(cl(j)

i − cl(j)

i−1) +ϕt

2

∣

∣

∣

∣

xi+

1

2

(cl(j)

i+1 − cl(j)

i )

]

+Uñ 2/

= τ

[

Dl(j)

ll

cl(j)

i+1 − cl(j)

i

hi+ 1

2

−D

l(j)

ll

cl(j)0 − c

l(j)

i−1

hi− 1

2

]

, 0 < i ≤ N2, j = A, B,

ÔeUéVîºéxi−1/2

Å Ç OUé ÓTé9OºéVîwÒS OUé ÓTé9Ø Ø[xi−1, xi]

hi−1/2 = xi − xi−1

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F(ζ) = 0,

+¹ñ /

UéVîºéζ = (c

s(A)

−N1, c

s(B)

−N1, c

s(A)

−N1+1, ..., c

s(A)

0 , cs(B)

0 , υph, c(A)

0 , c(B)

0 , c(A)

1 , ..., c(B)

N2)T

IUéÕÅOºéVî6&OºÅ êGé-PîºÒÓTéKÇÈÇkS¾ÒGîIO)Ué ÇÈÒØ ÆROÈÅ ÒwÒS +Uñ 2/ Å Çé0OÈé9îÈ×Å é9.GÇkS¾ÒØ Ø Ò½ÔÇ9Ù

F′(

(k)

ζ )δζ + F((k)

ζ ) = 0,

ÔeUéVîºéF

′

Å ÇQO)Ué ëGGÓTÒðÅ ×&OºîÈÅ,`δζ =

(k+1)

ζ −(k)

ζ , k

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δζ

[OUéÓTÒG×*PÒéVOºÇÒS¢ÔeUÅ ÓU OîºéwÅ ÓVîÈé9×¥é9OºÇÒSÓTÒ#ÓTéVOºîºOÈÅ ÒÇÒSOØ Ø Ç)P#éKÓTÅ é9Ç

δcs(j)

i , i = −N1, ..., 0, δcl(j)

i , i = 0, ..., N2, j = A, B

O¬Å OºéVî)S¹ÓTéÄî6&Oºéδυph

IUé ë3ÓVÒðÅ O¬×OÈîºÅ`¬Å Çé]éKwGÇVÙ

−N1

B

−N1

C 0 . . . . .

−N1

ξ . . . . . . . . . . . . . . . . 0−N1 + 1

A

−N1 + 1

B

−N1 + 1

C 0 . .

−N1 + 1

ξ . . . . . . . . . . . . . . . . 0

0 . . . . . .

−1

A

−1

B

−1

C

−1

ξ . . . . . . . . . . . . . . . . 0

0 . . . . . . . . . . . . .

0

A

0

B

0

C . . . . . . . . . . 0

0 . . . . . . . . . . . . . . . . . . .

1

ξ

1

A

1

B

1

C . . . . . 0

0 . . . . . . . . . . . . . . . . . . .

N2 − 1

ξ . . . . .

N2 − 1

A

N2 − 1

B

N2 − 1

C

0 . . . . . . . . . . . . . . . . . . .

N2

ξ . . . . . 0

N2

A

N2

B

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i

A,

i

C =

(

∗ 0

0 0

)

, −N1 ≤ i < 0,

(

∗ 0

0 ∗

)

, 0 < i ≤ N2,

i

B =

(

∗ 0

∗ ∗

)

, −N1 ≤ i < 0,

(

∗ 0

0 ∗

)

, 0 < i ≤ N2,

i

ξ =

(

∗

0

)

, −N1 ≤ i < 0 ,

(

∗

∗

)

, 0 < i ≤ N2 .

0

A

O 0

C

îÈé2 × 5

ðØ ÒÓ6ÉÇV 0

B

Å Ç5 × 5

ðØ ÒÓ6É5Ù

0

A,

0

C =

∗ 00 ∗

0 00 00 0

,

0

B =

∗ 0 ∗ ∗ 00 ∗ ∗ 0 ∗

0 0 0 ∗ ∗

∗ ∗ 0 ∗ ∗

∗ ∗ 0 ∗ ∗

.

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δcs(A)

i , i = −N1, . . . , 0

δc(j)

i , j = A, B, i = 0, . . . , N2

δυph

Å fO)Ué S¾ÒGØ Ø Ò½ÔÅ XS¾ÒGîÈ×

δcs(A)

i = αsi+1δc

s(A)

i+1 + βsi+1 + γ

si+1δυph, −N1 ≤ i ≤ −1,

δc(j)

i = αl(j)

i−1δc(j)

i−1 + βl(j)

i−1 + γl(j)

i−1δυph, 1 ≤ i ≤ N2, j = A, B .

Ò3é'ÓTÅ éVOºÇα

s(A)

i , βs(A)

i , γs(A)

i , i = −N1, . . . , 0, α(j)

i , β(j)

i , γ(j)

i , i = 0, . . . , N2,

j = A, B

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(δcs(A)

0 , δcs(B)

0 ,

δυph, δc(A)

0 , δc(B)

0 )T OºÇÇÈÒØ ÆROÈÅ Ò Å ÇÒðNOºOÅ éKAÅ îÈéKÓ_OºØ ÊÕðÊÕ×&OºîÈÅ,` Å êGéVî6ÇDÅ Ò1IUé

ðÓ6É.ÇDÆðÇOºÅOºÆROÈÅ ÒYPOî)O"ÒSBO)Ué¥×é0O)UÒ$XÅ êé9Çδc

s(j)

i , i = −N1, . . . , 0, δcl(j)

i , i =0, . . . , N2, j = A, B

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cs(A)

, cs(B)

, υph,

c(A)

, c(B)

ÇÇO6Oî)OÈÅ NXé9ÇVOÈÅ ×&OÈé!S¾ÒGîcs(A)

, cs(B)

, υph, c(A)

, c(B)

Î"éOÈé9îÈ×Å &OºÅ ÒG ÒSÓVÒé 'ÓTÅ éVOºÇ

αsi

β

si

γ

si

i = −N1, . . . , 0

α

l(j)

i

β

l(j)

i

γl(j)

i , i = 0, . . . , N2

j = A, B

wîÈéKÆÓ_OºÅ ÒG OºÒ*O)UéÕØ Å Ë3ÆÅ ?gOÇDÒGØ Å Å OÈé9îVS¹GÓTéñ"Å îºé9Ó_O×OÈîºÅ`Å êé9îºÇÈÅ ÒG6S¾ÒGîÒðRO6OÅ Å X

δcs(A)

0 , δcs(B)

0 , δυph, δc(A)

0 , δc(B)

0

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lsllϕt

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Â J#$&" F?MVK' ( "9J#(E D-H M9K J9MVG D

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lsllυph

Å PNPîºÒd`Å ×OÈÅ ÒÒS +¹ñ ÍH/)3 +Uñ ñ2/Å é`RPØ Å ÓTÅ,O ÔòÊ3Å,OÈÅ ÒØOºîºÇ)P#ÒGîVOOÈéVîº×Ç îÈéPNPîºÒd`Å ×OÈé9ðÊOUé"ÆP3ÔÅ ÇºÓUéV×éG3 Òé 'ÓTÅ éVOºÇ éPéV#Å X ÒÅ OÈéVî)S¹ÓVéîºOÈéυph

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cs(A)

i = σsi+1c

s(A)

i+1 + ηsi+1, −N1 ≤ i ≤ −1 ,

cl(j)

i = σl(j)

i−1cl(j)

i−1+ η

l(j)

i−1, 1 ≤ i ≤ N2, j = A, B .

+¹Ì ÍH/

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cs(A)

0 , cs(B)

0 , c(A)

0 , c(B)

0 .

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Å ÇÓVOØ ÓTÆØ OÈé91]ÇDé9×Å¸Ö³Å ×QPØ Å ÓTÅ,O-Oºé9ÓUÅ Ë3Æé OUéïéVÔIOºÒOØ XÒGîÈÅ,O)U×uÅ ÇPNPØ Å éK¬ÒGØ Ê&OeOUéPNUGÇDé!OÈî6O#ÇDÅ,OÈÅ Ò¬Å OÈé9îVS¹GÓTéG

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Ds(A) = 5 · 10−12 cm

2

s

, Dl(A/B) = 5 · 10−5 cm

2

s

.

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2/(2D

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N1 = 5000OÅ +OUé¢Ø Å ËGÆÅ Å Ç

N2 = 500

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T0

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O OÈé9×*PéVî6&OºÆîºéT1 < T0

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T = T0 −α t

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T1

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A

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B

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%1&'&)(+*d),*-.-/10 . 0,*/Z.2 +43,*65C.798-4*:;:6=<>/=*@? . -7. 2A:!0B*R06 2A7 ßV¾ àHº »Þ½ ¾ #P¸<>> º=ã ÊË ß6É½ º > ¾HãÞ¾ ·Ç1¸ @º=ã GQ>> º=ãR;Âº á PÀ½ Ç ßV¾ Ê > ãÞº ÇÞÇ º º9½ »ÞÉº9ãá º9»8;º=º »8;¾.ã¯¾P» »Þ½ 7B ã¯¾ ¸ ¸ Ç ¾Hã á º9»8;º=º ãÞ¾» »Þ½ 7B ãÞ¾ ¸ ¸+m S4"º mÇE· ã = ßVº G®> ãÞº ÇÞÇ ½ 7BÇ É¾ºP 5 Éº > ¾ã¯¾ ·Ç"¸ @Hº9ã½ Ç3> ãE»¯½ ¸ ¸ @ S ¸ ¸ º ;Ò½È»¯É 9 · ½ À1 Éº Ê ½C"» · ãÞº»ÞÉº=¾ã @B½ Ç·Ç º »Þ¾ º Ç ßVã¯½ á º»ÞÉº 9¾<;x¾=M9 · ½ Ç ½ ' º = ¾Hã ÊËPá¸ º > ¾ã¯¾ ·ÇëÊË »Þº9ã¯½ P¸ Ç9¦»ÞÉ½ ÇeÇ º=ßV»Þ½ ¾ ;Âº > ã¯º Ç º » B º º=ã ¸?B ¾¦àHº9ã ½ 7B ºD · »¯½ ¾ Ç= ;ÒÉ½ ß6É ã¯º ·Ç º »¯¾ Ç ½ Êâ·¸ »Þº»ÞÉº > ãÞ¾ß9º Ç¯Ç º Ç ½ J>#?> º9ã > ãÞº ÇÞÇ%Ê ß6É½ º Ç=3 ¾¾ ½ »¯ãÞ¾ · ßV»Þ½ ¾ Ç ½ »¯¾ ÊË »ÞÉº ÊË »¯½ ß P¸Ê ¾ º ¸ ÇÒ º Ç ßVã¯½ á ½ 7BË ã @½ 7B ¾= >> º=ã ã¯º B ½ àHº ½ I 1¿ ä<O

ºV» ·#Ç ßV¾ Ç ½ º=ã Ç¯PÊF>¸ º¾?= à¾ ¸ ·Ê ºº ¸ º Ê º »V

¾?= mÊ ½C"» · ãÞº½ > ¾ã¯¾ ·Ç¸ @Hº9ã 1 »¨ß9¾ Ç ½ Ç »B¾=Ò»ÞÉã¯º9º > É Ç º Ç=Ì 9 · ½ ½ ã > É Ç º Ç ½ ºD"º

”f”P

”a”

PâÇ ¾ ¸ ½ > É HÇ º ;ÒÉ½ ß6Éß9¾ Ç ½ Ç » Ç ¾=S á ã¯¾ ·#ÇÇ ¾ ¸ ½ ÊË »Þº=ãÞ½ P¸5P# ½ Ç ½ º"º ”s”

"Ç"¸ ; @ Ç ;º HÇÞÇÞ·Ê º »ÞÉº Ç ½ º¾?=ë»ÞÉ½ Ç àH¾ ¸ ·Ê ºº ¸ º Ê º »"½ Ç¤¸ ã B ºß9¾ ÊF> ãÞº ;Ò½ »ÞÉ»ÞÉº Ç ½ º¨¾?= > ¾ã¯º ÇP#ÇÞÊ¸ ¸ ßV¾ ÊF> ã¯º ;Ò½ »ÞÉ»ÞÉº Ç ½ ºâ¾=»ÞÉº¨º »Þ½ ã¯º Ç @ Ç »¯º Ê.E º º ¾»Þº¨»ÞÉº¨àH¾ ¸ ·Ê º Ç ¾ß9ß ·/> ½ º á @ ; »¯º9ã 5 ½ ã Ç ¾ ¸ ½ ÊË »Þº=ãÞ½ P¸ëá @

Vf , Va

PVs

ã¯º ÇA> ºLß»Þ½ àº ¸ @ Éº ²#PJ §Ë£F6³Þ©H¬V´¶G¦ ¾?=eº ß6É.ßV¾ Ê > ¾ º »Ò½ Ç¤ º S º «á @ Ì

φα =Vα

V

, α = f, a, s .

æ"· º »¯¾Ëà¾ ¸ ·Ê º ßV¾ Ç º=ãÞà »Þ½ ¾ »¯Éº= ¾ ¸ ¸ ¾<;Ò½ 7B ß9¾ Ç »Þã ½ » Ç É¾ ·¸ á º Ç¯ »¯½ Ç Sº ÀÌ

φf + φa + φs = 1 ,

G¶Ä ¿ P»ÞÉ ·Ç ½ »â½ ÇBÇÞ·IH ßV½ º »»¯¾;Òã¯½È»¯º»8;¾ºD · »¯½ ¾ Ç = ¾ã ºV»¯º9ã Ê ½ »Þ½ ¾ ¾?=»8;Â¾à¾ ¸ ·Ê º= ã ß»Þ½ ¾ #Ç ¾?=»¯Éº Ê ½C"» · ãÞº Éº > ¾ã¯¾ Ç ½È»8@

φ¾?=e»ÞÉº > ¾ã¯¾ ·ÇÊË »Þº9ã¯½ P¸ ½ Ç" º S º «á @

φ = 1 − φs .

!L¥£9J F¨©¦´£9³+ ¤©£V³³Þ£¡6¡ Ä?L îêLê¨ï ±êHíîììíï ±ê9ì ì5ì=ê"Ç¯ÇE·Ê ½ 7B »ÞÉ »»ÞÉº Ç ¾ ¸ ½ ¨P# 9 · ½ > É HÇ º Çë ã¯ºÂ½ »Þã¯½ #Ç ½ ß ¸ ¸ @½ ßV¾ ÊF> ã¯º Ç¯Ç ½ á¸ º G ½ º »ÞÉº½ »Þã¯½ Ç ½ ß º Ç ½È»¯½ º Ç ¾=À»ÞÉº Ç ¾ ¸ ½

ρs

P 9 · ½ ρf

ã¯º"ßV¾ Ç » P » P ;Âº"¾ á » ½ »¯Éº¸ ¾ß ¸ÀÊËÇ¯Ç ßV¾ #Ç º=ãÞà »¯½ ¾ ºD ·# »Þ½ ¾ #Ç ½ »ÞÉº %·¸ º=ãÞ½ P = ã PÊ º;Â¾ã¯¼ Ì

∂φα

∂t

+ ∇ · (φαvα) = 0 , α = s, f .

GWÄ Ä P ½ ßVº »ÞÉº ½ ãÂ½ Ç¤ ßV¾ ÊF> ãÞº ÇÞÇ ½ á¸ º ¸ ½ED · ½ À ;Âº B º9»Â»¯Éº= ¾ ¸ ¸ ¾<;Ò½ 7B ºD · »Þ½ ¾ 1Ì

∂

∂t

(

φaρa

)

+ ∇ · (φaρava) = 0 .

ã¯º »Þ½ 7B »¯Éº ½ ã HÇ P ½ º P¸'BHHÇ9 ;Âºâ¾ á » ½ »ÞÉººD · »Þ½ ¾ ¾?= Ç » »Þº= ¾ã»ÞÉº B3Ç> É Ç º Ì

ρa =1

Cp

pa ,

É ·#Ç »ÞÉº ÊËÇ¯Ç ßV¾ Ç º9ã¯à »Þ½ ¾ ºD · »¯½ ¾ ¾=»¯Éº ½ ã > É HÇ º ½ Ç B ½ àº á @∂

∂t

(

φapa

)

+ ∇ · (φapava) = 0 .

G¶Ä Å P ¾ Ç ½ º9ã¯½ 7B »ÞÉº ÇE¸ ¾<; 9¾<;vã¯º B ½ Ê º P ½ BH ¾ã¯½ /B »ÞÉº ;Âº ¼«»¯½ Ê º º > º # º »

º V5º=ß» Ç9 ;º ¾ á » ½ »ÞÉº ÊËÇ¯Çá¸ ßVº ºD · »Þ½ ¾ Ç ½ »ÞÉº Ç » »Þ½ ¾ ãA@'ß Ç º

∇ · (φαvα) = 0 , α = s, f ,

∇ · (φaphva) = 0 .

G¶Ä ä*P

!5ì éëî#"Wî#eï $%&eîì'±ê º ;Ò½ ¸ ¸ ¾P»ßV¾ #Ç ½ º9ã B º º9ã P¸#Ê ¾ Ê º » ·Êsá#P¸ P ß9º¤ºD · »Þ½ ¾ Ç=3 ¾Z= ¾ß ·#Ç ¾ 9¾<; Ç½ > ¾ã¯¾ ·ÇÊ º ½ »¯Éº Ç ½ ÊF>¸ ½W=Q@½ 7B Ç¯ÇÞ·ÊF> »Þ½ ¾ Ç" ã¯º ·#Ç º À ¾ Ç ½ º=ãÞ½ 7B »ÞÉº ÇE¸ ¾<;¸ ½ D · ½ 9¾<; «Ç¯ÇE·Ê ½ 7B »ÞÉ »º"ßVº Ç¯Ç ½ »¯º9ã ß»¯½ ¾ = ¾Hã¯ß9º Çá º9»8;º=º »ÞÉº Ç ¾ ¸ ½ «P¸ ½ D · ½ Ç ãÞº > ãÞ¾ > ¾HãE»¯½ ¾ P¸ »Þ¾»¯Éº"àHº ¸ ¾ßV½ »8@ ½WV5º9ã¯º ß9º Ç

vα −vs, α = a, f,;Âº;Òã¯½È»¯º

»ÞÉº B º º=ã ¸Ê ¾ Ê º » ·Ê áP¸ P ß9º ºD · »¯½ ¾ ÇÇZGWÇ º=ºJI ¿ OQP Ì

φα(vα − vs) = −Kα

µα

∇pa, α = a, f ,

GWÄ K P

;ÒÉº9ã¯ºµα

ã¯º»¯Éºâà½ Ç ß9¾ Ç ½È»8@ß9¾3º H ßV½ º » ÇPKα

ãÞº»ÞÉº > º=ã Ê º Pá ½ ¸ ½ »8@.»Þº Ç ¾ã Ç9 D · »¯½ ¾ ÇTG¶Ä K P ã¯ºË»ÞÉº »ÞÉã¯º9º( > É Ç ºJ9¾<; æ ã¯ß@ ¸ ; = ¾ãB»ÞÉº.ß HÇ º ¾?= Ê ¾¦à½ 7B> ¾HãÞ¾ ·ÇÂÊ º ½ ·Ê.

Éº > º9ã Ê º á ½ ¸ ½ »8@'ßV¾º H ßV½ º » ÇKα = Kα(φ, S)

º > º ¾ »¯Éº > ¾HãÞ¾ Ç ½ »8@φ

¾?=»ÞÉº > ¾ã¯¾ ·ÇÂÊË »¯º9ã¯½ ¸1P »ÞÉº ÇÞ » · ã »¯½ ¾

S¾?=\; »Þº=ã ;ÒÉº=ãÞº

S½ Ç¤ º S º «á @

S =φf

φ

.

æ ½Vº=ãÞº » Ê ¾ º ¸ ÇÒ ãÞº ·Ç º »¯¾ º Ç ßVã¯½ á º»ÞÉº > º9ã Ê º Pá ½ ¸ ½È»8@'¾=1»ÞÉº > ¾ã¯¾ ·#ÇÊ º ½ ·ÊG®Ç º9º I ¿ ä<OQP P# »ÞÉº@ B º º=ã ¸ ½ º»¯Éº ;Âº ¸ ¸ 3¼ ¾<; ¾ º @ Â ã Ê ºD ·# »Þ½ ¾ I Å O

Ä N¦Ã ®£ ¡ £V ©¡ ´¶P©¦¡ ìHí±î#" í¦ï ê ÉºËãÞ¾ ¸ ¸ Ç ºD"º=ãE» ¸ ã B º = ¾ã6ßVº Ç ¾ »ÞÉº ¸ ½ D · ½ nmÇ ¾ ¸ ½ C> É HÇ º Ç9 Éº Ç º= ¾Hã¯ß9º Ç ãÞº» ¼Hº ½ »¯¾ ß=ßV¾ · » á @ ·Ç ½ /B »ÞÉºÂß ¸ Ç¯Ç ½ ß ¸P º=ã L?B É½ > ã¯½ ß9½ >#P¸HÇ ½ Ç ß ·Ç¯Ç º ½ I ¿ ä<O » ½ Ç ¾P»¯º ½ I <O»ÞÉ »3= ¾HãÉ½ B É ¸ @ß9¾ ÊF> ã¯º Ç¯Ç ½ á¸ º > ¾ã¯¾ ·ÇÊË »¯º9ã¯½ ¸ Ç »ÞÉº º9ã =B É½> ãÞ½ ß9½ >¸5Ê @ ¾P» á º >7>¸ ½ ß Pá¸ º P ½ #Ç »¯º ¾?=±½È» ¨Ê ¾ã¯º B º º=ã ¸ = ¾Hã Ê Ç É¾ ·¸ á º ·Ç º 1

¾«ßV¾ º=ßV»"»ÞÉº º9ã =B É½ > ãÞ½ ß9½ >¸ ;Ò½ »ÞÉ»ÞÉº¨¾P»¯Éº9ãºD · »Þ½ ¾ Ç ¾?=%»ÞÉº Ê ¾ º ¸¾?= ÊB·¸ »Þ½ > É HÇ º 9¾<;\½ »¯Éº º = ¾Hã ÊËá¸ º > ¾ã¯¾ ·ÇÊ º ½ ·Ê« ½È»"½ Ç ºLßVº Ç¯Ç¯ ã @»Þ¾ ÇA> ºLß Á½W=Q@ «Ê ¾ º ¸ = ¾Hã »ÞÉº º = ¾ã ÊË »¯½ ¾ ¾=»¯Éº > ¾ã¯¾ ·Ç ÊË »¯º9ã¯½ ¸WÀÊ ¾ Ç » >> º=ã Ç »¯Éºà3½ Ç ßV¾º ¸ Ç »Þ½ ß Ê ¾ º ¸ ½ Ç¤·Ç º À

»¯ÉºF= ¾ ¸ ¸ ¾<;Ò½ 7BÇ º=ßV»Þ½ ¾ Ç ;Âº'ßV¾ Ç ½ º=ã »8;¾.ºD" Ê >¸ º Ç ¾?= Ç ½ ÊF>¸ ½Sº mÊ ¾ º ¸ Ç=;ÒÉ½ ß6É ãÞº¾ á » ½ º = ã¯¾ Ê »¯É½ Ç1B º º9ã P¸ ºD · »Þ½ ¾ Ç=

1& . / 2 +76 . ) . 2 2 / /-R+72-4- +;*R0 . ) *: »ë½ ÇÇ » »¯º ½ I ¿ ä<O»¯É »º9àH¾ ¸ · »Þ½ ¾ ¾?= = ·¸ ¸ @ Ç¯ » · ã »Þº 1 ¾ ºÂß9ãÞ½ »Þ½ ß ¸ ¸ @ º9»Þº9ã Ê ½ º Ç»ÞÉºº V5º=ß»¯½ àHº º ÇÞÇ ¾?=5»ÞÉº ã @½ 7B > ãÞ¾ß9º Ç¯Ç9 ¾ãºD" PÊF>¸ º »ÞÉº > ãÞº ÇÞÇ = º ¸ » Ç º9º »Þ¾ á º·Ç¯ » · ã »Þº ½ nÇ ¾ Ê º > ãE» ¾?=%»ÞÉºã¯¾ ¸ ¸ ½ 7B ¾ º ½ ¾ã º9ã»Þ¾ á º á¸ ºB»¯¾» ¼Hº ·7>Ç ½ B ½S#ß P » PÊ ¾ · »Ò¾?= ; »Þº=ã1= ãÞ¾ Ê »ÞÉº >#?> º9ã

Éº Ê »ÞÉº ÊË »Þ½ ß ¸Ê ¾ º ¸ ½ Ç º9àHº ¸ ¾ > º ½ I ¿ ä<O ;ÒÉ½ ß6É¨½ ÇëáÇ º ¾ »8;¾ > É Ç º9¾<; ºD · »¯½ ¾ Ç ½ ¾ º ½ Ê º Ç ½ ¾ 11 »B½ Ç·Ç º »¯¾ > ã¯º ½ ß»»¯Éº'º9àH¾ ¸ · »Þ½ ¾ ¾=,= ·¸ ¸ @ÇÞ » · ã »¯º ãÞº B ½ ¾ ÇP# »¯¾¨º Ç »¯½ ÊË »Þº »ÞÉº Ç ½ º Ç ¾?=»¯Éº Ç¯ » · ã »Þ½ ¾ ãÞº B ½ ¾ I ¿ ä?O

º ¾P»¯º+»¯É » ¾9¾<; á ¾ ·# ã @ßV¾ # ½È»¯½ ¾ Ç± ã¯º ·Ç º = ¾Hãd9 · ½ > É HÇ ºÂ½ »ÞÉ½ ÇÊ ¾ º ¸WH· ã B ¾ ¸ ½ Ç »¯¾ B º º=ã ¸ ½ º»ÞÉº Ê ¾ º ¸ ½ ¾ã º=ãe»¯¾ ½ ß ¸ ·# º ; »Þº=ã ¸ ¾ Ç¯Ç º ÇëP»Þ¾ßV¾ Ç ½ º=ã »ÞÉº @ Ê ½ ß Ç ¾=»¯Éº ÇÞ » · ã »¯½ ¾ · º9ãà ãÞ½ ¾ ·Ç Ç ßVº ã¯½ ¾ Ç ¾=b; »¯º9ã¸ ¾ ÇÞÇ º Ç=

ð#ð #í "®îì Òì' $ " »ë½ ÇëÇ¯ÇE·Ê º »¯É » 3> ¾HãÞ¾ ·Ç±¸ @Hº9ã½ ÇëÊ ¾¦à½ 7B ½ »ÞÉº > ¾ Ç ½ »Þ½ àº

x

½ ã¯º=ßV»Þ½ ¾ 1¦ Éºb= ¾ ¸ Á¸ ¾<;Ò½ 7BáHÇ ½ ß Ç¯ÇÞ·ÊF> »Þ½ ¾ Ç% ã¯º ·Ç º »Þ¾ Ç ½ ÊF>¸ ½=Q@¨»ÞÉº ÊB·¸ »Þ½ ½ Ê º Ç ½ ¾ P¸ ºD · »Þ½ ¾ Ç¾?= ÊHÇÞÇ ß9¾ Ç º9ã¯à »Þ½ ¾ CGWÄ äP PÊ ¾ Ê º » ·Ê á¸ ßVº GWÄ K P Ì

• Éº

y

½ ã¯º=ß»¯½ ¾ XG = ¾ ¸ ¸ ¾<;Ò½ 7B »¯Éº ãÞ¾ ¸ ¸5 "º Ç P%½ ÇÂ º B¸ º=ßV»Þº Ç ½ ß9º"»¯Éº ãÞ¾ ¸ ¸ Ç ãÞº ·7>»Þ¾ 10

Ê ¸ ¾ 7B#•

Éº; »Þº9ã19#¾<;R½ »¯Éºz

½ ã¯º=ßV»Þ½ ¾ ½ Ç¤ º BH¸ ºLß»Þº À•

Éº ½ ã > É Ç º ½ Ç º B¸ º=ßV»Þº À ½ º ½È»¤½ Ç¤Ç¯ÇÞ·Ê º »ÞÉ »φa = 0, pa = 0

îêLê¨ï ±êHíîìì ºß9¾ Ç ½ º9ãÂ»ÞÉº9#¾<;Rã¯º B ½ Ê º ½ »ÞÉºãÞº B ½ ¾ 1Ì

D = (x, z) : xa ≤ x ≤ xb, A(x) ≤ z ≤ B(x) ,

;ÒÉº9ã¯ºA(x), B(x)

ã¯º »¯Éº ¸ ¾<;º=ã ·/>7> º=ã á ¾ · ã¯½ º Ç ¾="»¯Éº > ¾HãÞ¾ ·Ç¨¸ @Hº9ã ãÞº Ç > º=ßV»Þ½ àº ¸ @ ë º9»

d(x) = B(x) − A(x)á º »¯Éº »¯É½ ß6¼ º Ç¯Ç ¾=Ò»ÞÉº > ¾HãÞ¾ ·Ç¸ @º=ã

!L¥£9J F¨©¦´£9³+ ¤©£V³³Þ£¡6¡ Ä N¿ ½ ßVº ;ºßV¾ Ç ½ º=ã¤»ÞÉº ÊËHÇÞÇ 9 · "¾ ¸ @.½ »¯Éº

x

½ ã¯º=ß»¯½ ¾ 1 »ÞÉº » ¼½ /B«ÇEÊËP¸ ¸à¾ ¸ ·Ê º [x, x + h] ;ºß ;Òã¯½ »Þº »ÞÉº ½ Ç ßVã¯ºV»¯º ÊHÇÞÇÂá¸ ßVº ºD ·# »Þ½ ¾

−d φfvf

∣

∣

x+h+ d φf vf

∣

∣

x= hJ ,

;ÒÉº9ã¯ºJ

º Ç »Þ½ ÊË »¯º Ç ; »Þº=ã ¸ ¾ ÇÞÇ º Ç »¯ÉãÞ¾ ·7B Éâ»ÞÉº ¸ ¾<;Âº9ã B·/>7> º=ã á ¾ · ã¯½ º Ç ¾?=»¯Éºà¾ ¸ ·Ê º º ¸ º Ê º » = ¾9¾<; á ¾ · ãA@ ß9¾ ½È»¯½ ¾ ÇÒ ã¯º HÇÞÇÞ·Ê º 1 »ÞÉº

J = 0

æ ½ à½ ½ 7Bá ¾»ÞÉ Ç ½ º Ç ¾?=»ÞÉºâºD ·# »Þ½ ¾ á @h

P » ¼½ 7B »¯Éº ¸ ½ Ê ½ »h → 0

;Âº¾ á » ½ »¯Éº¾ º Áý ½ Ê º #Ç ½ ¾ ¸5ÊHÇÞÇ ß9¾ Ç º9ã¯à »¯½ ¾ ºD · »Þ½ ¾ 1Ì

−(

d φf vf

)

′

= J .

GWÅ ¿ P º ¾»Þº»ÞÉ »

J = 0 ½ Ç » ¼Hº ½ I ¿ ä<O v ½Vº=ãÞº » ÊËHÇÞÇÒá#P¸ P ß9ººD · »¯½ ¾ ß P.á º¾ á » ½ º ½= ?> ãÞ½ ¾ã¯½ Ç¯ÇE·Ê > »Þ½ ¾ Ç

ã¯ºâß6É 7B º À ¾ãºD" PÊF>¸ º ¸ ºV» ·Ç ßV¾ #Ç ½ º9ã ß Ç º ;ÒÉº »ÞÉº º=ãÞ½ à »¯½ àHºB¾?=»¯Éº9 · "½ »ÞÉº

x

½ ã¯º=ßV»Þ½ ¾ ¾º Ç¤ ¾P» º > º ¾ »ÞÉºãÞº Ê ½ ½ 7B ß9¾3¾Hã ½ »Þºz

Éº ½ »Þº B ã »¯½ 7B »ÞÉº Ç » »Þ½ ¾ # ã @ ÊHÇÞÇ ß9¾ Ç º9ã¯à »¯½ ¾ ºD · »Þ½ ¾ GWÄ äP%½ »ÞÉºã¯º B ½ ¾

∆D = (x, z) : x1 ≤ x ≤ x2, x2 = x1 + h, A(x) ≤ z ≤ B(x)

;º B º9»x2∫

x1

B(x)∫

A(x)

∂

∂x

(

φfvf

)

dz dx +

x2∫

x1

(

φfvzf

)∣

∣

B(x)

A(x)dx = 0 .

Éº »Þº=ã Ê

J1 =1

h

x2∫

x1

(

φfvzf

)∣

∣

B(x)

A(x)dx

º S º Ç ; »¯º9ã ¸ ¾ Ç¯Ç º Ç G ¾ã½ ßVã¯º Ê º » P»ÞÉã¯¾ ·/B É »ÞÉº ¸ ¾<;º=ã Pn·7>7> º9ã ÇE· ãA= ß9º Ç ¾=»ÞÉº S ½ »Þº Á àH¾ ¸ ·Ê º "Ç ½ 7B »¯Éº Ç¯ÇE·Ê > »Þ½ ¾ YB ½ àHº á ¾¦àHº ½ »Þº B ã »Þ½ 7B »¯Éº Sã Ç »½ »Þº B ã P¸5á @ > ãE» Ç ;º B ºV» ½ Ç ßVã¯ºV»¯º ÊHÇÞÇ ß9¾ Ç º9ã¯à »¯½ ¾ ºD · »Þ½ ¾

−(d φfvf )∣

∣

x2

+ (d φf vf )∣

∣

x1

+

x2∫

x1

d′

φfvf dx = hJ1 .

e ¼3½ 7B »ÞÉº ¸ ½ Ê ½È»h → 0

B ½ àHº Ç »ÞÉº ½WV5º9ã¯º »Þ½ P¸ÊHÇÞÇ ß9¾ Ç º9ã¯à »Þ½ ¾ ºD · »Þ½ ¾

−(

d φf vf

)

′

+ d′

φf vf = J1 .

îí¦ï "Wî ¾ Ê º » ·Ê á¸ ßVººD · »Þ½ ¾ GWÄ K P á ºLßV¾ Ê º Ç

φf (vf − vs) = −Kf

µf

dpf

dx

,

GWÅ Ä P

;ÒÉº9ã¯ºvf , vs

PKf (φ, S)

ãÞºãÞº P¸ à ¸ · º = · ß»¯½ ¾ Ç º > º ½ 7B ¾ »ÞÉºx

ã BH·ÁÊ º »

Ä NPÄ ®£ ¡ £V ©¡ ´¶P©¦¡·áÇ »Þ½ » · »¯½ /BG®Å Ä PÂ½ »Þ¾ ÊËÇ¯Ç ßV¾ Ç º=ãÞà »Þ½ ¾ ºD · »¯½ ¾ GWÅ ¿ P,;Âº¾ á » ½ « ßV¾ Á

àº=ßV»Þ½ ¾ ½WV ·Ç ½ ¾ ºD · »¯½ ¾ = ¾Hã · ¼ ¾<; pf (x)

−(

d

Kf

µf

p′

f

)

′

+(

d φf vs

)

′

= −J .

º ¾»Þº¨»ÞÉ »½ I ¿ ä<O = ß»¯¾ãd

½ Ç ¾ Ê ½È»Þ»Þº ½ »¯Éº Sã Ç »»Þº9ã Ê ¾ »ÞÉº ¸ º =» Á É Ç ½ º¾?=e»ÞÉ½ Ç ºD ·# »Þ½ ¾

Éº¾ á » ½ º ºD · »¯½ ¾ P¸ Ç ¾ º > º Ç ¾ S

φf

Ò ½ »Þ½ ¾ ¸ ßV¾ #Ç »¯½È» · »Þ½ àºãÞº ¸ »Þ½ ¾ Ç± ã¯º ·Ç º »Þ¾ ß ¸ ¾ Ç º+»¯Éº Ç @ Ç »Þº Ê.<U @ ·Ç ½ 7B »ÞÉº ºS ½ »Þ½ ¾ ¾?=»¯Éº ÇÞ » · ã »¯½ ¾ ;º ºD" > ã¯º Ç¯Ç

φf

Çφf = S φ

ã¯¾ Ê ºD" > º9ã¯½ Ê º » Ç ;º¼ ¾<;^»¯ÉºËã¯º ¸ »Þ½ ¾ á º9»8;º=º ß > ½ ¸ ¸ ã @ > ãÞº ÇÞÇÞ· ã¯º

pc =pa − pf

P«Ç¯ » · ã »Þ½ ¾ S

GWÇ º=º º B I ¿ OQP

S = g(pc).

¾ · ã¤ß HÇ ºpa = 0

»ÞÉº=ãÞº= ¾ã¯º;º ¾ á » ½ »ÞÉº º=º º ãÞº ¸ »Þ½ ¾ S = g(pf )

·áÇ »Þ½ » · »¯½ /B »¯Éº Ç ººD" > ã¯º Ç¯Ç ½ ¾ Ç ½ »¯¾B»¯ÉºßV¾ àHº=ß»¯½ ¾ ½WV ·Ç ½ ¾ ºD ·# »Þ½ ¾ ;º

B ºV»»¯Éº ¾ ¸ ½ º ã ½Vº=ãÞº »¯½ ¸ ºD · »Þ½ ¾ ½ »ÞÉº · ¼ ¾<; pf (x)

Ì

−(

d

Kf

(

φ, g(pf ))

µf

p′

f

)

′

+(

d φ g(pf )vs

)

′

= −J .

GWÅ Å P

U ¾ · ãA@'ß9¾ ½È»¯½ ¾ Ç ;º=ãÞº ºS º ½ I ¿ ä<O Ç = ¾ ¸ ¸ ¾<; Ç=Ì

pf (−xb) = g−1(S0), d

Kf

µf

p′

f

∣

∣

x=xb

= 0 ,

;ÒÉº9ã¯ºxb

½ Ç Ç ¾ Ê º ¸ ã B º ·Êâá º=ã ÀÇÞ· ß6É»¯É » ¾«½ 9 · º ß9ºâ¾=xb

¾ »ÞÉº Ç ¾ ¸ · »Þ½ ¾ pf

½ Ç ¾ áÇ º9ã¯àº À

±ê9ì ì'eì'.í"®îìeê ½ ßVº ¾ · ã B ¾ P¸ ½ Ç »¯¾â½ àº Ç »¯½ B3 »Þº"»ÞÉº ½ 9 · º ßVº¾?= ; »Þº=ã ¸ ¾ Ç¯Ç º Ç ¾ »¯Éº ÇÞ » · ã »¯½ ¾ @ PÊ ½ ß Ç= ;º ·Ç º »ÞÉº Ç¯PÊ º > ã Ê º9»Þã¯½ º = · ß»¯½ ¾

g(pf )HÇ ½ I ¿ ä<O

g(pf ) =

11

1 − s∞

+(

pf

a

)n + s∞

, pf ≤ 0 ,

1, pf > 0 ,

;ÒÉº9ã¯º > ã PÊ º9»Þº9ã Çs∞

, a, n;Âº9ã¯º Ç º ¸ ºLß»Þº »Þ¾ S»»ÞÉººD" > º=ãÞ½ Ê º » ¸" » Â Éº

¾ á » ½ º «ÇÞ » · ã »¯½ ¾ > ã¯º Ç¯ÇÞ· ãÞº3= · ßV»Þ½ ¾ g(pf ) ½ ÇÒÇ É¾<; ½ ½ BÀ¿©¦

ßV¾ #Ç »¯½È» · »Þ½ àºÒã¯º ¸ »¯½ ¾ = ¾HãKf

½ Ç ºD" > ã¯º Ç¯Ç º ¨Ç ¾ ¸ ½ º ã\= · ßV»Þ½ ¾ ¾= > ¾ã¯¾ ÁÇ ½ »8@ P#«Ç¯ » · ã »Þ½ ¾ CG®Ç º9ºFI ¿ ä<OQP Ì

Kf (φ, S) = k0

φ3

1 − φ2S

3.4,

GWÅ ä*P

!L¥£9J F¨©¦´£9³+ ¤©£V³³Þ£¡6¡ Ä N¦Å

;ÒÉº9ã¯ºk0

½ Ç »ÞÉº > º=ã Ê º á ½ ¸ ½ »8@J= ßV»Þ¾Hã º ¾P»¯º»¯É »«½ I ¿LÃ O Ç ½ Ê ½ ¸ ã á· » ½V5º9ã¯º »º" > ãÞº ÇÞÇ ½ ¾ ; HÇ'·Ç º »Þ¾

º Ç ßVã¯½ á º»ÞÉ½ ÇÒ ¾ ¸ ½ º ã1= · ß»Þ½ ¾

Kf (φ, S) = k0

φ3

(1 − φ)2S

3.4.

GWÅ K P

»ÞÉºß Ç º'¾=b= ·¸ ¸ÇÞ » · ã »¯½ ¾ ¾?= ; »Þº=ã bG®Å K PãÞº · ßVº»¯¾»¯ÉºY;Ò½ º ¸ @ ·Ç º º Ê¨Á> ½ ã¯½ ß P¸ ¾ º @ Â ã ÊË ºD · »Þ½ ¾ 1 ;ÒÉ½ ß6ÉnãÞº ¸ »Þº Ç > ¾ã¯¾ Ç ½È»8@ H> º9ã Ê º á ½ ¸ ½ »8@ º9àHº9ãÞ»ÞÉº ¸ º ÇÞÇ= ½ ¾ã º9ã%»Þ¾ßV¾ ÊF> ãÞº¾ · ãÂã¯º ÇÞ·¸ » Ç ;Ò½ »ÞÉCI ¿ ä<O ;ºÉ àº ·Ç º = ¾Hã Êâ·¸ G®Å ä*P½ P¸ ¸À3·Ê º=ãÞ½ ß ¸ ºD" > º9ã¯½ Ê º » Ç9

-100 -80 -60 -40 -20 0p [hPa]

0

0.2

0.4

0.6

0.8

1

S

0 10 20 30 40 50epsilon [%]

0

1

2

3

4

5

6

7

8

E [

MPa

]

K M K±NK9ñþõEK9ñGJôLF ; ûHõ C òòþõ C ú þ3F3>ÞñGÈô=F

s∞

= 0.11, a = −1.2, n = 2ö?ÀûHõ C òò; òýñõEK=GJF¨ú þHF3>¯ñGJô=F#ÿ

!" í'ì í îì' íeê îì3íWî#" ºV» HÇÞÇÞ·Ê º'»¯É »J;Âº¼ ¾<;s»¯Éº = · ßV»Þ½ ¾

d(x) ;ÒÉ½ ß6É º Ç ßVã¯½ á º Ç »¯Éº«»ÞÉ½ ß6¼ º Ç¯Ç

¾?=+»¯Éº > ¾ã¯¾ ·#Ç"¸ @º=ã 5P¸ º9»d0

á º¨»ÞÉºB»¯É½ ß6¼ º ÇÞÇ ¾=+»ÞÉº · ß9¾ ÊF> ã¯º Ç¯Ç º > ¾ã¯¾ ·Ç¸ @Hº9ã Éº¨à¾ ¸ ·Ê º

V (x) ¾?=»¯Éºâº ¸ º Ê º »¾= > ¾HãÞ¾ ·Ç¤ÊË »¯º9ã¯½ ¸ ½ Ç > ãÞ¾ > ¾HãE»¯½ ¾ P¸ »Þ¾d(x)

5 º9» º ¾»Þº á @Vs(x) = Cds(x) »¯Éºà¾ ¸ ·Ê º ¾ß9ß ·7> ½ º .á @'»¯Éº Ç ¾ ¸ ½ > É HÇ º

"Ç¯ÇE·Ê ½ 7B »ÞÉ »¨»¯Éº Ç ¾ ¸ ½ > É Ç º½ Ç¨ ¾P» º= ¾ã Ê º ¾ ¸ @m»ÞÉº > ¾ã¯º ÇA> ß9º½ ÇßV¾ ÊF> ã¯º Ç¯Ç º À ;Âº B ºV»»¯ÉººD ·¸ ½ »8@

ds(a) = ds(x) .

æ ½ à½ ½ 7B ½ » á @d0

ãÞº ã¯ã 7B ½ 7BBÇ ¾ Ê º»Þº=ã ÊËÇ ;Âº ¾ á » ½ »¯ÉººD · »Þ½ ¾

1 − φ0 =(

1 − φ(x))d(x)

d0

,

= ãÞ¾ Ê ;ÒÉ½ ß6É»¯Éº > ¾ã¯¾ Ç ½È»8@φ(x) ß á ºßV¾ Ê >· »Þº

φ(x) = 1 −d0

d(x)

(

1 − φ0

)

.

Ä N ä ®£ ¡ £V ©¡ ´¶P©¦¡ Éº = ¾ ¸ ¸ ¾<;Ò½ 7B Ê ¾ º ¸ ; Ç > ã¯¾ > ¾ Ç º ½ I ¿ ä<OÒ»Þ¾ º9»Þº=ã Ê ½ º

d(x) ½ »ÞÉº º Á= ¾ã Ê º = º ¸ » · ß»Þ½ ¾

d = d(x; dmin) ºD" > ã¯º Ç¯Ç º Ç »¯Éº«»ÞÉ½ ß6¼ º Ç¯Ç ¾?=»ÞÉº > ¾ã¯¾ ·Ç¸ @Hº9ã HÇ" = · ß»Þ½ ¾ ¾?=

x

P »¯Éº Ê ½ ½ Êâ·Ê) ½ Ç » P ß9º á ºV»8;Âº9º »ÞÉºâãÞ¾ ¸ ¸±ÇE· ãA= ß9º Çdmin

Ì

d(x; dmin) =

d0, x ≤ xl(dmin) = −

√

R2 −

(

1

2(d0 − dmin) − R

)2,

2R + dmin − 2√

R2 − x

2, xl(dmin) ≤ x ≤ xr(dmin) .

»¯ÉºFSã Ç »½ »Þº9ã¯à P¸ »¯Éº > ¾HãÞ¾ ·Ç ÊË »¯º9ã¯½ ¸ ½ Ç· ßV¾ ÊF> ã¯º Ç¯Ç º À ½ »¯Éº Ç º=ßV¾ ¾ º½È»= ¾ ¸ ¸ ¾<; Ç »ÞÉº B º9¾ Ê ºV»Þã @¾=Ò»ÞÉºãÞ¾ ¸ ¸ÇE· ãA= ß9º ÇB· »Þ½ ¸ÂÇ »Þã¯º Ç¯Ç

τzz

á ºLßV¾ Ê º Ç º9ã¯¾ »x = xr(dmin)

# Éº ¾ ¸ ½ º ã º ¸ à½ Æ ¾H½ B » ¸ ;\½ Ç¤·Ç º »Þ¾'» ¼Hº½ »¯¾ ß=ßV¾ · »à3½ Ç ßV¾º ¸ Ç »Þ½ ßV½ »8@Ë¾?=±»ÞÉº= º ¸ » Ì

τzz(x) = E(x) + vsΛd

dx

E(x) ,

;ÒÉº9ã¯ºΛ

½ Ç à½ Ç ß9¾3º ¸ HÇ »¯½ ß»¯½ Ê º¤ßV¾ Ç » P » P Éº Ç »¯ãÞº ÇÞÇ Ç »Þã ½ = · ß»Þ½ ¾ E

½ Ç S»E»¯º »Þ¾ » FB ½ àº ½ I ¿ ä?O Ì

E(ε) = 35ε2.3 [MPa] .

Éº¾ á » ½ º = · ßV»Þ½ ¾ ½ ÇÒÇ É¾<; ½ ½ B1¿V ¾ã

x > xr(dmin) »ÞÉº Ç »Þã ½ ε = ε(x; dmin) = 1 − d(x; dmin)/d0

½ Ç º9»Þº9ã ÁÊ ½ º = ã¯¾ Ê

E(x) ;ÒÉ½ ß6É ÇÞ »Þ½ Ç S#º Ç »ÞÉº½ ½ »Þ½ P¸ à ¸ · º > ã¯¾ á¸ º Ê

d

dx

E(x) = −1

vsΛE(x) ,

E

(

xr(dmin))

= E

(

ε(xr; dmin); dmin

)

.

GWÅ L P

É ·#Ç ;º B ºV» Ì

d(x; dmin) = d0

(

1 − ε(x))

, x > xr(dmin).

Éº > ã PÊ º9»Þº9ãdmin

½ Ç ¾ á » ½ º = ã¯¾ Ê »ÞÉº º=ã L?B É½ºD · »¯½ ¾ 11 ÉºJ= ¾Hã¯ß9º ÇFf

Fs

ß á ºßV¾ ÊF>· »Þº .Ç

Ff (dmin) =

xr(dmin)∫

xl(dmin)

pf dx, Fs(dmin) =

xr(dmin)∫

xl(dmin)

(

E(x; dmin)+vsΛd

dx

E(x; dmin))

dx ,

»ÞÉº=ãÞº= ¾ã¯º ;Âº B º9»Â»¯Éº ¾ ¸ ½ º ãÒºD · »Þ½ ¾ = ¾ãdmin

Ì

xr(dmin)∫

xl(dmin)

(

pf + E(x; dmin) + vsΛd

dx

E(x; dmin))

dx = F .

GWÅN P

!L¥£9J F¨©¦´£9³+ ¤©£V³³Þ£¡6¡ Ä N?K ðÀð Wê=ïí3ìîì'ì' $ " ºV» ·#Ç ½ »Þã¯¾ · ß9º»¯Éº · ½= ¾ã Ê B ãÞ½

ωh =

xi : xi = −xb + ih, i = 0, 1, . . . , N, xN = xb

.

»ÞÉº,= ¾ ¸ ¸ ¾<;Ò½ 7B ;º,;Ò½ ¸ ¸ º ¾»Þº%»ÞÉº ½ Ç ßVã¯ºV»¯º >7> ã¯¾#"½ ÊË »¯½ ¾ Ç ¾?=ß9¾ »Þ½ · ¾ ·#Ç = · ß Á»Þ½ ¾ Çë·Ç ½ 7BÇÞ·áÇ ßVã¯½ > » ”h”

º B »ÞÉº > ãÞº ÇÞÇÞ· ã¯º = · ß»¯½ ¾ pf

;Ò½ ¸ ¸á º ?>7> ãÞ¾#"½ Ê »Þº á @'»ÞÉº= · ß»Þ½ ¾

pf,hi = pf,h(xi) = pf,h(xi, dmin) .

ÉºÉ@ ãÞ¾ Ç » »Þ½ ß > ãÞº ÇÞÇÞ· ã¯º= · ß»¯½ ¾ ½ Ç¤ ºS º HÇ = ¾ ¸ ¸ ¾<; Ç=Ì

ph = Spf + (1 − S)pa .

ì'3í¦îì' î#"í®ì' #í ê" !%&±îì ð >7> ã¯¾#"½ ÊË »Þ½ 7B ½ »Þº B ã P¸ Ç ½ G®Å N P á @ »ÞÉº »¯ã > º ¾½ ¸ ã ·¸ º

Sh(dmin) =

nR∑

i=nL+1

phi + ph,i−1 + E(εhi) + E(εh,i−1)

2h

+ vsΛ

(

E(εh,nR) − E(εh,nL

))

,

;º B º9»Ò»ÞÉº ½ Ç ßVã¯ºV»¯º ¾ ¸ ½ º ãºD ·# »Þ½ ¾

Sh(dmin) − F = 0 .

Éº¾ á » ½ º ' ¾ ¸ ½ º ãÂºD · »Þ½ ¾ ß Pá º Ç ¾ ¸ àHº Ëá @ ÊËP @½ »Þº9ã »¯½ àHº ¸B ¾ã¯½È»¯É ÊËÇ9 ½ ßVº= · ßV»Þ½ ¾

Sh½ ÇÒÊ ¾ ¾»Þ¾ ½ ß P¸¶ ;º ·Ç º»ÞÉº á ½ Ç º=ßV»Þ½ ¾ «Ê º9»ÞÉ¾ »Þ¾FS

dmin

ì'" «êLï #í ê'" nì ±í3êLêí$%&±îì' æ"· ã¯½ 7B º ß6É á ½ Ç º=ßV»Þ½ ¾ ½ »Þº=ã »Þ½ ¾ G ¾ã ¦ ´£V³ ½ »Þº=ã »Þ½ ¾ Pd;Âº Ç É¾ ·¸ BÇ ¾ ¸ àº " ¾ ¸ ½ º ã> ãÞ¾ á¸ º Ê. ;ÒÉ½ ß6É ?>7> ã¯¾#"½ ÊË »Þº Ç »ÞÉº á ¾ · ãA@mà P¸ · º > ã¯¾ á¸ º Ê = ¾ãâ»ÞÉº > ã¯º Ç¯ÇÞ· ãÞº= · ß»¯½ ¾ 1

ºV» ·Ç ½ »Þã¯¾ · ß9º»¯Éº= ¾ ¸ ¸ ¾<;Ò½ 7B¨ ¾» »¯½ ¾ ¾=RS ½È»¯º ½Vº=ãÞº ßVº Ç9Ì

δ−

phi =phi − ph,i−1

h

, δ+phi =ph,i+1 − ph,i

h

.

>7>¸ @½ /B »ÞÉºHS ½ »Þº Á à¾ ¸ ·Ê º Ê º9»ÞÉ¾ ;Âº ?>7> ã¯¾#"½ ÊË »Þº.ºD · »Þ½ ¾ G®Å Å P Pá ¾ · ãA@ ßV¾ ½ »Þ½ ¾ Çá @»ÞÉº = ¾ ¸ ¸ ¾<;Ò½ 7B ß9¾ Ç º9ã¯à »¯½ àHºRS ½È»¯º Á ½Vº=ãÞº ßVº Ç ß6Éº Ê º G®Ç º9ºI L O P

−ai+ 1

2

(ph) δ+phi + ai− 1

2

(ph) δ−

phi + wig(phi) − wi−1g(ph,i−1) = −Ji h ,

ph0 = g−1(S0), aN−

1

2

(ph) δ−

phN = 0 ,

;ÒÉº9ã¯º;º ·Ç º ¾P» »¯½ ¾

Ä N<L ®£ ¡ £V ©¡ ´¶P©¦¡

ai(ph) = d(xi)Kf

(

φhi, g(phi))

µf

, wi = d(xi)φhivs .

½ ßVº ;Âº ã¯º½ »Þº9ã¯º Ç »Þº ½ Ê ¾ ¾»Þ¾ ¾ ·Ç>7> ã¯¾#"½ ÊË »¯½ ¾ Ç= »¯Éº"ßV¾ àº=ßV»Þ½ ¾ »Þº9ã Ê ½ Ç?>7> ã¯¾#"½ ÊË »Þº .á @ ·7> ;Ò½ # ½ /B = ¾ã ÊB·¸ Éº=ãÞº= ¾ã¯º»ÞÉº»Þ¾» ¸ »¯ã · ß »¯½ ¾ º9ã¯ã¯¾ã¤¾=»ÞÉº ½ Ç ßVã¯ºV»Þ½ = »Þ½ ¾ ½ Ç ¾ ¸ @B¾=¾Hã º=ã O(h)

H Éº ß9ß · ã ß @¾?=»ÞÉº¤¾ á » ½ º ¨Ç ¾ ¸ · »Þ½ ¾ ½ Ç º Ç »¯½ ÊË »Þº m©.¡´£V³GP³5á @ ·Ç ½ 7B »¯Éº ~Ò·7B º ã ·¸ º

Éº ¾ ¸ ½ º ã ½ Ç ßVã¯ºV»Þº > ãÞ¾ á¸ º Ê ½ Ç¤¸ ½ º ãÞ½ º .á @ »ÞÉº = ¾ ¸ ¸ ¾<;Ò½ 7B ½ »Þº=ã »Þ½ àº P¸ ÁB ¾ã¯½ »ÞÉ Ê.Ì

−ai+ 1

2

(psh) δ+p

s+1

hi + ai− 1

2

(psh) δ

−p

s+1

hi + wig′(ps

hi)ps+1

hi

−wi−1g′(ps

h,i−1)ps+1

h,i−1= −wig(ps

hi) + wi−1g(psh,i−1)

+wig′(ps

hi)pshi − wi−1g

′(psh,i−1)p

sh,i−1 − Ji h ,

ps+1

h0= g

−1(S0), aN−

1

2

(psh) δ

−p

s+1

hN = 0 .

ð ð ìLîì' ±î#" Hí 5ì=ê 3·Ê º=ãÞ½ ß ¸ ºD" Ê >¸ º Ç ;Âº.» ¼Hº > ã Ê º9»Þº=ã ÇË ßV¾º H ß9½ º » Ç = ã¯¾ Ê I ¿ ä<O Éº; »Þº9ãà½ Ç ß9¾ Ç ½È»8@

µf; HÇÇ º9»»Þ¾ 4.7×10−4

$Ç9P »¯ÉºÒºD" > ã¯º Ç¯Ç ½ ¾ = ¾ã»¯ÉºÒã¯º ¸ »Þ½ àº> º=ã Ê º á ½ ¸ ½ »8@

Kf

»¯Éº = ß»¯¾ãk0

; Ç±Ç º9»»Þ¾k0 = 10−10

m2¦ ÉºÂà½ Ç ßV¾º ¸ Ç »Þ½ ßë»¯½ Ê º

ßV¾ #Ç » P »Λ; ÇÇ ºV»Â»Þ¾ 0.0004 s

3 Éº > ã PÊ ºV»¯º9ã Ç ¾= >> º=ã ÊË ß6É½ º > ¾ã¯¾ ·#Ç¸ @Hº9ã1;º=ãÞº» ¼º «HÇ

d0 = 2.5 mm, R = 100 cm, F = 70 kN/m, S0 = 0.5, φ0 = 0.52 .

-0.15 -0.125 -0.1 -0.075 -0.05 -0.025 0 0.025 0.05x [m]

0.5

0.6

0.7

0.8

0.9

1

S

v=0.01

v=0.001

v=0.0001

v=0.00001

-0.1 -0.075 -0.05 -0.025 0 0.025 0.05x [m]

0.5

0.6

0.7

0.8

0.9

1

S

v=0.01

v=0.1v=1

v=5

K M NK9ñþHõEKVñGJôLF¨ûõô3ø C òëú ô=õ%ùG C õ C F¦ñòôLøJGÈù C øÈô¦>¯GJñG C ò 3K

1× 10−5 ≤ vs ≤ 0.01

ö

0.01 ≤ vs ≤ 5.

½ ã Ç » = ¾ ¸ ¸ ¾<;Ò½ 7BP¸ @ Ç ½ Ç > ãÞº Ç º »¯º ½ I ¿ ä<O ;Âº HÇÞÇÞ·Ê º»ÞÉ » ¾ 9¾<; á ¾ ·Á ã @ßV¾ ½ »Þ½ ¾ Ç ã¯ºà P¸ ½ À ½ º

J = 0# Éº ÇÞ » · ã »¯½ ¾ ¾='; »Þº9ã > ãÞ¾S ¸ º Ç = ¾Hã Ç ¾ ¸ ½

> É Ç ºBàHº ¸ ¾ßV½ »Þ½ º Ç¤á ºV»8;Âº9º vs = 0.0001

P0.1 m/s

ã¯º Ç É¾<; ½ ½ B#5Ä#©¦À Éº= ·¸ ¸ÀÇ¯ » · ã »Þ½ ¾

S = 1 ¾ º ã¯½ Ç º Ç ¾ ¸ @ » ÇE·IH ß9½ º » ¸ @ ¸ ã B º Ç ¾ ¸ ½ Y> É Ç º¤àHº ¸ ¾ß9½È»8@

!L¥£9J F¨©¦´£9³+ ¤©£V³³Þ£¡6¡ Ä NN ¾ã ÇEÊË¸ ¸ àº ¸ ¾ßV½ »Þ½ º Ç »ÞÉº > ã¯º Ç¯ÇE· ãÞº > ãÞ¾S ¸ º Ç% ãÞº"ß ¸ ¾ Ç º"»Þ¾B»ÞÉº ßV¾ Ç » P » > ã¯º Ç¯ÇÞ· ãÞº

pf0

GWÇ º=º'ºD · »Þ½ ¾ G®Å Å P= ¾HãJ = 0

Pvs

ÇEÊË¸ ¸ P P# ÇÞ » · ã »¯½ ¾ > ãÞ¾S ¸ º ÇB ãÞºàº9ã @ ÇÞÊ ¾¾»ÞÉ

· ã"ã¯º ÇÞ·¸ » Ç" ãÞº ½WV5º9ã¯º »3= ãÞ¾ Ê ã¯º ÇÞ·¸ » Ç > ãÞº Ç º »¯º ½ I ¿ ä?O ;ÒÉº9ã¯º½ »½ ÇÇ » »Þº »ÞÉ » = ¾Hã ÇÞÊËP¸ ¸Ç ¾ ¸ ½ àHº ¸ ¾ß9½È»¯½ º Ç »ÞÉºb; »Þº=ã Ê ¾ · » > ã ß»Þ½ ß ¸ ¸ @ ãÞº ÊË ½ Ç±· ß6É 7B º »º ß6É > ¾½ »ë¾= > ¾ã¯¾ ·Ç±ÊË »¯º9ã¯½ ¸P »ÞÉº=ãÞº= ¾ã¯ºÂ»ÞÉº Ç¯ » · ã »Þ½ ¾ ½ ß9ãÞº Ç º Çë ã¯¾ ·#»ÞÉºß9º »Þº=ãÒ¾?=±»ÞÉºã¯¾ ¸ ¸ ½ 7B' ¾ º »

x = 0 ãÞº9º=ßV»Þ½ 7B »ÞÉº º=ßVã¯º HÇ º¾?=±»ÞÉº > ¾HãÞ¾ Ç ½ »8@

φ½ »ÞÉº ãÞ¾ ¸ ¸ ½ /B ¾ º e· ß6É > ãÞ¾S ¸ º Ç ß ¾P» á º'¾ á » ½ º = ã¯¾ Ê »ÞÉº > ãÞº Ç º »Þº

Ê »ÞÉº ÊË »Þ½ ß ¸Ê ¾ º ¸W+Ç ½ ßVº»ÞÉº > ã¯º Ç¯ÇÞ· ãÞºãÞº ÊË ½ ÇP¸ Ê ¾ Ç »ËßV¾ Ç » P » »¯Éº9ã¯º = ¾HãÞº»ÞÉº3; »Þº9ã PÊ ¾ · » Ç É¾ ·¸ á º"ã¯º ½ Ç »Þã¯½ á· »Þº Ë ß9ß9¾ã ½ /B »¯Éº Ç¯ » · ã »Þ½ ¾ > ã¯º Ç¯ÇÞ· ãÞººD · »¯½ ¾ 1

· ã ÊË ½ B ¾ P¸ ¾?=»¯É½ ÇÇ º=ßV»Þ½ ¾ ½ Ç »Þ¾½ ß ¸ · º1; »Þº=ã ¸ ¾ ÇÞÇ º Ç ½ »¯¾"»¯Éºb= ¾ã ÊB·¸ »¯½ ¾ ¾?=Ò»ÞÉº ÊË »¯Éº ÊË »¯½ ß P¸ÂÊ ¾ º ¸P »¯¾ ßV¾ Ç ½ º=ã»¯Éº @ Ê ½ ß Ç ¾=Ò»ÞÉº ÇÞ » · ã »¯½ ¾ · º=ã à ã¯½ ¾ ·Ç'Ç ßVº ã¯½ ¾ Ç ¾?=Z; »Þº=ã ¸ ¾ Ç¯Ç º Ç9 º HÇÞÇÞ·Ê º.»ÞÉ » ; »¯º9ã ¸ ¾ Ç¯Ç º Ç » ¼º>¸ ßVº Ê ¾ Ç » ¸ @ »Ò»¯ÉºßVº »Þº9ãÒ¾=»¯ÉºãÞ¾ ¸ ¸ ½ 7B ¾ º GWÇ º=ºFI ?O P º Ç ßVã¯½ á º ½È» HÇ

J(x) = J0φf (x)S(x)ε(x) .

-0.1 -0.075 -0.05 -0.025 0 0.025 0.05x [m]

0.2

0.4

0.6

0.8

1

S

J0 = 0J0 = 0.0001J0 = 0.00035J0 = 0.001

M NHK9ñþõEK9ñGJôLF¨ûHõô3ø C òú ô=õùG C õ C F¦ñ+òôLøJGÈù C øJô¦>6G ñG C ò6ÿ

º«É àº S4"º »ÞÉº Ç ¾ ¸ ½ > É Ç º àHº ¸ ¾ß9½È»8@n»¯¾v = 1 m/s

Ç ½ Êâ·¸ »Þº »ÞÉº> ãÞ¾ß9º Ç¯Ç = ¾ã ½WV5º9ã¯º »¤à P¸ · º Ç ¾?=

J0

Éº¾ á » ½ º .ÇÞ » · ã »¯½ ¾ T> ã¯¾?S ¸ º Ç¤ ã¯º Ç É¾<; ½ ½ B#%Å º Ç º9º »ÞÉ »J= ¾ã ÇÞ·IH ßV½ º » ¸ @ ½ »Þº Ç ½ àº ; »¯º9ã ¸ ¾ Ç¯Ç º Ç »¯Éº Ç¯ » · ã »Þº ¾ º Ç ½ ÇÞ>7> º ã 5 »½ Ç ½ »Þº=ãÞº Ç »¯½ /B »¯¾ ¾P»¯º»ÞÉ » á ¾ · » KPÃ > º9ã6ßVº » Ç ¾?=b; »Þº9ã½ Ç» ¼Hº ; @F= ¾ã

J0 = 0.0013 » ?B ã¯º9º Ç ;º ¸ ¸ ;Ò½È»¯ÉãÞº ÇE·¸ » Ç > ãÞº Ç º »Þº ½ I <O ;ÒÉº9ã¯º

= ·¸ ¸eÇÞ » · ã »¯½ ¾ ¾=R; »Þº=ãÒ½ Ç ¾P»"ßV¾ Ç ½ º=ãÞº ½ »ÞÉº ÊË »¯Éº ÊË »Þ½ ß P¸ÀÊ ¾ º ¸W

& 3 -.* * ? 3,- : * 0 . 0,*/ . 2 ? -8 *-@?X-4*:;: 0B-'4.336 ) * »ÞÉ½ Ç Ç ºLß»¯½ ¾ ;º¨ßV¾ Ç ½ º=ã¤»ÞÉº ÊË »¯Éº ÊË »¯½ ß P¸±Ê ¾ º ¸¶ ;ÒÉ½ ß6É; Ç> ã¯¾ > ¾ Ç º .á @¤½ ¸ » · º I O

Ä N ç ®£ ¡ £V ©¡ ´¶P©¦¡ ð#ð #í "®îì Òì' $ " Éº= ¾ ¸ ¸ ¾<;Ò½ 7B¨ÊË ½ .HÇÞÇÞ·ÊF> »¯½ ¾ Ç ã¯º HÇÞÇÞ·Ê º »Þ¾ á ºà P¸ ½ ÀÌ

• ½ Ê º º > º º »"º V5º=ßV» ÇÒ ãÞº ÇÞÊ¸ ¸1 ß á º º BH¸ ºLß»¯º À

• Éº

y

½ ã¯º=ß»¯½ ¾ XG = ¾ ¸ ¸ ¾<;Ò½ 7B »¯Éº ãÞ¾ ¸ ¸5 "º Ç P%½ ÇÂ º B¸ º=ßV»Þº Ç ½ ß9º"»¯Éº ãÞ¾ ¸ ¸ Ç ãÞº ·7>»Þ¾ 10

Ê ¸ ¾ 7B#•

Éº19¾<; ¾= á ¾P»¯É ¸ ½ D · ½ Ç9 ½ º ; »¯º9ã â ½ ã ½ Ç\>· ã¯º ¸ @»¯ã Ç àHº9ã ÇÞ¸ ½ »ÞÉºÒãÞº Ç »= ã Ê º ¾=»ÞÉº Ê »¤º »Þº=ãÞ½ 7B »ÞÉº ½ >1# É½ Ç¤Ê º PÇ »¯É »

vxα = v

xs , α = a, f.

ºZ;Ò½ ¸ ¸·Ç º Ç ½ ÊF>¸ º ¾» »¯½ ¾ Ç ¾?=±àº ¸ ¾ßV½ »Þ½ º Ç

c = vxs , vα = v

zα, α = a, f, s .

»ÞÉº > ã¯º Ç¯Ç«Ç ºLß»Þ½ ¾ ¾?= e>?> º9ã ÊË ß6É½ º ; »¯º9ã«½ Ç.Ç D · º9º º ¾ · ».¾?=»¯Éº>?> º9ãT;º á ½ »Þ¾ »ÞÉºH= º ¸ » »ÞÉ ·Ç ;Âºß9¾ Ç ½ º9ãË»¯Éº9¾<; ½ »ÞÉºH= ¾ ¸ ¸ ¾<;Ò½ 7B ã¯º B ½ ¾ D = Dw ∪Df

;ÒÉº9ã¯ºDw, Df

ã¯º»ÞÉº >> º=ã ;º á'P = º ¸ »ãÞº B ½ ¾ Ç= ã¯º ÇA> ºLß»Þ½ àº ¸ @ Ì

Df =

(x, z) : −L

2≤ x ≤

L

2, −γf (x) ≤ z ≤ 0

,

Dw =

(x, z) : −L

2≤ x ≤

L

2, 0 ≤ z ≤ γw(x)

.

½È»¯É»ÞÉº Ç º Ç¯ÇÞ·ÊF> »Þ½ ¾ Ç ;Âº B º9»d= ã¯¾ Ê GWÄ äP GWÄ K P5»ÞÉº = ¾ ¸ ¸ ¾<;Ò½ /B¤ÊËÇ¯Çá¸ ßVººD · »¯½ ¾ Ç

c

∂φα

∂x

+∂

∂z

(

φαvα

)

= 0, α = s, f,

c

∂

∂x

(

φaph

)

+∂

∂z

(

φaphva

)

= 0,

G ä ¿ P

P »ÞÉº Ê ¾ Ê º » ·Ê áP¸ P ß9º ºD · »¯½ ¾ Ç

φα(vα − vs) = −Kα

µα

∂ph

∂z

, α = a, f,

∂ps

∂z

− ph

∂φs

∂z

=φ

2aµa

Ka

(

va − vs

)

+φ

2f µf

Kf

(

vf − vs

)

.

G ä Ä P

·ÊÊ ½ 7B·7>¸ ¸ ºD · »¯½ ¾ Ç ¾?= G ä Ä P ;º B ºV»Ò»ÞÉ »

∂pT

∂z

= 0 ,

;ÒÉº9ã¯º»ÞÉº »¯¾P» ¸ > ã¯º Ç¯ÇE· ã¯º½ Ç¤ º S º .Ç

pT = pa + pf + ps = ps + (1 − φs) ph .

"Ç ½ 7B »¯Éº B º º=ã ¸ ½ º º9ã =?B É½ > ã¯½ ßV½ >¸ ºZ;ÂºZ;Òã¯½È»¯º »ÞÉº > ã¯º Ç¯ÇÞ· ãÞº á¸ ßVººD ·¦Á»Þ½ ¾ ½ . = ¾ã Ê

!L¥£9J F¨©¦´£9³+ ¤©£V³³Þ£¡6¡ Ä N

pT (x) = pst(s) + f

(

φs(x, z))

ph(x, z) ,

G ä Å P

;ÒÉº9ã¯ºs = 1−

φs0

φs

½ Ç »ÞÉº Ç »¯ã ½ âPpst

½ Ç »ÞÉº Ç »Þã · ß» · ã ¸ > ãÞº ÇÞÇÞ· ã¯º+½ »ÞÉº Pá#Ç º ßVº¾?= É*@ ã¯¾ Ç » »Þ½ ß > ãÞº ÇÞÇÞ· ã¯º » ½ ÇË ¾»Þ½ ßVº ½ I <O¤»¯É » > ¾ã¯¾ ·#ÇÊ »Þº=ãÞ½ P¸ ß P Ç É¾<;É*@ Ç »Þº=ãÞº Ç ½ Ç ÇÞ· V5º9ã > º9ã Ê º » º= ¾ã Ê »Þ½ ¾ 1 Éº ßVº»ÞÉº Ç »Þã · ß» · ã P¸"> ã¯º Ç¯ÇE· ã¯º"½ Ç> ã PÊ ºV»¯ãÞ½ º HÇ

pst = pst0

r

1 − r

, r =

s

s0

, during compression,

s − εsm

s0 − εs0

, during expansion,

G ä ä*P

;ÒÉº9ã¯ºsm

½ Ç »¯Éº ÊË "½ Êâ·Ê Ç »Þã ½ « ß6É½ º9àº á @ ß9º9ãÞ» ½ > ¾H½ » »%ãÞº ÊË ½ Ç »Þ¾ B ºV»%»ÞÉº¤ºD · »¯½ ¾ = ¾ã»¯Éº Ç ¾ ¸ ½ > É Ç º¤àº ¸ ¾ßV½ »8@

vs

3·ÊÊ ½ 7B·7>P¸ ¸ »¯ÉãÞº=º ÊËÇ¯Ç ßV¾ #Ç º=ãÞà »¯½ ¾ ºD · »Þ½ ¾ ÇG ä ¿ P ;º B º9»"»ÞÉº Sã Ç »¾ã º=ã ½Vº=ãÞº »Þ½ P¸ºD · »¯½ ¾ = ¾Hã

vs

Ì

∂

∂z

(

φf uf + φaua + vs

)

= −φa

ph

(

c

∂ph

∂x

+ (ua + vs)∂ph

∂z

)

,

;ÒÉº9ã¯ºuα = vα − vs, α = f, a

HÇ ½ 7B »¯Éº¤ã¯º ¸ »¯½ ¾ G ä Å P »¯Éº ÊËÇ¯Ç ß9¾ Ç º9ã¯à ¦Á»Þ½ ¾ ºD ·# »Þ½ ¾ = ¾ãe»ÞÉº Ç ¾ ¸ ½ > É HÇ º,;ºÂßV¾ àHº9ãÞ»1»ÞÉº

x º=ãÞ½ à »Þ½ àº%¾?=»ÞÉºÉ@ ã¯¾ Ç » »¯½ ß

> ãÞº ÇÞÇÞ· ã¯º½ »Þ¾»¯Éº ¼ ¾<; x º=ãÞ½ à »Þ½ àº ¾=»ÞÉº »Þ¾P» P¸> ãÞº ÇÞÇÞ· ã¯º Ì

(

1+φa

ph

φsq

)∂vs

∂z

= −φa

ph

(

c

f(φs)

∂pT (x)

∂x

−qua

∂φs

∂z

)

−∂

∂z

(

φfuf +φaua

)

,

G ä K P

;ÒÉº9ã¯ºq =

phf′(φs) + p

′

st(φs)

f(φs).

Éº B ã ½ º »¾?=»¯ÉºÉ@ ãÞ¾ Ç » »¯½ ß > ãÞº ÇÞÇÞ· ã¯º½ ÇÒÇ¯ÇÞ·Ê º »Þ¾ á ºß9¾ Ç » »Ò½ »¯Éº= º ¸ »

∂ph(x, z)

∂z

=∆ph

γf (x), (x, z) ∈ Df ,

G ä L P;ÒÉº9ã¯º

∆ph½ Ç »ÞÉº »¯¾P» ¸ > ã¯º Ç¯ÇE· ã¯º ã¯¾ > ßVã¯¾ Ç¯Ç »ÞÉº= º ¸ »

±îí î# ï 5ì' &®ì ï ®ì' ±ê »¯Éº ÇE· ã = ßVº ¾?=±»ÞÉº ·7>/> º=ã¤ãÞ¾ ¸ ¸

Γw =

(x, z) : −L

2≤ x ≤

L

2, z = γw(x) ,

;º Ç¯ÇÞ·Ê º»¯É » ¾ 9¾<; á ¾ · ãA@ ß9¾ ½È»¯½ ¾ ÇÒ ã¯ºà ¸ ½

vα = vs, α = a, f, (x, z) ∈ Γw .

G ä N P »¯Éº ¸ ¾<;Âº9ã ÇÞ· ãA= ßVº

Ä ç Ã ®£ ¡ £V ©¡ ´¶P©¦¡

Γf =

(x, z) : −L

2≤ x ≤

L

2, z = γf (x)

»ÞÉºÉ@ ãÞ¾ Ç » »Þ½ ß > ã¯º Ç¯ÇÞ· ãÞº"½ ÇbB ½ àº 'P »ÞÉº3; »Þº=ã ' ½ ã Ê ¾¦àHº = ã¯º9º ¸ @¾ · »¾?=À»¯Éº= º ¸ » Ì

ph(x, z) = phf .

Éº;º á Y= º ¸ »¤½ »Þº9ãA= ß9ºΓwf

Γwf =

(x, z) : −L

2≤ x ≤

L

2, z = 0

½ Ç S."º À »ÞÉº9ã¯º = ¾ã¯ºvs = 0, (x, z) ∈ Γwf .

ÉºZ; »Þº=ã . ½ ãb9¾<; ÇÒ ã¯º HÇÞÇÞ·Ê º »Þ¾ á ºßV¾ »Þ½ 3· ¾ ·Ç ¾ »¯Éº½ »¯º9ãA= ßVºΓwf

ðÀð Wê=ïí3ìîì'ì' $ " »ÞÉ½ Ç¨Ç ºLß»Þ½ ¾ ;º > ãÞº Ç º » S ½È»¯º3àH¾ ¸ ·Ê º Ç ß6Éº Ê ºJ= ¾Hã Ç ¾ ¸ à½ 7B »ÞÉº º Ç ßVã¯½ á º Pá ¾¦àº Ê »ÞÉº ÊË »Þ½ ß ¸ÀÊ ¾ º ¸ ¾= >> º=ã > ã¯º Ç¯ÇÊË ß6É½ º

º º S º ½ Ç ßVã¯ºV»Þº Ê º Ç Éº Ç ;ÒÉ½ ß6É ã¯º @ Ê ½ ß ¸ ¸ @ > »Þº »Þ¾ »ÞÉº Ê ¾¦à½ 7Bá ¾ · ãA@'¾=»¯Éº Ç ¾ ¸ ½ T> É Ç º Éº Ê º Ç É.½

x ½ Ê º Ç ½ ¾ ½ Ç1B ½ àHº «á @

ωτ =

xn : x

n = xn−1 + τ

n−1, n = 1, 2, . . . , N, x

0 = −L

2, x

N =L

2

.

Éº Ê º Ç Én½ z ½ ãÞºLß»Þ½ ¾ ½ Ç P¸ Ç ¾ ¾ Á¶· ½W= ¾Hã Ê) ½ » º > º #Ç ¾ »¯Éº > ¾ Ç ½ »Þ½ ¾

¾?= á ¾ · ãÞ½ º Çγf

Pγw

Ì

ωh(xn) =

znj : z

nj = z

nj−1 + h

nj−1, j = −J/2, . . . , J/2

,

;ÒÉº9ã¯ºz

n−J/2 = −γ

nf , z

n0 = γ

nwf , z

nJ/2 = γ

nw .

¤º9ã¯º P ½ »¯Éº = ¾ ¸ ¸ ¾<;Ò½ 7B ;º ·Ç º»ÞÉº ¾» »¯½ ¾ u

nj = u(xn

, zj)= ¾Hã P @ ½ Ç ß9ãÞº9»Þº

= · ß»¯½ ¾ u

ºV» Ç¯ÇÞ·Ê º»ÞÉ »»ÞÉº Ç ¾ ¸ · »Þ½ ¾ ½ Ç ¼ ¾<; = ¾ã

x = xn

½ º À ½ Ç ß9ãÞº9»Þº:= · ß»¯½ ¾ Çφ

ns , φ

nf , v

ns , v

nf , z

n ã¯º B ½ àHº 1± ÉºYS ½È»¯º3àH¾ ¸ ·Ê º ½ Ç ßVã¯ºV»Þ½ = »Þ½ ¾ ½ Çâ º S º á @

»ÞÉº= ¾ ¸ ¸ ¾<;Ò½ /BâÊË ½ Ç »¯º >Ç ¾=e»¯Éº P¸EB ¾HãÞ½ »ÞÉ Ê«

£V´£9³§â5©¦´¶G¦ F´£§Ë£V¡Eω

n+1

h

»¯ÉºZ;Âº á ãÞº B ½ ¾ »¯Éº ¾ º Ç¤ »tn+1

ãÞº ºS º .HÇ

zn+1

j = znj + τ

nv

nsj

c

, j = 1, 2, . . . , J/2 .

G ä ç*P »¯Éº= º ¸ »¤ã¯º B ½ ¾ «P. »Â»¯ÉºZ;Âº á = º ¸ »¤½ »¯º9ãA= ßVº»¯Éº ¾ º ÇÒ ãÞº ¾» Ê ¾¦à½ 7BÌ

zn+1

j = z0j , j = −J/2, , . . . , 0 .

!L¥£9J F¨©¦´£9³+ ¤©£V³³Þ£¡6¡ Ä ç ¿ ¡9¬V³Þ£9´£§Ë©¡6¡B¬ P¡9£9³²©P´WGPm£9 ©¦´¶G¦ FDP³ ¡¦J®¥AF6³Þ©¬9´¶G¦ ½ ßVº B ãÞ½ ¾ º Ç ãÞº Ê ¾¦à½ 7B ;Ò½ »ÞÉ'àHº ¸ ¾ß9½È»8@

vs

»ÞÉº Ç ¾ ¸ ½ àH¾ ¸ ·Ê º = ã ß»¯½ ¾ à ¸ · º Ç ã¯º B ½ àHº á @

φn+1

sj = φnsj

hnj + h

nj−1

hn+1

j + hn+1

j−1

, −J/2 < j < J/2 ,

G ä P

φn+1

s,−J/2= φ

ns,−J/2

hn−J/2

hn+1

−J/2

, φn+1

s,J/2= φ

ns,J/2

hnJ/2−1

hn+1

J/2−1

.

¡9¬V³Þ£9´£§Ë©¡6¡B¬ P¡9£9³²©P´WGPm£9 ©¦´¶G¦ FDP³ %©¦´£V³ F6³¯©¬9´WGP ¤·Ê º9ã¯½ ß P¸%Ê º9»ÞÉ¾ Ç = ¾ã Ç ¾ ¸ à½ 7B ºD ·# »Þ½ ¾ #Ç º Ç ßVã¯½ á ½ /B »¯ÉºJ9 · ½ 9¾<; > ã¯¾ á¸ º ÊËÇ½ Ê ¾¦à½ /B ßV¾¾ã ½ »¯º Ç @ Ç »¯º Ê ã¯º > ã¯º Ç º »Þº ½ ÊËP @ >#?> º9ã Ç9Ç º9ºJI ä e¿¿P¿LÄ O ¤º9ã¯º ;º ;Ò½ ¸ ¸1?>/>¸ @»¯Éº ½ Ç ß9ãÞº9»Þ½ = »¯½ ¾ Ê ºV»¯É¾ À ;ÒÉ½ ß6É ; ÇÂ·Ç º ½ I L O ½ ã Ç »1;º¾ á » ½ »¯Éº½ »¯º B ã P¸ = ¾Hã Êâ·¸ »Þ½ ¾ ¾?=±»ÞÉº ÊËÇ¯Çá¸ ßVº ºD · »Þ½ ¾

c

∂φf

∂x

+∂

∂z

(

φfvf

)

= 0 .

»¯º B ã »Þ½ 7B ½ »¤½ »¯Éºº ¸ º Ê º » ã @Ëà¾ ¸ ·Ê º [zs(x), zf (x)]P# ·Ç ½ 7B »ÞÉººD ·#P¸ ½È»8@

d

dx

zf (x)∫

zs(x)

φf (x, z) dz =

zf (x)∫

zs(x)

∂φf

∂z

dz +dzf (x)

dx

φf (x, zf (x)) −dzs(x)

dx

φf (x, zs(x)),

B ½ àº Ç »¯Éº½ »¯º B ã P¸ÀÊËHÇÞÇÂá¸ ßVº ºD · »Þ½ ¾

c

d

dx

zf (x)∫

zs(x)

φf (x, z) dx + ff (x) − fs(x) = 0 ,

G ä ¿LÃ P

;ÒÉº9ã¯ºfα(x) =

[

vf

(

x, zα(x))

− vs

(

x, zα(x))]

φf

(

x, zα(x))

.

e ¼3½ 7B º ¸ º Ê º » ã @và¾ ¸ ·Ê º [zj− 1

2

(x), zj+ 1

2

(x)] >7>¸ @½ 7B »¯Éº S ½ »Þº Á

à¾ ¸ ·Ê º Ê º9»ÞÉ¾ ;Âº >7> ã¯¾#"½ ÊË »¯º¤»¯Éº ½ »Þº B ã P¸ ºD · »Þ½ ¾ CG ä ¿=Ã P á @»¯Éº= ¾ ¸ ¸ ¾<;Ò½ 7BßV¾ #Ç º=ãÞà »¯½ àHºº" >¸ ½ ßV½ » S ½ »Þº( ½V5º9ã¯º ß9º Ç ß6Éº Ê º

hn+1

j− 1

2

φn+1

fj − hnj− 1

2

φnfj

τn

+ F

(

wnj+ 1

2

, φnf,j+1, φ

nfj

)

− F

(

wnj− 1

2

, φnfj , φ

nf,j−1

)

= 0,

;ÒÉº9ã¯º»ÞÉº ·Ê º9ã¯½ ß P¸ 9 · "F

(

wj+ 1

2

, φj+1, φj

) ½ Ç¤ ºS º .HÇ = ¾ ¸ ¸ ¾<; Ç I L O Ì

F

(

wj+ 1

2

, φj+1, φj

)

=1

2wj+ 1

2

(φj+1 + φj) −1

2|wj+ 1

2

| (φj+1 − φj) ,

wnj+ 1

2

= vnf,j+ 1

2

− vns,j+ 1

2

, hnj− 1

2

=h

nj + h

nj−1

2.

Ä ç Ä ®£ ¡ £V ©¡ ´¶P©¦¡ Éº á ¾ · ãA@ ß9¾ ½È»¯½ ¾ ÇÒ ã¯º ?>7> ã¯¾#"½ ÊË »Þº 'á @

hn+1

−

J

2

φn+1

f,− J

2

− hn−

J

2

φnf,− J

2

2τn

+ F

(

wn−

J

2+ 1

2

, φnf,− J

2+1

, φnf,− J

2

)

− Fn−

J

2

= 0,

hn+1J

2−1

φn+1

f, J

2

− hnJ

2−1

φnf, J

2

2τn

− F

(

wnJ

2−

1

2

, φnf, J

2

, φnf, J

2−1

)

= 0,

;ÒÉº9ã¯ºF

n−

J

2

ºS º Ç »ÞÉº = ãÞº=º¾ · » *9¾<; P ¾3½ 9¾<; 9 · " Ì

Fn−

J

2

=

0, if wn−

J

2

> 0,

wn−

J

2

φnf,− J

2

, if wn−

J

2

≤ 0 .

Éº ÇÞ » · ã »¯½ ¾ ¾?=\; »Þº=ãb= ã ß»Þ½ ¾ .Ç É¾ ·¸ .P¸ ; @ ÇÂÇÞ »Þ½ Ç =Q@'»ÞÉºß9¾ ½ »Þ½ ¾ #Ç

0 ≤ Sn+1

fj ≤ 1, zj ∈ ωh(xn+1) .

Éº9ã¯º = ¾HãÞº % =»¯º9ãJS # ½ /Bφ

n+1

f= ã¯¾ Ê »ÞÉº ½ Ç ß9ãÞº9»Þº ÊËÇ¯Ç'á¸ ßVººD · »¯½ ¾ 1%P

ãÞ»Þ½S#ßV½ P¸À¸ ½ Ê ½ »Þº=ãÒ½ Ç¤H ½ »Þ½ ¾ ¸ ¸ @ >7>¸ ½ º ÀÌ

φn+1

fj :=

φn+1

fj , if 0 ≤ Sn+1

fj ≤ 1,

Sn+1

fj φn+1

j , otherwise .

£( ²¦©¦J £V¡ =F ´ £¨©¦³F6³¯©¬9´WGP Éº º; à P¸ · º Ç ¾=e»¯Éº ½ ã1= ã ß»¯½ ¾ « ãÞº¾ á » ½ º = ã¯¾ Ê G¶Ä ¿ P

φn+1

aj = 1 − φn+1

sj − φn+1

fj , zj ∈ ωh(xn+1) .

G ä ¿H¿ P ª5£ µ¥P³ ¡´©¦´¶®¬ ³Þ£V¡6¡6 ³¯£â£9 ©¦´¶G¦ Éº º; à P¸ · º Ç ¾=e»¯ÉºÉ@ ãÞ¾ Ç » »Þ½ ß > ãÞº ÇÞÇÞ· ã¯º ãÞº ¾ á » ½ º = ã¯¾ Ê G ä Å P

pn+1

hj =pT (xn+1) − p

n+1

st,j

f(sn+1

j ), zj ∈ ωh(xn+1) ,

G ä ¿Ä P

;ÒÉº9ã¯º»ÞÉº Ç »Þã · ß» · ã P¸ > ã¯º Ç¯ÇE· ãÞº½ Ç ß9¾ ÊF>· »¯º = ã¯¾ Ê G ä ä*P

pn+1

st,j = pst0

rn+1

j

1 − rn+1

j

.

ª5£B£ ²¦©¦J £V¡1=F³Þ£9 ©P´W²¦£²¦£9=¬9´¶®£¡ Éº º; à ¸ · º Ç ¾?=±àº ¸ ¾ßV½ »Þ½ º Ç ã¯º ¸ »¯½ àHº"»¯¾â»¯Éº Ç ¾ ¸ ½ T> É Ç º ã¯º¾ á » ½ º = ã¯¾ Ê »¯Éºæ ã¯ß@ ¸ ; Ì

φn+1

αj uαj = −K

n+1

αj

µα

pn+1

h,j+1− p

n+1

hj

hn+1

j

, α = a, f, zj ∈ ωh(xn+1) .

!L¥£9J F¨©¦´£9³+ ¤©£V³³Þ£¡6¡ Ä ç Å ª5£B£ ²¦©¦J £V¡1=FB©P²¦£VL¬9´W®£V¡ Æ º ¸ ¾ß9½È»8@ ¾?=»¯Éº Ç ¾ ¸ ½ T> É HÇ º

vn+1s

½ Ç ¾ á » ½ º á @ Ç ¾ ¸ à½ 7B¨ ½ Ç ßVã¯ºV»Þ½ º ºD ·# »Þ½ ¾ G ä K P Éº »¯ÉºZ; »Þº=ã P« ½ ãÒàº ¸ ¾ßV½ »Þ½ º Ç ãÞº B ½ àº á @

vn+1

αj = vn+1

sj + un+1

αj , α = a, f, zj ∈ ωh(xn+1) .

»ÞÉ½ ÇÊ ¾ º ¸ »¯Éº > ãÞº ÇÞÇÞ· ã¯ºBºD · »Þ½ ¾ ½ ÇÇ ¾ ¸ àº ºD" >¸ ½ ß9½È» ¸ @ G ;º ¾P»¯º »ÞÉ » ½ »ÞÉº > ã¯º9à½ ¾ ·Ç¨Ê ¾ º ¸ »ÞÉº > ã¯º Ç¯ÇÞ· ãÞº«ºD · »Þ½ ¾ ; HÇ B¸ ¾ á¸ ½

x ½ ã¯º=ßV»Þ½ ¾ P · ß6É

Ç ½ ÊF>¸ ½S#ß »Þ½ ¾ ¾?=±»ÞÉº Ê ¾ º ¸ ; Çb> ¾ ÇÞÇ ½ á¸ º · º »Þ¾¨»8;¾ HÇÞÇÞ·ÊF> »Þ½ ¾ #Ç9Ì

• Éº > ã¯º Ç¯ÇÞ· ãÞºË½ Ç¨ º = ¾Hã ÊË »¯½ ¾ ã¯½ àHº ¾ ¸ @n»ÞÉº ½Vº=ãÞº ßVº á º9»8;º=º »¯Éº»Þ¾P» P¸M> ãÞº ÇÞÇÞ· ã¯º G ;ÒÉ½ ß6É ¾º Ç+ ¾» º > º ¾

zP »ÞÉº Ç »Þã · ß» · ã ¸/> ã¯º Ç¯ÇÞ· ãÞºÒ½ Ç

» ¼Hº ½ »¯¾ ß9ß9¾ · » •

Éº 9¾<;R½ Ç >· ã¯º ¸ @Ë»¯ã Ç àHº9ã ÇÞ¸W ÉºßV¾ #Ç »¯½È» · »Þ½ àº.ã¯º ¸ »Þ½ ¾ á ºV»8;Âº9º »ÞÉº ÇÞ » · ã »¯½ ¾ ß ?> ½ ¸ ¸ ãA@ > ã¯º Ç¯ÇÞ· ãÞº

½ ÇË ¾P» ·Ç º ½ »ÞÉº Ê ¾ º ¸ »Þ¾ ß ¸ ¾ Ç º«»ÞÉº Ç @ Ç »Þº Ê« »ÞÉ ·ÇJB ã H ½ º » Ç ½ »¯Éº½ ½È»¯½ ¸ ½ Ç »Þã¯½ á· »Þ½ ¾ ¾=À»ÞÉº ÇÞ » · ã »¯½ ¾ Ë ã¯º ¾P» ÇÞÊ ¾¾P»ÞÉº á @¨»ÞÉº > ãÞº ÇÞÇÞ· ã¯º ¾ã%º" PÊF>¸ º ¸ º9» ·ÇÇ¯ÇE·Ê º.»¯É »»ÞÉº > ¾HãÞ¾ ·ÇËÊË » ½ Ç' ¾P»«ß9¾ ÊF> ã¯º Ç¯Ç º

pT (x) = 0P »¯Éº

= ¾ ¸ ¸ ¾<;Ò½ 7B ½ ½ »Þ½ P¸ ßV¾ # ½È»¯½ ¾ ÇÒ ã¯º à P¸ ½ ÀÌ

v0s = 0, v

0f = 0, v

0a = 0, φ

0aj = 0, φ

0sj = 0.3, zj ∈ ωh(x0) ,

φ0fj =

0.6 if j = j0 ,

0.3 if j 6= j0 .

Éº »ÞÉº ; »Þº9ã ½ Ç »Þã¯½ á· »Þ½ ¾ ãÞº ÊË ½ Ç±· ß6É 7B º = ¾Hã P @x

n ∈ ωτ

½ º φ

nf = φ

0f

ð ðÀìLîé "®ì î#±î#" 1ê®ê »¯É½ Ç¤ÇE·áÇ ºLß»¯½ ¾ ;ÂºßV¾ Ç ½ º=ã»ÞÉº Ç » á ½ ¸ ½ »8@'¾?=±»ÞÉº¾ á » ½ º S ½ »Þº(3àH¾ ¸ ·Ê º >Á> ãÞ¾#"½ ÊË »Þ½ ¾ I O5»ÞÉº Ç » á ½ ¸ ½ »8@ P¸ @ Ç ½ Ç ½ ÇÂ ¾ º Ç º > ã »Þº ¸ @= ¾ãÂº ß6É ºD · »¯½ ¾ 1 ½ ßVº ÊËÇ¯Ç ßV¾ #Ç º=ãÞà »¯½ ¾ ºD · »Þ½ ¾ Ç1G ä P P G ä ¿LÃ P ã¯ºÉ@ > º9ã á ¾ ¸ ½ ß ;Âº B ºV»±»ÞÉ »ºD" >¸ ½ ß9½È» ?>7> ã¯¾#"½ ÊË »Þ½ ¾ Ç+ ãÞº Ç » Pá¸ º¤½= ½ Ç ß9ãÞº9»Þº Ç »¯º >Ç

τn

Ç¯ »Þ½ Ç =Q@â»¯Éº = ¾ ¸ ¸ ¾<;Ò½ 7B º ÇEÁ»Þ½ Ê »Þº Ç

vns τ

n< min

jh

nj , v

nf τ

n< min

jh

nj .

º ¾»Þº»ÞÉ » ÇÞ· ß6É P¸ @ Ç ½ Ç ½ Ç ¾P» ÇÞ·IH ßV½ º »Ò½ ¾Hã º=ã»Þ¾ > ã¯¾¦àº»ÞÉº Ç » á ½ ¸ ½ »8@¾?=5»ÞÉº Ç @ Ç »Þº Ê ¾=5ºD · »¯½ ¾ Ç93 º9»ßV¾ Ç ½ º=ã Ç ½ ÊF>¸ ½WS#º ËÊ ¾ º ¸ ¾?=5»8;Â¾ > É Ç ºb9¾<; ;ÒÉ½ ß6É.½ ÇÒ º Ç ßVã¯½ á º á @'»¯Éº ÊËÇ¯Ç ßV¾ #Ç º=ãÞà »¯½ ¾ ºD ·# »Þ½ ¾ #Ç

∂φs

∂x

+∂

∂z

(

φsvs

)

= 0,

∂φf

∂x

+∂

∂z

(

φf vf

)

= 0,

P »ÞÉº æ ã6ß @ ºD · »¯½ ¾

Ä çPä ®£ ¡ £V ©¡ ´¶P©¦¡

φf (vf − vs) = −Kf

µf

∂ph

∂z

.

·ÊÊ ½ 7BË·/>«ÊËHÇÞÇ ß9¾ Ç º9ã¯à »Þ½ ¾ ºD · »Þ½ ¾ Ç ;Âº B ºV»Ò»¯É »

∂

∂z

(

φsvs + φfvf

)

= 0,

¾ãvs + φf (vf − vs) = C,

;ÒÉº9ã¯ºC

½ Ç ßV¾ Ç » P » 1e ¼3½ 7B ½ »¯¾ ß9ß9¾ · » »ÞÉº æ ã¯ß@ºD · »Þ½ ¾ 1 ;Âº B º9»»¯ÉºÊHÇÞÇá#P¸ P ß9ººD · »¯½ ¾ ¾=»ÞÉº Ç ¾ ¸ ½ > É Ç º

∂φs

∂x

+ C

∂φs

∂z

= −∂

∂z

(

φs

Kf

µf

∂ph

∂z

)

.

G ä ¿LÅ P

» ; HÇ'Ç É¾<; Pá ¾¦àº«»ÞÉ »'»ÞÉºÉ@ ãÞ¾ Ç » »Þ½ ß > ã¯º Ç¯ÇE· ãÞº.½ Ç ºS º HÇ' = · ßV»Þ½ ¾ ph = P (φs)

=P

′

< 0 ;Âº B º9»b= ã¯¾ Ê G ä ¿=Å P ># ã Pá ¾ ¸ ½ ßºD · »Þ½ ¾

∂φs

∂x

+ C

∂φs

∂z

=∂

∂z

(

φs

Kf

µf

|P ′(φs)|∂φs

∂z

)

.

G ä ¿ ä*P

Éº »ÞÉº ¸B ¾ã¯½È»¯É Ê > ã¯º Ç º »Þº á ¾¦àHºâ½ Ç ºD · ½ à P¸ º » »Þ¾ ½ Ç ß9ãÞº9»Þ½ = »¯½ ¾ ¾= G ä ¿ ä*Pá @'»ÞÉºº" >¸ ½ ßV½ » +·¸ º9ã Ç ß6Éº Ê º

φn+1s − φ

ns

τ

+ C ∂z,h φns = ∂z,h

(

φs

Kf

µf

|P ′(φns )| ∂z,h φ

ns

)

,

;ÒÉº9ã¯º∂z,h

ºS º Ç S ½È»¯º ½WV5º9ã¯º ßVºB¾ > º=ã »Þ¾Hã »3= ¾ ¸ ¸ ¾<; Ç = ãÞ¾ Ê »ÞÉº Ê "½ Êâ·Ê> ãÞ½ ß9½ >¸ º G®Ç º9ºJI ¿=Å OQP»ÞÉ »Ò»ÞÉ½ Ç¤Ç ß6Éº Ê º ½ ÇÒÇ » Pá¸ º ¾ ¸ @ ½=

τ ≤ c minj

h2j .

=P

′

> 0 »ÞÉº »¯Éº > ã¯¾ > ¾ Ç º aÊË »ÞÉº Ê »Þ½ ß ¸Ê ¾ º ¸FG ä ¿=Å P á º=ß9¾ Ê º Ç ½ ¸ ¸ Á

ßV¾ # ½È»¯½ ¾ º GWÇ ½ Ê ½ ¸ ã »¯¾ Ç ¾ ß ¸ ¸ º Éº » ßV¾ · ßV»Þ½ ¾ ºD · »¯½ ¾ ;Ò½ »ÞÉm»¯ÉºË½ àHº9ã Ç º ½ ã¯º=ßV»Þ½ ¾ ¾?=±»Þ½ Ê ºP

∂φs

∂x

+ C

∂φs

∂z

= −∂

∂z

(

φs

Kf

µf

P′(φs)

∂φs

∂z

)

.

P Ç > º=ß9½ ¸¤·Ê º9ã¯½ ß P¸Ê ºV»¯É¾ Ç'Ç É¾ ·¸ xá º ·Ç º »¯¾XS ½È» Ç'Ç ¾ ¸ · »Þ½ ¾ I ç?O ¾ãºD" PÊF>¸ º »¯ÉºÉ@ ãÞ¾ Ç » »Þ½ ß > ãÞº ÇÞÇÞ· ã¯º= · ß»¯½ ¾ = ãÞ¾ Ê G ä Å P

ph(x, z) =pT (x) − ps(s)

1 − φs(x, z)

; HÇ"·Ç º ½ ·Ê º9ã¯½ ß P¸ ºD" > º9ã¯½ Ê º » Ç > ãÞº Ç º »Þº ½ I ?O À ½ BH· ãÞº¨ä»ÞÉº B ã ?> É½ ß¾?=±»ÞÉ½ Ç = · ßV»Þ½ ¾ ½ Ç1> ãÞº Ç º »Þº 1 ;ÒÉº

x = 0, pT (x) =5

2

(

1 + cos(2πx

L

))

.

!L¥£9J F¨©¦´£9³+ ¤©£V³³Þ£¡6¡ Ä ç K º Ç º9º»ÞÉ » = ¾ã

s ∈ [0.25, 0.35] »ÞÉºÉ@ ã¯¾ Ç » »Þ½ ß > ã¯º Ç¯ÇÞ· ãÞº B ã¯¾<; ÇÂ·7>1¸ º H ½ /B»Þ¾ P ½ ¸ ¸ÈÁ> ¾ Ç º :> ã¯¾ á¸ º Ê«¦ Éº=ãÞºÂ½ Çë ¾º" > º=ãÞ½ Ê º » ¸3 » »ÞÉ » ÇE· ß6É > Éº ¾ Ê º ; HÇ ¾ áÇ º9ã¯àº ºD" > º9ã¯½ Ê º » P¸ ¸ @

0.2 0.3 0.4 0.5S

6.8

6.9

7

7.1

7.2

P(S)

M A D C D<HùõôLòýñEKVñGJ>ûHõ C òòþõ C ú þHF3>¯ñGJôLFâú ô=õx = 0.

*2 *d-4*d)4 *: Q ÿ a øJø C F#öÿ' C DHG C K9Fùÿ A õEK9F3ü C F3òýñ C GJF#ÿ "!#$&%'#(=ÿ

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(

1 + ζ(α, t))

õ#Î Ü9n ÎyÎ=z Ô#§»Ü Ð ü¨ü o Ü n ÎMo½öÞÎ=ÎyÐ ¨3Ü Î9ö9o § ó9Î>o ü ö § õ Ô õõ Ó ÎÐ ¨ ô üÓ§ öó üü ö © Ð ¨§»Ü Î Ò(r, α)ó ü¨¨ Î=ó Ü Î ©7r Ð Ü n Ð ÜúÒ Ï ü»Ý Ð ¨× ó9Î ¨3Ü Î=ö Crn Î=öÞÎ

ζ(α, t) Ð ÒíÜ9n Î © Ð ÏÎ ¨Ò Ð ü¨Ó Î ÒúÒ Ð ¨3Ü Î=öpo § óUÎô#Î=ö ÜúÔ öÞõ §»Ü Ð ü¨ü o Ü n Îyó9Ð ö6ó ÔÓ§ öõ Ô õõ Ó Î

r = RÐ ¨Ü n Î s Î Ó Î ßÁCn§r ó9Î Ó ÓW §PÍ Ð ¨×iÜ9n Î

Í Ð ¨ Î=Ï §PÜ Ðó õ üHÔ¨©§ ö u ó ü¨© Ð Ü Ð üH¨ o½ö ü Ï Ü n Î § öúó u Îz Ô§PÜ Ð ü¨§ ó9ó üÔ¨3Ü Ð ¨× o ü ö Ü n ÎÏ §×¨ Î Ü ÐóEo ü ö6óUÎ Ò v ÷4w§P¨#©Ó Ð ¨ Î § öÞÐ989Ð ¨× Ð Ü)r Ð Ü n öÞÎ Ò ôÎ=ó Ü+ÜÞü

ζ

r Î ü õ Üú§ Ð ¨Ü n Îo üÓ Ó ü4r Ð ¨×ôö ü õ Ó Î9ÏhÐ ¨ Ü9n Î © Ð ÏÎ ¨Ò Ð ü¨Ó Î ÒúÒ o ü öúÏ à

∂ζ

∂t

= −Bg sin α

∂ζ

∂α

− 2Bg cosα ζ +∂p

∂r

∣

∣

∣

r=1

,

²Ú Õ

p|r=1 = −

∂2ζ

∂α2− (2Bg cosα + a)ζ +

Bm

2h2

∫ π

−π

[

ζ(τ, t) − ζ(α, t)

| sin (α − τ)/2|

−

ζ(τ, t)√

sin2 (α − τ)/2 + h2/4

]

dτ,

²Ú Ú

É#¾iº ¥´ µ®#° µ9Ì½¸Uº¿6µ£µ9Ì°W´Ì 6º»°²±À¶»®¯ ÚPý

∆p = 0 if r > 1, p = c ln r + O(1) if r → +∞ .

WÚ ÷ s Î9öúÎ

c = c(t)× Î ¨ Î=ö §Ó Ó"u'© Î=ô#Î ¨©Ò£üH¨

t

§P¨© Ð Ü Ð Ò¥© Î Ü Î=öÞÏÐ ¨ Î © õ u'Ü n ÎÖÎz Ô§Ó Ð Üu∫ π

−π

ζ(τ, t) dτ = 0WÚ Û

§P¨© Îz Ô§PÜ Ð ü¨ÒWÚ Õ §¨© WÚ ÷ »ª o Ü Î=ö Ò Ð Ïô Ó Î¦ó ü Ïô ÔÜ6§»Ü Ð üH¨Ò Ð Ü ó §P¨ õÎ Òpnü4r£¨·Ü9n#§»Ü

c =1

2π

∫ π

−π

cos τ ζ(τ, t) dτ.

¨ Ð Ü Ð §Ó1Ý»§PÓ Ô Î ÒpnüÔÓ© õÎ Ò ô#ÎLóUÐ | Î © o ü öζ

o Ô¨ ó Ü Ð üH¨.§PÜt = 0

à

ζ(α, 0) = ζ0(α) .

²Ú Ù s Î9öúÎ Ü n Î o üÓ Ó ü4r Ð ¨× © Ð ÏÎ ¨#Ò Ð ü¨Ó Î ÒúÒ ô § ö § ÏÎ Ü Î9ö Ò>r Î9öúÎÈÐ ¨3Ü ö ü©Ô óUÎ ©à

r = rR

Ð ÒÜ9n Îö §© Ð ÔÒ=

h = hR

Ü9n Î Ü n Ð ó Í¨ Î ÒÞÒ'ü o Ü9n Î s Î Ó Î ØCn§r ó9Î Ó ÓW Bm

Ü n ÎÏ §P×¨ Î Ü Ðó¬íüH¨©Ë¨Ô Ïyõ#Î=ö v Ú ÷4w ù

Bg

Ü n Î × ö §Ý Ð Üú§PÜ Ð ü¨§Ó%¬ü¨©Ë¨Ô Ïyõ#Î=ö ùp = p(reiα

, t)Ü n Î

Î Î=ó Ü Ð Ý ÎyôöÞÎ ÒÞÒÞÔ öúÎyÐ ¨Ü n ÎÏ §×¨ Î Ü Ðó Ó Ðz Ô Ð © üÔÜ6Ò Ð © Î Ü9n Îõ Ô õõ Ó Î 1_n Îió üH¨ÒEÜú§P¨3ÜaÐ Ò¥© Î Ü Î=öÞÏÐ ¨ Î © §Ò

a = 1 −

2Bm

h2

2 +

∫ π/2

0

cos 2τ dτ

√

sin2τ + h

2/4

,

²Ú

ü ö£õ u Î Ó Ó Ð ô Ü Ð ó:o Ô¨ ó Ü Ð üH¨Ò

a = 1−

2Bm

h2

(

2 −

2E

k

+(2

k

− k

)

K

)

,

rn Î9öúÎ

K =

∫ π/2

0

dτ

√

1 − k2 sin2

τ

, E =

∫ π/2

0

√

1 − k2 sin2

τ dτ

§ öúÎ o ÔÓ Ó Î Ó Ó Ð ô Ü Ð ówÐ ¨3Ü Î × ö §PÓÒËü o Ü n Î | ö ÒÁÜ§P¨©ZÒ Î=ó ü¨© Í Ð ¨#© öÞÎ Ò ôÎ=ó Ü Ð Ý Î Ó"u k =

1√

1 + h2/4

.

Ð ¨ óUÎ£ó ü Î'ó9Ð Î ¨HÜ6Ò ü o5ôö ü õ Ó Î9Ï WÚ Õ Ø ²Ú Ù ©ü¨üPÜ+© Î=ô#Î ¨© Î;xô Ó Ð ó9Ð ÜúÓyuBü¨yÜ Ð ÏÇÎ r ÎöúÎ9ôöúÎ Ò Î ¨3Ü¦Ü9n Î ÒÞüÓ ÔÜ Ð ü¨ Ð ¨«Ü n Î(o ü öúÏ

ζ = eλt

u(α), p|r=1 = eλt(Pu)(α),

WÚ2~ rn Î9öúÎ

PÐ Ò"§ÇÓ Ð ¨ Î § ö ü ôÎ9ö §»ÜÞü ö _n Ð Ò¥Ó Î §H©Ò¦ÔÒ¥ÜÞüÜ9n Î(o üÓ Ó ü4r Ð ¨×'Ò ô#Î=ó Ü ö §PÓ ôö ü õ Ó Î=Ï

Ð ¨ Ü9n Î Ò Î ÜΩ

ü o Ür Ð ó9Îó ü¨3Ü Ð ¨ÔüÔ#ÒEÓ"uÈ© Ð Î=öÞÎ ¨3Ü Ð § õ Ó Î 2πôÎ9öúÐ ü© Ðó:o Ô¨ ó Ü Ð ü¨Ò=à

−λu = Bg sin α

du

dα

+ 2Bg cosα u −

1

4π

(v.p.)

∫ π

−π

(Pu)(τ) − (Pu)(α)

sin2 (τ − α)/2dτ

−

Bm

2π

∫ π

−π

cos τ u(τ) dτ ,

WÚ ý

Ú ø Ì´J±¯ 1Â±ÀµU°²´Ã=±Àµ9°²±¯Äµ6µ9Ì6¯

(Pu)(α) = −

d2u

dα2− (a + 2Bg cosα) u −

Bm

2h2

π∫

−π

u(τ) dτ

√

sin2 (τ − α)/2 + h2/4

+Bm

2h2

∫ π

−π

u(τ) − u(α)

| sin (τ − α)/2|dτ,

²Ú

r Ð Ü n«Ü n Î·ó ü Ïô Ó Î=ÏÎ ¨3Üú§ ö u ó ü¨© Ð Ü Ð ü¨∫ π

−π

u(α)dα = 0.

WÚ Õ=ø

ÌÞ¶L¶Á¸ ¢ z Ô§PÜ Ð ü¨ WÚ Mo üÓ Ó ü4r¥Ò o½ö ü Ï WÚ Ú §P¨©Ü n ÎöúÎ9ôöúÎ Ò Î ¨3Ü6§»Ü Ð ü¨#ÒWÚ~ %ªõ üHÔ¨© Î © n#§ öúÏ üH¨ Ðóo Ô¨ ó Ü Ð ü¨

p(reiα) o ü ör > 1 ó §P¨ õ#ÎËöúÎ9ôöúÎ Ò Î ¨3Ü Î © õ u Ü n Î

ó nCr§ ö 8Ð ¨3Ü Î × ö §PÓ² "Ð Î9öúÎ ¨3Ü Ð §»Ü Ð ¨×p

r Î × Î Ü

∂

∂r

p(reiα) = −

1

π

∫ π

−π

(1 + r2) cos (α − τ) − 2r

(

1 − 2r cos (α − τ) + r2)2

p(eiτ ) dτ,

rn Ð ó n ó §P¨ õ#Î Ü ö §¨Ò o ü öÞÏÎ © o ü ör > 1

§HÒ

∂

∂r

p(reiα) = −

1

π

(v.p.)

∫ π

−π

(1 + r2) cos (α − τ) − 2r

(

1− 2r cos (α − τ) + r2)2

[p(eiτ ) − p(eiα)] dτ,

WÚ ÕHÕ Ò Ð ¨ óUÎ

∫ π

−π

(1 + r2) cos (τ − α) − 2r

(1 − 2r cos (τ − α) + r2)2

dτ = 0 .

_n Î Ó §HÒÁÜ Îz Ô§Ó Ð Üu ó §¨ õÎ ü õ Üú§ Ð ¨ Î © ÔÒ Ð ¨×«Ü9n Î ÒÞÔ õ ÒEÜ Ð ÜÞÔÜ Ð üH¨ξ = exp

(

i(α − τ))

§P¨© öÞÎ Ò Ð ©Ô Î Ò~Ü9n Î ü ö u o ü öó ü Ïô Ó Î;xBÐ ¨3Ü Î × ö §Ó Ò=c§Í Ð ¨×Ü9n Î Ó Ð ÏÐ Ür → 1+0 Ð ¨ WÚ ÕÕ

× Ð Ý Î Ò

∂p

∂r

∣

∣

∣

r=1

=1

4π

(v.p.)

∫ π

−π

(Pu)(τ) − (Pu)(α)

sin2 (τ − α)/2dτ.

ü ö Ô¨ õ üÔ¨© Î ©>n#§ öúÏ üH¨ Ðóo Ô¨ ó Ü Ð ü¨Ò Òú§»Ü Ð Ò o u Ð ¨×!WÚ ÷ r Î%Ï Ô#ÒÁÜ§©©B§ ó üH¨ÒEÜú§P¨3Ü

c

Üúü ∂p

∂r

∣

∣

r=1

Ð ¨WÚ Õ P_n Ð Ò ó üH¨ÒÁÜ6§P¨3Ü ó §P¨ õ#Î © Î Ü Î=öÞÏÐ ¨ Î © õ u Ð ¨HÜ Î × ö §PÜ Ð ¨× ²Ú Õ §P¨©ÔÒ Ð ¨×WÚ Û _n Ð Ò¦× Ð Ý Î ÒÔÒ²Ú ý

_b j ')& g lpk*" +_h +pjIe<lph)d h10 2 gh14+/&(' _b _bpa

¨lÜ9n Ð ÒyÒ Î=ó Ü Ð ü¨ r Î'ôöúÎ Ò Î ¨3ÜB§P¨ Î 'óUÐ Î ¨3Ü ÏÎ Ü nü© o ü ö ¨3Ô ÏÇÎ=öÞÐó §Ó%ÒÞüÓ ÔÜ Ð ü¨Ëü o Ü n ÎÒ ôÎ=ó Ü ö §PÓ ôö ü õ Ó Î9Ï ²Ú ý Ø WÚ Õ=ø 5 Ü Ð Ò õ §Ò Î ©üH¨¨üH¨Òú§»ÜÞÔ ö §PÜ Î © § ôôö ü xÐ Ï §PÜ Ð ü¨Òü o Î9Ð × Î ¨ o Ô¨ ó Ü Ð üH¨Ò_r Ð Ü9n Ü9n Î fn Î9õ uÒpn Î Ý ô üÓ"u3¨ü ÏÐ §Ó Ò%ØÈ§P¨#§PÓ ü×üHÔÒ%ÜÞüKnü4r Ð ÜIr¦§Ò©ü¨ ÎÐ ¨ v ýw o ü ö Ü9n Î §»Ü n Ð Î Ô o Ô¨ ó Ü Ð üH¨Ò=~_n ÎiÏÎ Ü nü© Ð ÒÖ© Ð Î=öÞÎ ¨3Ü o ü öÖÎ Ý Î ¨l§P¨©ü©© Î=Ð × Î ¨ o Ô¨ ó Ü Ð üH¨Ò

u+(α)

u−(α)

rn Î9öúÎ

u+(α) =

u(α) + u(−α)

2, u

−(α) =u(α) − u(−α)

2.

É#¾iº ¥´ µ®#° µ9Ì½¸Uº¿6µ£µ9Ì°W´Ì 6º»°²±À¶»®¯ Ú Õ Î9ô § ö §PÜ Ð üÿü o#Î Ý Î ÿ§P¨#©Bü©© ô § ö ÜúÒ ü o WÚ ý Ø ²Ú Õ=ø × Ð Ý Î ÒcÜríüÒ ô#ÎLó Ü ö §PÓ ôö ü õ Ó Î=Ï Ò=à

−λ±

u± = Bg sin α

du±

dα

+ 2Bg cosα u±

− (v.p.)

π∫

−π

(Pu±)(τ) − (Pu

±)(α)

4π

(

sin2 (τ − α)/2) dτ

−

Bm

2π

∫ π

−π

cos τ u±(τ) dτ,

À÷ Õ

(Pu±)(α) = −

d2u±

dα2

− (a + 2Bg cosα)u±

−

Bm

2h2

π∫

−π

u±(τ) dτ

√

sin2 (τ − α)/2 + h2/4

+Bm

2h2

∫ π

−π

u±(τ) − u

±(α)

| sin (τ − α)/2|dτ,

W÷ Ú r Ð Ü n«Ü n Î·ó ü Ïô Ó Î=ÏÎ ¨3Üú§ ö u ó ü¨© Ð Ü Ð ü¨

∫ π

−π

u±(α) dα = 0.

À÷ ÷ ¨ Ü n ÎÈó §Ò Î ü oÎ Ý Î ¨ Î=Ð × Î ¨ o Ô¨ ó Ü Ð üH¨Òr Î Ô#Ò Î fn Î9õ uÒpn Î Ý ô üÓ"u3¨ü ÏÐ §Ó Òü o Ü n Î

| ö ÒEÜ Í Ð ¨© õ ÔÜ Ð ÿÜ9n Î£ó §Ò Î ü o ü©© Î=Ð × Î ¨ o Ô¨ ó Ü Ð üH¨Ò9 fn Î=õ uÒ9n Î Ý ô üHÓyu¨ü ÏÇÐ §PÓÒ ü o Ü n ÎÒ ÎLó ü¨#©Í Ð ¨©l§ öúÎ Ô#Ò Î ©1ª"Ò·§ öúÎ ÒÞÔÓ Ür Î ü õ Üú§ Ð ¨Üríü Î=Ð × Î ¨Ý»§PÓ Ô Î §¨© Î9Ð × Î ¨3Ý Î=ó ÜÞü öôö ü õ Ó Î9Ï Ò o ü ö

N ×NÏ §PÜ öúÐ ó9Î Ò

A±

N

H_n Î § ôôö ü xÐ Ï §PÜ Î ÒEüHÓ ÔÜ Ð ü¨Ò ü o ²Ú ý Ø WÚ ÕLø ó §P¨ õÎÇöúÎ9ôöÞÎ Ò Î ¨HÜ Î © õ u ÏÇÎ §P¨Òü o Ü n Î ÒEüHÓ ÔÜ Ð ü¨Òü o Ò ôÎ=ó Ü ö §PÓ ôö ü õ Ó Î=Ï Ò o ü ö Ü9n Î Ò ÎÏ §PÜ öúÐ ó9Î Ò=¨× Î ¨ Î9ö §PÓ² 3Ü9n Î § ó9ó Ô ö § ó uiü o § ôôö ü xÐ Ï §»Ü Ð üH¨Ò Ð ¨ ó9öÞÎ §Ò Î Ò_rn Î ¨

NÐ Ò Ð ¨ß

óUöúÎ §Ò Î ©1 N Äz¿6¶»¾ ´°º»°²±À¶»®¶Á¸µUÃ»µU®µ9±Hµ9®=¸6´®5¿U°²±À¶»®¯

u+(α)

µ6ºÅ»¯¥° ¶B°³¼µ ¸U¶ ¶±³®'¯µ6¿U°²ÌÞº ÌÞ¶ µ9¾

(A+

N + λ+EN)~w = 0, ~w = (w1, w2, . . . , wN )T

,

W÷ Û

A+

N = Bg

[

diag(x2k − 1)∆

(1)

N + 2diag xk

]

−G+

NPN ,

G+

N =1

2N

[

2H+

N − diag(x2k − 1)(∆

(1)

N )2 − diag xk∆(1)

N

]

,

PN = diag(x2k − 1)(∆

(1)

N )2 + diag xk∆(1)

N

− diag(a + 2Bg xk) +Bm

h2

−4diag xk∆(1)

N

+π

N

[M+

N − diag s+

k ∆(1)

N ]

.

s Î9öúÎdiag µk

Ð Ò¥§© Ð §P×Hü¨§Ó Ï §PÜ öúÐ#x r Ð Ü9n Î Ó Î=ÏÇÎ ¨3ÜúÒµk, k = 1, 2, . . . , N

xk = − cos(2k − 1)π

2N

, k = 1, 2, . . . , N,

s+

k =

N∑

j=1,j 6=k

(xj − xk)Φ(xk , xj , 0),

Ú HÚ Ì´J±¯ 1Â±ÀµU°²´Ã=±Àµ9°²±¯Äµ6µ9Ì6¯

Φ(x, ξ, h) =

[

1 − xξ + h2/2 +

√

(ξ − x)2 + (1 − xξ)h2 + h4/4

(ξ − x)2 + (1 − xξ)h2 + h4/4

]1/2

,

W÷ Ù

∆(1)

NÐ Ò£Ü9n Î fn Î=õ uÒ9n Î ÝÈ© Ð Î=öÞÎ ¨HÜ Ð §PÜ Ð ü¨ Ï §»Ü öúÐyx r Ð Ü n Î Ó Î9ÏÎ ¨3ÜúÒ

δ(1)

kj =T

′

N (xk)

(xk − xj)T ′

N(xj), if j 6= k, δ

(1)

kk =T

′′

N (xk)

2T′

N(xk),

H+

N

§P¨©M

+

N

§ öÞÎÏ §»Ü öÞÐóUÎ Òfr Ð Ü9n Î Ó Î=ÏÇÎ ¨3ÜúÒ

h+

kj =1− xjxk

(xj − xk)2, if j 6= k, h

+

kk =x

2k

4(1 − x2k)

−

N2− 1

3,

m+

kj = Φ(xk , xj , 0) − Φ(xk , xj , h), if j 6= k ,

m+

kk = −Φ(xk , xk, h) −

N∑

j=1,j 6=k

Φ(xk , xj , 0).

Î Üλ

+m

§P¨©~w(m) = (w1(m), . . . , wN (m))T

, m = 1, 2, . . . , N,õ#Î Ü9n Î·Î=Ð × Î ¨Ý»§PÓ ß

Ô Î Òí§P¨© ó ü öÞöúÎ Ò ô ü¨© Ð ¨× Î9Ð × Î ¨Ý Î=ó ÜÞü ö Ò ü o W÷ Û H_n Î ¨Ü n Î § ôôö ü xÐ Ï §»Ü Î¦Î=Ð × Î ¨ o Ô¨ ó ßÜ Ð ü¨

u+m(α) ó ü öÞöúÎ Ò ô üH¨© Ð ¨×·Üúü

λ+m

ó §¨ õÎÖöÞÎ=ôöúÎ Ò Î ¨3Ü Î ©È§Ò o üHÓ Ó ü4r¥Ò

u+m(arccosx) ≈ TN(x)

N∑

k=1

wk(m)

(x − xk)T ′

N (xk), x = cosα,

À÷

ü ö

u+m(α) ≈ cosNα

N∑

k=1

(−1)N+kwk(m) sin[(2k − 1)π/(2N)]

cosα + cos[(2k − 1)π/(2N)].

W÷~

ÌÞ¶L¶Á¸ 7 ÎyÐ ¨3Ü ö ü©Ô óUÎ ¨ Î rÝ»§ öÞÐ § õ Ó Î Òx = cosα, ξ = cos τ

Ð ¨ À÷ Õ Ø À÷ ÷ o ü öλ

+; u+(α)

c§P¨©ü õ Ò Î9ö Ý Î Ü9n§PÜ ö § ôÐ ©Ó"u ó üH¨Ý Î9ö × Î ¨3ÜÒ Î9öúÐ Î Ò o ü ö Ü9n ÎÎ9Ð × Î ¨ o Ô¨ ó Ü Ð ü¨u

+(α)§ ó9ó ü ö © Ð ¨×iÜÞü

cos kα, k = 0, 1, . . .

Ü ö §P¨#Ò o ü öÞÏ Ò Ð ¨3ÜÞü.Ò Î=öÞÐ Î Ò!r Ð Ü n öúÎ Ò ô#ÎLó ÜÜÞü

Tk(x) Î ¨üPÜ Ð ¨×

u+(arccosx) = u(x)

§P¨©§ o Ü Î9ö ÒÞü ÏÎyÎ Ó Î=ÏÎ ¨3Üú§ ö u Ï §P¨ Ð ßô ÔÓ§»Ü Ð ü¨#Ò9 <r Î ü õ Ü6§ Ð ¨

−λ+u(x) = Bg[(x2

− 1)u′(x) + 2xu(x)]

− (v.p.)1

π

∫ 1

−1

[(Pu)(ξ) − (Pu)(x)](1 − xξ)dξ

√

1 − ξ2 (ξ − x)2

.

W÷ ý

_n Î Ó Ð ¨ Î § ö ü ô#Î=ö §PÜÞü öP

Ð ÒB© Î | ¨ Î © õ uÜ n ÎiÎz Ô§Ó Ð Üu (Pu)(arccosx) = (Pu)(x)

_n Î9öúÎ;o ü öÞÎ

(Pu)(x) = (x2− 1)u′′(x) + xu

′(x) − (a + 2Bg x)u(x)

+Bm

h2

∫ 1

−1

[u(ξ) − u(x)]Φ(x, ξ, 0) − u(ξ)Φ(x, ξ, h)√

1 − ξ2

dξ,

W÷

É#¾iº ¥´ µ®#° µ9Ì½¸Uº¿6µ£µ9Ì°W´Ì 6º»°²±À¶»®¯ Ú ÷rn Î9öúÎ

Φ(x, ξ, h) Ð Ò£© Î | ¨ Î © õ uÀ÷ Ù # Ð ¨ óUÎ

(v.p.)

∫ 1

−1

dξ

(ξ − x)√

1 − ξ2

= 0 ,

r Î n§Ý Î

(v.p.)

1∫

−1

[(Pu)(ξ) − (Pu)(x)(1 − xξ)√

1 − ξ2 (ξ − x)2

dξ = −x

1∫

−1

(Pu)(ξ) − (Pu)(x)√

1 − ξ2 (ξ − x)2

dξ

+ (1 − x2)

∫ 1

−1

(Pu)(ξ) − (Pu)(x) − (ξ − x)(Pu)′(x)√

1− ξ2 (ξ − x)2

dξ.

¡¨ÈÜ n Î üPÜ n Î9ö n§¨©∫ 1

1

u(ξ) − u(x)√

1 − ξ2

Φ(x, ξ, 0) dξ = −4xu′(x)

+

∫ 1

−1

u(ξ) − u(x) − (ξ − x)u′(x)√

1 − ξ2

Φ(x, ξ, 0) dξ .

¢+Ý»§PÓ Ô§»Ü Ð ¨×À÷ ý ØÀ÷ §PÜxk

Crn Ðó n«§ öÞÎ589Î=ö ü Î Òíü o Ü9n Î fn Î=õ uÒ9n Î Ý ô üÓ"u¨ü ÏÐ §PÓTN (x)

Î Ý»§PÓ Ô§»Ü Ð ¨×B§PÓ Ó#© Î=öÞÐ Ý»§»Ü Ð Ý Î Ò%§ ó9ó ü ö © Ð ¨×Üúü·Ü n Î © Ð Î9öúÎ ¨3Ü Ð §»Ü Ð üH¨ Ï §PÜ öúÐ#x∆

(1)

N

§P¨© § ôôö ü xÐ Ï §PÜ Ð ¨× Ð ¨3Ü Î × ö §PÓÒ õ u'Ü9n Î(o üHÓ Ó ü4r Ð ¨× z Ô§H© ö §»ÜÞÔ öÞÎ`o ü öúÏ ÔÓ§

∫ 1

−1

f(ξ)√

1 − ξ2

dξ ≈

π

N

N∑

j=1

f(− cos[(2j − 1)π/(2N)]),À÷ ÕLø

r Î ü õ Üú§ Ð ¨ÇÜ9n Î£Î9Ð × Î ¨Ý»§PÓ Ô Î §¨© Î9Ð × Î ¨Ý Î=ó Üúü ö ôö ü õ Ó Î9Ï o ü öÏ §»Ü öÞÐyxA

+

N

¢%§ ó n Î9Ð × Î ¨ßÝ§Ó Ô Î

λ+m

ó ü öúöÞÎ Ò ô ü¨#©Ò+ÜúüÜ9n ÎÖÎ=Ð × Î ¨Ý ÎLó Üúü ö~w(m)

mrn Ð ó n n§Ý Î Ü n Î·ó ü ÏÇô ü¨ Î ¨3ÜúÒ

wk(m) ≈ u+m(arccosxk), k = 1, . . . , N .

_n Î9öúÎ;o ü öÞÎ o ü öúÏ ÔÓ§Ò W÷ §¨© À÷~ § öÞÎ ü õ Üú§ Ð ¨ Î ©l§ÒÜ n Î ~§× ö §P¨× ÎyÐ ¨3Ü Î=öÞô üÓ§»ßÜ Ð ü¨Ò=

N Äh¿6¶»¾ ´°º»°²±À¶»®l¶Á¸B¶LÅÅ«µU±3µU®L¸6´®¿9°W±À¶P®¯u−(α)

µ6ºHÅ¯·°¶i°½¼µí¸U¶ ¶±³®'¯µ6¿U°²ÌÞº ÌÞ¶ µ9¾¸U¶»Ì·¾iº»°²Ì±À¿6µ¯

(A−

N + λ−

EN)~v = 0, ~v = (v1, v2, . . . , vN )T,

À÷ ÕHÕ

A−

N = Bg

(

diag(z2k − 1)∆

(2)

N + 3diag zk

)

−G−

NQN ,

G−

N =1

N + 1

[

H−

N −

1

2diag(z2

k − 1)(∆(2)

N )2 − (N −

7

2)diag zk∆

(2)

N

]

,

QN = diag(z2k − 1)(∆

(2)

N )2 + [3 − 4Bm/(3h2)]diag zk∆

(2)

N

− diag(

2Bg zk + a − 1 −

4Bm

h2

)

+π

N + 1

[

−diag s−

k ∆(2)

N + M−

N

]

.

Ú PÛ Ì´J±¯ 1Â±ÀµU°²´Ã=±Àµ9°²±¯Äµ6µ9Ì6¯s Î9öúÎ

aÐ Ò ó §Ó ó ÔÓ §PÜ Î © o½ö ü Ï ²Ú

zk = − cos(

kπ

N + 1

)

, k = 1, 2, . . . , N

§ öúÎ 89Î=ö ü Î ÒBü o Ü n Î fn Î9õ uÒpn Î Ý ô üHÓyu¨ü ÏÇÐ §PÓíü o Ü n Î Ò Î=ó ü¨© Í Ð ¨©UN(x)

∆

(2)

NÐ Ò

fn Î9õ uÒpn Î ÝÈ© Ð Î9öúÎ ¨3Ü Ð §»Ü Ð üH¨ Ï §»Ü öÞÐyx r Ð Ü9n Î Ó Î9ÏÎ ¨HÜ6Ò

δ(2)

kj =U

′

N(zk)

(zk − zj)U ′

N (zj), if j 6= k, δ

(2)

kk =U

′′

N (zk)

2U′

N(zk),

H−

N

§P¨©M

−

N

§ öÞÎÏ §»Ü öÞÐóUÎ Òfr Ð Ü9n Î Ó Î=ÏÇÎ ¨3ÜúÒ

h−

kj =1 − z

2j

(zj − zk)2, if j 6= k, h

−

kk = −

zk

4(1 − z2k)

−

N2 + 2N + 3

3,

m−

kj = (1 − z2j )[Ψ(zk, zj , 0) − Ψ(zk, zj , h)], if j 6= k;

m−

kk = −(1 − z2k)Ψ(zk, zk, h) −

N∑

j=1,j 6=k

(1 − z2j )Ψ(zk, zj , h) .

Ô¨ ó Ü Ð ü¨Ψ

Ð Ò£© Î | ¨ Î © §HÒ

Ψ(x, ξ, h) = g(x, ξ, h)[

1 − xξ +h

2

2+ g(x, ξ, h)

]1/2

,

rn Î9öúÎg(x, ξ, h) = [(ξ − x)2 + (1 − xξ)h2 + h

4/4]1/2

,

s−

k =N

∑

j=1,j 6=k

(1 − z2j )(zj − zk)Ψ(zk, zj , 0).

s Î9öúÎλ−

m

§P¨©~v(m) = (v1(m), . . . , vN (m))T

, m = 1, 2, . . . , N,

§ öúÎ Ü9n ÎÖÎ=Ð × Î ¨Ý»§PÓ ßÔ Î Ò§P¨© ó ü öúöÞÎ Ò ô ü¨© Ð ¨× Î=Ð × Î ¨Ý ÎLó Üúü ö Òíü o+ôö ü õ Ó Î9Ï À÷ ÕÕ _n Î § ôôö ü xÐ Ï §»Ü Ð ü¨.ü oÎ9Ð × Î ¨ o Ô¨ ó Ü Ð üH¨

u−

m(α) ó §P¨ õÎÖöÞÎ=ôöÞÎ Ò Î ¨3Ü Î ©«§Ò=à

u−

m(arccosx) ≈√

1 − x2

UN (x)

N∑

k=1

vk(m)

(x − zk)U ′

N (zk), x = cosα,

À÷ ÕLÚ

rn Î9öúÎzk = − cos[kπ/(N + 1)]

ü ö

u−

m(α) ≈sin (N + 1)α

N + 1

N∑

k=1

(−1)N+k+1vk(m) sin[kπ/(N + 1)]

cosα + cos[kπ/(N + 1)].

À÷ ÕL÷

ÌÞ¶L¶Á¸ tª ôö üü o ü o Ü9n Ð Ò ó §Ò Î£Ð Òí§P¨§Ó üH×üHÔÒ r Ð Ü nÜ9n Î"ó §Ò Î ü oÎ Ý Î ¨ Î9Ð × Î ¨ o Ô¨ ó Ü Ð ü¨Ò=¡¨Óyu¨ü4r o ü ö

λ−

§P¨©u−(α) Ð ¨ W÷ Õ W÷ Ú r ÎÇÏ ÔÒEÜÔÒ Î Ü9n Î ÒÞÔ õ ÒEÜ Ð ÜÞÔÜ Ð üH¨Òü o

Ý§ öÞÐ § õ Ó Î Òx = cosα, ξ = cos τ, u

−(α)/ sin α = v(cos α).W÷ Õ9Û

É#¾iº ¥´ µ®#° µ9Ì½¸Uº¿6µ£µ9Ì°W´Ì 6º»°²±À¶»®¯ Ú HÙ¨ÒÁÜ Î §H©ü o Ü9n Î Ó Ð ¨ Î § ö ü ô#Î=ö §PÜÞü ö

Pr Î ü õ Ü6§ Ð ¨Ü9n Î Ó Ð ¨ Î § ö ü ô#Î=ö §PÜÞü ö

Q rn Ðó n Ð Òí© Î ß

| ¨ Î © õ u Î=z Ô§PÓ Ð Üu (Pu−)(α) =

√

1 − x2(Qv)(x)

§P¨#© ÒÞÔ õ ÒEÜ Ð ÜÞÔÜ Ð ü¨ÒW÷ Õ9Û #¡©©Î9Ð × Î ¨ o Ô¨ ó Ü Ð üH¨Ò ó §P¨ õ#Î"öÞÎ=ôöÞÎ Ò Î ¨3Ü Î © õ u ö § ôÐ ©Ó"u ó üH¨Ý Î9ö × Î ¨3Ü Ò Î9öúÐ Î Òr Ð Ü n öÞÎ Ò ôÎ=ó ÜÜÞüsin kα, k = 1, 2, . . .

Ð ¨ óUÎ

sin[(n + 1)x]

sin x

= Un(x), n = 0, 1, . . . ,

r Î ü õ Ü6§ Ð ¨W÷ ÕLÚ ØW÷ Õ=÷ +õ uiÔÒ Ð ¨×yÒÞÔ õ ÒEÜ Ð ÜÞÔÜ Ð üH¨Ò Ð ¨ À÷ Õ À÷ Ú §¨© Î Ý§Ó Ô#§»Ü Ð ¨×Ð Ü o ü ö 89Î9ö ü Î Òü o Ü9n Î fn Î9õ uÒpn Î Ý ô üÓ"u¨ü ÏÐ §PÓü o Ü9n Î Ò Î=ó ü¨©«Í Ð ¨#©

UN(x)

Ü Ð Ò© Ð 'ó ÔÓ Ü%Üúü Î ÒEÜ Ð Ï §»Ü Î Ü9n Î ü öúÎ Ü Ð ó §PÓ ÓyuyÜ9n ÎÎ9öúö ü ö ü o Ü n Î"ôöúÎ Ò Î ¨3Ü Î ©¨Ô ÏÎ9öúÐ ó §PÓÏÇÎ Ü9nü©Ò=_n Î9öúÎ;o ü öÞÎ 'r ÎíÑ Ô©× Î § õ üHÔÜíÜ n ÎÎ9öúö ü ö Ò õ u ÏÎ §P¨Òü o Ü9n Î`o üÓ Ó ü4r Ð ¨×BÜríüôö ü ó9Î ©Ô öúÎ Ò=à

Õ_n Îôö ü õ Ó Î9Ï Ð Ò¥ÒÞüÓ Ý Î © r Ð Ü9n© Ð Î=öÞÎ ¨3ÜN

Ú¨ Ü9n Îó §Ò Î

Bg = 0 +¨Ô ÏÎ=öÞÐó §Ó öÞÎ ÒEÔÓ ÜúÒÈ§ öúÎ.ó ü Ïô § öÞÎ © r Ð Ü9n Ü n ÎÎ;x § ó Ü

ÒEüHÓ ÔÜ Ð üH¨1à

λ+m = λ

−

m = −m ϕ(m), u+m(α) = C1 cosmα, u

−

m(α) = C2 sin mα ,

À÷ ÕÙ rn Î9öúÎ

ϕ(m) = m2− 1 +

4Bm

h2

[

1 +

∫ π/2

0

sin[(m + 1)τ ] sin[(m − 1)τ)] dτ

√

sin2τ + h

2/4

−

∫ π/2

0

sin2mτ

sin τ

dτ

]

,

À÷ Õ

C1

§P¨©C2

§ öúÎ § öúõÐ Ü ö § ö u ó ü¨#ÒÁÜ6§P¨3ÜúÒ=+ª_Ý»§PÓ Ð © Ð Üu ü o W÷ ÕLÙ §¨© À÷ Õ ó §¨ õÎÝ Î9öúÐ | Î © õ uÈÒEÔ õ ÒÁÜ Ð ÜúÔÜ Ð ü¨ Ð ¨3ÜÞüÀ÷ Õ ØW÷ Ú

Ib j ')& g lpk*" + &Cj +e

ü ö Ü9n Î+õöÞÎ Ý Ð Üu r ÎôöÞÎ Ò Î ¨HÜ1üH¨Ó"u¨Ô ÏÎ9öúÐó §PÓH§ ôôö ü xÐ Ï §PÜ Ð ü¨Òü o Ü n Î+Î9Ð × Î ¨Ý§Ó Ô Î Òü oÜ9n Î Ò ô#ÎLó Ü ö §PÓ ôö ü õ Ó Î=Ï WÚ ý ØWÚ Õ=ø û¥Ô ÏÇÎ=öÞÐó §Ó ÏÇÎ Ü9nü©Ò¦× Ð Ý Î ¨ Ð ¨«Ü9n ÎôöúÎ Ý Ð üÔÒÒ ÎLó Ü Ð ü¨l§PÓ Ó ü4rÔÒÜÞü ó ü Ïô ÔÜ Î Ü n Îió ü öÞöúÎ Ò ô üH¨© Ð ¨× Î=Ð × Î ¨ o Ô¨ ó Ü Ð üH¨ÒÖ§PÓÒEü.§P¨#©Ü n Î ¨ÜÞü ó §PÓ ó ÔÓ §PÜ Î Ü n Î ÒEüHÓ ÔÜ Ð ü¨ü o WÚ Õ Ø WÚ Ù ¥õ u Ï §P¨ Ð ô ÔÓ§»Ü Ð üH¨ÒÜu ôÐ ó §PÓ Óyu.ÔÒ Î © o ü öÜ9n Î üÔ öúÐ Î=ö¦ÏÇÎ Ü9nü©

oBm = Bg = 0

Ü9n Î ¨ Ð Ü o üÓ Ó ü4r¥Ò o½ö ü Ï À÷ Õ Ü9n§PÜ

λ+m = λ

−

m = −m(m2− 1) , m ≥ 1 .

_n Î¥ôöúÎ Ò Î ¨3Ü Î ©Ç¨3Ô ÏÇÎ=öÞÐó §Ó "Î × Î ¨ Î9ö §»Ü Î §»Ü öÞÐyx +ÏÎ Ü9nü©1 rn Ðó n Ð Ò õ §Ò Î ©Çü¨¨ü¨Òú§»ÜúÔ ö §»Ü Î ©È§ ôôö ü xÐ Ï §»Ü Ð üH¨Ò%ü o Î=Ð × Î ¨ o Ô¨ ó Ü Ð üH¨Ò_r Ð Ü nfn Î9õ uÒpn Î Ý ô üÓ"u¨ü ÏÐ §PÓÒ9 × Ð Ý Î Ò Îx § ó Ü öúÎ ÒÞÔÓ Ü6Ò Ð ¨ Ü n Ð Ò ó §Ò Î

ü3Ò Ð Ü Ð Ý Î Î9Ð × Î ¨3Ý»§Ó Ô Î Ò«§ öÞÎ Ü9n ÎÏ üHÒEÜ Ð ¨3Ü Î=öÞÎ ÒÁÜ Ð ¨× o ü ö § ôô Ó Ðó §PÜ Ð ü¨Ò Ò Ð ¨ ó9Î Ü n ÎÐ ¨ÒEÜú§ õÐ Ó Ð Üu Ð ¨RÜ9n ÎlÐ ¨3Ü Î9ö9o § ó9Î ©<u¨§ ÏÐ ó Ò«ü o Ü9n ÎöúÐ Ò Ð ¨× ×3§Ò õ Ô õõ Ó ÎËó §¨|© Î Ý Î Ó ü ô

Ú Ì´J±¯ 1Â±ÀµU°²´Ã=±Àµ9°²±¯Äµ6µ9Ì6¯ Q G2O$ C G *LC F8=K & C LLH 75 8=J ; H 75 Q ?=J "

Bg = 0, Bm = 5, h = 1,/+DHGJ>ÞD

K; C =$EK=GJF C " H½S 5 Q VJ K=F $ÁD C_FG ÖC $ÁD, /+G$ÁDN = 20

K=F N = 80.

N = 20 N = 80

m (3.16) λ+

m λ−

m λ+

m λ−

m

Q ?5 ?L?=?L?=? ?5 ?=?L?L?=? ?5 ?=?L?=?L? ?5 ?L?L?=?L? ?5 ?L?=?L?L?7 Q 5 : S :9@LV Q 5 : S :9@LV Q 5 : S :U@V Q 5 : S :U@V Q 5 : S :9@VS T²75 :=?=8L< S T²75 :9?L8L< S T²745 :9?L8=< S T 745 :=?L8=< S T²75 :=?=8L< S@T 7 Q 5 Q @LVLV=< T²7 Q 5 Q @VL<=?]T²7 Q 5 Q @»:=7=? T²7 Q 5 Q @V=VL<]T²7 Q 5 Q @LVLVL< T²V=?5 : Q <L8L: TWVL?5 :=7 QLQ 8]TWVL?5 :=7L7=: Q TWVL?5 : Q <=8L8]TWV=?5 : Q <L8L8V"T Q 7985 QLQ :9@=@T Q 7=85 Q 7L7 Q 8T Q 7=85 Q 7=VL:L:|T Q 7985 QLQ :9@VT Q 7985 QLQ :9@L8:¥T²7L79<5 :=VL: Q 8uT²7=7=<5 :98L? Q <T 7=7=<5 :9<L?=<L?T²7L79<5 :=VL:L7 S T²7L79<5 :=VL:L7=:8"T S : Q 5 < Q ?=VuT S : Q 5 <=8 Q ? S T S :L745 ?=?7U@ T S : Q 5 < QLQ :T S : Q 5 < Q 7 <"T V=?5 8LVL:9@ Q T VL?5 <=V:L7 S T VL?5 8L: Q ?LVT V=?5 8LVL:=V=@ÖT V=?5 8LVL:L:98

Q ?TW8L?L75 VLV S <L:TW8=?75 :=:=8 :T²8=?745 89@»798=@TW8L?L75 VLV9@ S <TW8L?L75 VLV9@V9@

ü¨ ó ü öúöúÎ Ò ô üH¨© Ð ¨× Î=Ð × Î ¨ o ü öÞÏ Ò=%¡¨ Ü n Î üPÜ n Î9ö n§¨© Î § ó n Î9Ð × Î ¨3Ý»§Ó Ô Î õÎ=ó ü ÏÎ Òô ü3Ò Ð Ü Ð Ý Î!o ü ö ÒÞÔ 'óUÐ Î ¨ÓyuÓ§ ö × Î Ý»§PÓ Ô Î ü o

Bm

_n Ð Ò ó üH¨ ó Ó ÔÒ Ð üH¨ o üÓ Ó ü4r¥Ò o½ö ü Ï W÷ Õ ¨Bc§ õ Ó Î Õfr ÎíôöÞÎ Ò Î ¨3Ü~Ü9n Î | ö ÒEÜcÜ Î ¨ Î=Ð × Î ¨Ý»§PÓ Ô Î Ò~ü o Ü n Îôö ü õ Ó Î9Ï WÚ ý Ø ²Ú Õ=ø

Bg = 0, Bm = 5, h = 1 rn Ðó nË§ öúÎÇó §PÓ ó ÔÓ§»Ü Î © õ u À÷ Õ §P¨© ÏÎ Ü nü©Ò

r Ð Ü nÈü ö © Î=ö Ò%ü o Ü n ÎÏ §PÜ öúÐ ó9Î ÒN = 20, N = 80

HªÒEr Î Ò Î=Î 3Ü9n ÎÏ § xÐ Ï §Ó öúÎ Ó§»Ü Ð Ý ÎÎ9öúö ü ö ü o Ü n Î Î9Ð × Î ¨Ý»§PÓ Ô Î Ò ó §PÓ ó ÔÓ§»Ü Î © õ u ÏÎ Ü nü© Ð ÒBü o Ü9n Î ü ö © Î=ö 10−4 o ü öN = 20

§P¨©10−6 o ü ö

N = 80

Q G2O$ C G *=C F8LK & C )?H 745 8J ; H 75 Q ?=J Bg = Bm = 5, h = 1,

/+DHGJ>ÞDiK; C=$EK=GJF C >FG ÖC $ÁD, /+G2$ÁD

N = 20, N = 40K9F

N = 80.

N = 20 N = 40 N = 80

m λ+

m λ−

m λ+

m λ−

m λ+

m λ−

m

Q Q 5 ?7=:=8L8 Q 5 ?7=:=8 S Q 5 ?L7L:=8=8 Q 5 ?7L:98: Q 5 ?7=:=8L8 Q 5 ?7=:=8=87 Q ?5 :=? QLQ @ Q ?5 :9? QLQ 7 Q ?5 :9? Q ?L< Q ?5 :=? Q ?L< Q ?5 :=? Q ?L< Q ?5 :=? Q ?=<S TÀ@'5 8 SPQ 8 S TÀ@'5 8 SPQ < Q TÀ@'5 8 SQ :9< TÀ@5 8 SQ 8L? TW@5 8 SQ :=< TÀ@'5 8 SPQ :=<@qT²7=75 QLQ6S @»:)T 7=75 QLQ6S 8=8]T²7L75 Q=QS 7 Q T²7L745 QLQS 7 S T²7=75 QLQ6SQ <qT 7=75 QLQ6SQ < TWV Q 5 Q @LV S :)T²V Q 5 Q @L8L? Q TWV Q 5 Q @ ?9@]TWV Q 5 Q @ Q @TWV Q 5 Q @=@<LVqT²V Q 5 Q @=@<=VV"T Q 7985 SLS=S V=<ÖT Q 7=85 S=S 8 S <T Q 7=85 S 7=< Q ?T Q 7=85 S 7=< S <T Q 7985 S 798L87T Q 7985 S 798L89@:¥T²7L79<5 8L<9@<9@·T²7=7=<5 <=? :9VT 7=7=<5 8=87 @ÖT²7L7=<5 8L8 S 7 Q T²7L79<5 8L8 Q :=<T²7L79<5 8L8 Q 8 S8"T S :=75 ?=@LVLV T S :L75 ?=VL8 Q <T S :L745 ? Q 8L79@ÖT S :L745 ? Q < <T S :=75 ? Q V T S :=75 ? Q VLV S<"T V=?5 <LVL:L7 Q T V Q 5 ?=V 8=<T VL?5 <=?L< S 7T VL?5 < QLQ : S T V=?5 <L? 8L<T V=?5 <L?=VL? S

Q ?"TW8L?L75 8L? Q 79<ÖTW8=?75 8=V V=<T²8=?745 V=< SPQ @ÖTW8L?745 VL<: Q TW8L?L75 VL8=V:=VTW8L?L75 VL8L:=? Q

¨ § õ Ó Î Ú r ÎôöúÎ Ò Î ¨3ÜÜ9n Î | ö ÒEÜÖÜ Î ¨ Î9Ð × Î ¨Ý»§PÓ Ô Î Òü o ü o Ü n Îiôö ü õ Ó Î9Ï WÚ ý ØWÚ Õ=ø

Bg = 5, Bm = 1, h = 1 rn Ð ó n § öÞÎ'ó §PÓ ó ÔÓ§»Ü Î © õ uÜ n Î 5 ÏÎ Ü9nü©

r Ð Ü n ü ö © Î9ö Ò¦ü o Ü9n ÎÖÏ §»Ü öÞÐóUÎ ÒN = 20, N = 40

§¨©N = 80

#íü Ïô § öúÐ ¨× öúÎ ÒÞÔÓ ÜúÒü õ Ü6§ Ð ¨ Î © o ü ö

λ+

§P¨©λ−

mr Î Ò Î9Î Ü9n#§»Ü£Ü9n Î § ó=ó Ô ö § ó u o ü öN = 20

§P¨©N = 80 Ð Ò

Ü9n Î Òú§ ÏÎ §Ò Ð ¨ Ü9n Î § õ Ó Î Õ o ü öBg = 0

É#¾iº ¥´ µ®#° µ9Ì½¸Uº¿6µ£µ9Ì°W´Ì 6º»°²±À¶»®¯ Ú ,~

5 10 15 20Bm

-100

-50

50

100

1

2

34

5

WQ íA D C >&8 C >$ÁD C 0 O$ 0 8 CiC G *LC F8=K & C @λ = λ(Bm)

"Bg = 3

5

¨ Ð ×Ô öúÎ ÕÜ9n Î © Î9ôÎ ¨© Î ¨ óUÎ ü o Ü9n Î | ö ÒEÜ | Ý ÎiÎ=Ð × Î ¨Ý»§PÓ Ô Î ÒüH¨ Ü n ÎiÏ §P×H¨ Î Ü Ð ó¬íüH¨©«¨3Ô ÏBõÎ9ö

BmÐ Ò¥Ò9nü4r£¨ n Î9öúÎ

Bg = 3

_b h)d!k +9j Clh)d

_n ÎÏ §»Ü9n Î9Ï §»Ü Ð ó §PÓ Ï ü© Î Ó o ü ö Ü9n Î ©mu3¨#§ ÏÐ ó Òiü o Ü9n Î ×H§Ò õ Ô õõ Ó ÎöÞÐ Ò Ð ¨× Ð ¨Ü n ÎÝ Î9ö Ü Ðó §Ó s Î Ó Î ßCn#§r óUÎ Ó Ó | Ó Ó Î © r Ð Ü9n Ï §P×H¨ Î Ü Ð ó Ó Ð"z Ô Ð © Ð Ò o ü öÞÏ ÔÓ§»Ü Î ©«§HÒ£§Ò ôÎ=ó Ü ö §PÓôö ü õ Ó Î9Ï o ü ö Ü9n Î Ó Ð ¨ Î § öiÐ ¨3Ü Î × ö üPß© Ð Î=öÞÎ ¨HÜ Ð §Ó¦ü ôÎ9ö §»ÜÞü ö r Ð Ü n ô#Î=öÞÐ ü© Ð ó.õ üÔ¨©§ ö uó ü¨#© Ð Ü Ð üH¨Ò=H_n Î ü õ Üú§ Ð ¨ Î ©iÒ ô#ÎLó Ü ö §PÓ ôö ü õ Ó Î9Ï Ð ÒíÒEüHÓ Ý Î © õ uBÜ n Î Î × Î ¨ Î9ö §»Ü Î §»Ü öÞÐ ßóUÎ Ò ÏÎ Ü nü©Ò9 Zrn Ðó n § öÞÎyó ü¨#ÒÁÜ ö Ô ó Ü Î © õ u.ÔÒ Ð ¨× © Ð Î9öúÎ ¨3Ü Ð §PÜ Ð ü¨ Ï §»Ü öÞÐóUÎ Ò o ü ö Ü n Î~§P× ö §¨× Î«ôö ü ÑÁÎ=ó ÜÞü ö r Ð Ü9nRfn Î=õ uÒ9n Î ÝË¨ü© Î Ò=+_n Î § ó=ó Ô ö § ó uËü o ¨3Ô ÏÇÎ=öÞÐó §Ó ó §PÓ ßó ÔÓ§»Ü Ð üH¨Ò ó §P¨ õÎ ó ü¨3Ü ö üÓ Ó Î © Î §HÒ Ð Ó"u õ ulÝ»§ ö u Ð ¨×Ü n Î ü ö © Î9ö

N

ü o © Ð Î=öÞÎ ¨3Ü Ð §»Ü Ð ü¨Ï §PÜ öúÐ ó9Î Ò= Î ÒÞÔÓ ÜúÒ£ü o ¨Ô ÏÎ=öÞÐó §Ó Î;xô#Î=öÞÐ ÏÎ ¨3ÜúÒ"§ öúÎôöÞÎ Ò Î ¨HÜ Î © §P¨©«Ü n Î u ôö ü»Ý Î Ü n ÎÎ'óUÐ Î ¨ ó u'ü o Ü9n Î © Î Ý Î Ó ü ôÎ © ÏÇÎ Ü9nü©

&(0& g%& d!k*&

Q F:5 C F'G =F 5 N$EKHG G2$ 8PG>&, 0 F *=C ÁGJF *'5 "!#$ "%'& ](( H 7=J ] Q6S ?7;Q6S ?L8 ] Q <L8LV5

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²² Å é `Á c Â Å ½ ¾Dµ¾¯¾À² Å Á ìbéÀ·ê Á é `½ ¶ Á ¶·²´ ` a£·³Ûé `Á ½ ¶3é Á ³ Å · Â ÁAú ² Â µ=³Û¸¡µ=¶ ì Á ³wbµJµ Â ¾ a£·³ úRÁ ¾Õé¯·'µ ú éô·J¶ ½ é"µ=¶#ì'é¯·~¶ Á ´ Â ÁAú éôé `Á ´J³¯µ¸ ½ é(gsµ ú é ½ ·J¶

µ³Gé Á ¾ ½ µ¶ ú ··³6ì ½ ¶µ¡é Á ¾/g¾é Á Å(x, y, z) Qh `½ ú`½ ¾ ú ·¶¶ ÁAú é Á ìéÀ·äé `Á a£³6µ Å Á

x = ±ay = ±b

na ≤ b

s ½ ¾ ½ ¶3éÀ³¯·ì² ú9Á ì 9`Á Å Á µ¶ é `½ ú ¿3¶ Á ¾¯¾ ·aé `Á c Â Åεa½ ¾ Å ² ú` ¾ Å µ Â Â Á ³é ` µ¶é `Áú` µ³¯µ ú é Á ³ ½ ¾Gé ½ úÂ Á ¶´=é `

a ½ ÁJ

ε << 19`Á ¶é `Á c Â Å

a£³ ÁAÁ ¾ g ÅÅ Á éÀ³ ½ ú µ Â ¾G²³/a¦µ ú9Á ¾ ` µ¸ Á e ·3¾ ½ é ½ ·¶#¾z = ±h/2 h `Á ³ Á

h(x, y, t) = O(ε)ì Á é Á ³ Å ½ ¶ Á ¾Dé `Á c Â Å ¾ ` µ e ÁJ ½ ¶ ú9Á é `Á c Â Å ½ ¾¾/g ÅÅ Á é¯³ ½ ú h ½ é ` ³ Á ¾ e Á:ú éDé¯·é `ÁÅ ½ ìì Â Á e Â µ¶ Á

z = 0 #é `Á é Á Åfe Á ³6µ¡é¯²³ Á c ÁAÂ ì ½ ¾µ Â ¾À·¾/g ÅÅ Á é¯³ ½ úθ∗(z) = θ

∗(−z)µ=¶ì é `Á ¾/g ÅÅ Á éÀ³*g ú ·J¶ì ½ é ½ ·¶.·¶z = 0 ³ Á µJì¾ o qr Æ

θ∗

z = 0, for z = 0 .nTº ¹s

go²¾ ½ ¶´ é `Á ¾ g ÅÅ Á éÀ³*g µJ¾À¾À² Åfe é ½ ·J¶1!h Á ¾G·J²´ ` éé `Á é Á Åfe Á ³¯µ=éÀ²³ Á a£²¶ ú é ½ ·¶θ∗(x, y, z, t) µJ¾ôµ=¶.µ¾/g Å)e éÀ·=é ½ úÁ e µ=¶¾ ½ ·J¶

θ∗(z) = θ0 +

∝

∑

k=1

z2k

θ2k, −h

2≤ z ≤

h

2.

nº ºs9`½ ¾4a£·³ Å ¾Àµ=é ½ ¾(c Á ¾ ½ ì Á ¶3é ½ ú µ Â Â g nº ¹sµ=¶ì é `Á ¾ g ÅÅ Á éÀ³*g µJ¾À¾À² Åfe é ½ ·¶

éé `Á µ=ê¾ Á ¶ úRÁ ·a `Á µ¡é¾G·J²³ ú9Á ¾é `ÁDÁ ¶ Á ³¯´g ú ·¶#¾ Á ³À¸¡µ¡é ½ ·J¶ Â µh a£·³"é `ÁÛÂ ½Bb ² ½ ìc Â Å ½ ¾´ ½ ¸ Á ¶ ½ ¶äé `Á ½ ¶3é Á ´³6µ Â a£·J³ Å Æ

d

dt

∫

ρcθ∗

dv = −

∫ ,∗ ·

%ds ,

nº ûsh `Á ³ Á é `Á½ ¶3é Á ´³6µ¡é ½ ·¶ ½ ¾ì·J¶ Á ·¡¸ Á ³+é `Á h ` · Â Á c Â Å ¸J· Â ² Å Á µ=¶#ì ¾À²³ a¦µ úRÁ J³ Á ¾ e Á:ú Äé ½ ¸ Á9Â g Á ³ Á ,

∗

½ ¾é `Á `Á µ¡é!j#² µ¶ì % é `Á ²¶ ½ é·²é(hµ³¯ìä¶·J³ Å µ Â ¸ ÁAú éÀ·J³ôéÀ·sé `Ác Â Å ¾G²³ a¦µ úRÁ9`Á6`Á µ¡é9j² ½ ¾ô¾À² eSe ·J¾ Á ì éÀ·s·Jê Á gsé `Á 2 ·²³ ½ Á ³ Â µh , ∗ = −κ∇θ

∗½ ¶¾ ½ ì Á é `Á c Â Å µ=¶ì éÀ· ê Á ì² Á éÀ· ³6µì ½ µ¡é ½ ·J¶ h ½ é ` µ Å ê ½ Á ¶3és·¶hê#·é ` c Â Å a£³ Á9Á¾G²³/a¦µ ú9Á ¾ Æ

,∗ ·

%= β(θ∗4 − θ

4a) at z = ±h/2,

nTº ë.sh `Á ³ Á

β

½ ¾é `Á ³¯µJì ½ µ=é ½ ·¶ ú · Á'ú9½ Á ¶Jéµ¶ìθa

½ ¾ôé `Á µ Å ê ½ Á ¶3éé Á Å)e Á ³6µ¡é¯²³ Áu ¶Jé Á ´J³¯µ=é ½ ¶´ Áb ²µ=é ½ ·¶ nº ûs+µ Â ·J¶´é `Á c Â Å é `½ ú ¿¶ Á ¾¯¾

z ∈ [−h/2, h/2] é6µ=¿ ½ ¶´½ ¶3éÀ·.µ úAú ·J²¶3é)nTº ºsµ¶ì nº ësQh Á ·Jêé¯µ ½ ¶é `Á~Â · ú µ Â a£·³ Å ·a+é `Á§Á ¶ Á ³¯´g.êµ Â µ¶ úRÁÁb ²µ¡é ½ ·J¶ nT¾ ÁAÁ o º¼3qr s #rÂ Á µì ½ ¶´~é Á ³ Å ·aÝ·J³¯ì Á ³

O(ε)½ ¾ô´ ½ ¸ Á ¶µ¾ a£· Â Â ·h¾ Æ

ρch

∂θ0

∂t

+ s · ∇sθ0

= κ∇s(h∇sθ0) + 2β(θ4a − θ

40),

nº åsh `Á ³ Á s = (u0, v0)

½ ¾%é `Á ¾G²³ a¦µ úRÁ c Â Å ¸ ÁAÂ · ú9½ é(g¸ Á:ú é¯·³Õµ¶ì ∇sé `Á ¾À²³/a¦µ ú9Á ´³6µ Ä

ì ½ Á ¶3é w Á ¶·=é Á é ` µ¡é (u0, v0)µ=³ Á é `ÁÕÂ Á µì ½ ¶´·³6ì Á ³ é Á ³ Å ¾ ½ ¶Dµ¾ ½ Å ½ Â µ=³µ¾/g Åfe é¯·=é ½ úÁ e µ=¶#¾ ½ ·¶.µ¾ nTº ºsê²é4a£·³é `Á Â ·J¶´ ½ éÀ²#ì ½ ¶#µ Â ¸ ÁAÂ · ú9½ é(g nT¾ ÁAÁ nTº ¹s ½ ¶po ûrts

! ! *3% 0;m* &

Á ³ Á h Á ¾ ` µ Â Â ³ ÁAú µ Â Â é `Á ìSg3¶#µ Å ½ ú ¾ g¾Gé Á Å h `½ ú` hµ¾~ì Á ¸ Á9Â · e Á ì ½ ¶ o ûër u éì Á ¾ ú ³ ½ ê Á ¾é `Á c Â Å é `½ ú ¿¶ Á ¾À¾ôµ¶ìä¾G²³/a¦µ ú9Á ¸ Á9Â · úR½ é(g Á ¸J· Â ²é ½ ·¶·añ·³6ì Á ³

O(ε)Æ

û»Jº ±¯¡¤~¬¡Ù%®ôÚ# â6 J¡Þ¡ Ù1ã¯Ø¡Þ¡

ht + ∇s · (h s) = 0,

ρ

D s

Dt

=1

h

∇s · T,

nº s

h `Á ³ ÁT = −P + T

½ ¾é `Á ¾G²³ a¦µ úRÁ c Â Å ¾GéÀ³ Á ¾¯¾%é Á ¶¾À·³:P

½ ¾é `Á e ³ Á ¾¯¾G²³ Á é Á ¶#¾G·J³

P = −0.5σ

[

h∇2shIs + 0.5(∇sh)2Is −∇sh ⊗∇sh

]

+ 1.5hφ ,

µ=¶ì é `Á ¸ ½ ¾ ú ·²¾ô¾GéÀ³ Á ¾¯¾Õé Á ¶¾G·J³T

½ ¾ô´ ½ ¸ Á ¶ ê.g

T = 2µh

(∇s · vs) Is + 0.5[

∇svs + (∇svs)T]

.

Á ³ ÁIs

½ ¾+é `Á"½ ì Á ¶Jé ½ ú µ Â ¾G²³ a¦µ úRÁ é Á ¶¾À·³:φ = A

′

h−3

/(6πρ)½ ¾+é `Á e ·=é Á ¶3é ½ µ Â a£²¶ ú Ä

é ½ ·¶·a+¸¡µ=¶ì Á ³w µµ Â ¾!a£·³ úRÁ ¾9A

′ ½ ¾"é `Á µ Å µ=¿ Á ³ ú ·¶¾Gé¯µ¶3é:nA

′

∼ O(10−20J) sµ=¶ì T ¾é6µ=¶ì¾ a£·³'é¯³¯µ¶¾ e ·3¾ ½ é ½ ·¶ Ã b ²µ¡é ½ ·J¶¾ nº ssµ³ Á ·Jêé¯µ ½ ¶ Á ì µaé Á ³µ ¾ ½ Å ½ Ä

Â µ³µJ¾ g Åfe é¯·=é ½ ú µ=¶µ Â g¾ ½ ¾·a%é `Á a£² Â Â ý"µ¸ ½ Á ³ Ä éÀ·¿ Á ¾ Áb ²µ¡é ½ ·J¶¾>h ½ é ` µ eSe ³¯· e ³ ½ µ¡é Áê#·J²¶ìµ³/g ú ·¶ì ½ é ½ ·J¶¾·¶ma£³ ÁAÁ ¾G²³/a¦µ ú9Á ¾·a é `Á c Â Å

! %'& 0 &-%i 0 1%i 1'&

u a1é `ÁÂ Á ¶´é ` ·a1·¶ Á a£³¯µ Å Á"½ ¾ Å ² ú` ê ½ ´´ Á ³%é ` µ¶'é `Á ·é `Á ³Õ·J¶ Á ½T Á b a

Jé `Á ¶é `Á ÁzÁ:ú é¯¾ ½ ¶

yì ½ ³ ÁAú é ½ ·¶.µ=³ Á ¶ Á ´ Â ½ ´ ½ ê Â Á ·J¶é `Á ì=g¶µ Å ½ ú ¾ôµ¶ì ú ·· Â ½ ¶´µ¶ì'é `Á

é `Á ³ Å ·ì=g¶µ Å ½ ú e ³À·Jê Â Á Å ì Á e Á ¶ì¾ô·J¶ Â g ·¶ (x, t)9`Á ¾/g¾Gé Á Å nº ås nTº s ½ ¶½ é6¾ì ½ Å Á ¶¾ ½ ·J¶ Â Á ¾À¾ a£·J³ Å ¾ ½ Åfe Â ½ c Á ¾éÀ· Æ

∂h

∂t

+∂

∂x

(uh) = 0,nTº ¼s

∂u

∂t

+ u

∂u

∂x

=ε

We

∂3h

∂x3

+4

Re h

∂

∂x

(

h

∂u

∂x

)

+A

h4

∂h

∂x

,nº qs

∂T

∂t

+ u

∂T

∂x

=1

Pe h

∂

∂x

(

h

∂T

∂x

)

+Ra

Pe h

(

T4a − T

4)

,nº ys

h `Á ³ ÁRe = ρaU/µ

½ ¾é `ÁmôÁ g3¶· Â ì¾¶² Å ê Á ³:We = ReCa = 2ρaU

2/σé `Á w Á ê Á ³§¶² Å ê Á ³A

Caé `Áú µ e ½ Â Â µ=³*g¶² Å ê Á ³A

A = A

′

/(2πρU2a3ε3) é `Á ì ½ Ä

Å Á ¶¾ ½ ·J¶ Â Á ¾À¾ "µ Å µ=¿ Á ³ ú ·¶¾Gé¯µ¶Jé:Pe = RePr = ρcaU/κ

é `ÁäñÁAúRÂ Á é¶² Å ê Á ³:Ra = 2βaθ

3m/ε κ

é `Á ³6µì ½ µ=é ½ ·¶ ¶² Å ê Á ³:θm

é `Á ¾À· Â ½ ì ½ c ú µ¡é ½ ·¶ é Á Åfe Á ³6µ¡éÀ²³ Áê ÁAÂ ·h h `½ ú` é `Á Â ½Bb ² ½ ìc Â Å ê Á:ú · Å Á ¾ô¾G· Â ½ ì

9`Á ú` µ³¯µ ú é Á ³ ½ ¾Gé ½ ú ¾ ú µ Â Á ¾ ²¾ Á ìbéÀ· ì Á ³ ½ ¸ Á ¾ g¾Gé Á Å nº ¼s nº ys µ³ Á é `Á a£· Â ÄÂ ·h ½ ¶´ Æ

aa£·³ Â Á ¶´é `

Ua£·J³%¸ ÁAÂ · ú9½ é(g n ú µ e ½ Â Â µ³/g§·³%¸ ½ ¾ ú ·²¾s

a/Ua£·³%é ½ Å Á

εaa£·³

c Â Å é `½ ú ¿3¶ Á ¾¯¾Õµ¶ìθm

a£·³Õé Á Å)e Á ³6µ¡é¯²³ Á n `Á ³ Áx

tu

µ=¶#ìT

µ=³ Á ì ½ Å Á ¶#¾ ½ ·¶ Â Á ¾¯¾s ½ ¶ úRÁ ì²³ ½ ¶´é `Á ì Á ³ ½ ¸¡µ¡é ½ ·J¶·a¾/g¾Gé Á Å nº ¼s Ä nTº ysÕé `Á é Á ³ Å ¾·a

o(ε) ·J³¯ì Á ³` µ¸ Á ê ÁAÁ ¶ ½ ´¶·J³ Á ìàé `Á ¶äé `Á a£· Â Â ·h ½ ¶´ ½ ¶ Áb ²µ Â ½ é ½ Á ¾ô¾ ` ·J² Â ìäê Á ¸¡µ Â ½ ì Æ

Re ≤ ε−1

, We ≤ 1, A ≥ ε, Pe ≤ ε−1

, Ra ≥ εPe .

9`Á ê#·J²¶ìµ³/g ú ·¶ì ½ é ½ ·J¶¾ a£·³h

u

µ=¶#ìT

µ=³ Á é `Á a£· Â Â ·h ½ ¶´ Æ

£¢:¡© 3ø.D à±¯¯äÚ5¢+© ¤D û»û

u(0, t) = u(1, t) = 0,nº ¹A»s

∂T

∂x

(0, t) = 0, T (1, t) = Tg,nTº ¹J¹s

∂h

∂x

(0, t) = 0,

∂h

∂x

(1, t) = tanα,nTº ¹:ºs

h `Á ³ ÁTg

½ ¾é `Á ì ½ Å Á ¶¾ ½ ·¶ Â Á ¾À¾ a£³6µ Å Á é Á Åfe Á ³¯µ=éÀ²³ Á µ=¶#ì π

2− α

½ ¾Ûé `Á h Á éÀé ½ ¶´µ=¶´ Â Á h ½ é ` é `Á a£³6µ Å ÁJ9`Á ½ ¶ ½ é ½ µ Â1ú ·¶#ì ½ é ½ ·J¶¾ôµ=³ Á ´ ½ ¸ Á ¶ê.g Æ

h(x, 0) = 1, u(x, 0) = 0, T (x, 0) = T0,nTº ¹:ûs

h `Á ³ ÁT0

½ ¾é `Á ì ½ Å Á ¶¾ ½ ·¶ Â Á ¾À¾ ½ ¶ ½ é ½ µ Â é Á Åfe Á ³¯µ=éÀ²³ Á ·aÝé `Á c Â Å 9`Á Å µJ¾À¾ ú ·¶¾ Á ³¯¸¡µ¡é ½ ·J¶ ·aÝé `Á c Â Å ì²³ ½ ¶´ ½ é¯¾é `½ ¶¶ ½ ¶´ ½ ¾ Á e ³ Á ¾¯¾ Á ì µ¾ Æ

∫ 1

0

(h − 1)dx = 0 ,nº ¹9ë.s

½ é6¾ô¸¡µ Â ½ ì ½ é(g ½ ¾ e ³¯·¡¸ Á ì'ê.g ½ ¶Jé Á ´J³¯µ=é ½ ¶´ nº ¼sµ=¶#ì é6µ=¿ ½ ¶´ ½ ¶3éÀ·sµ ú9ú ·J²¶3é nTº ¹A»s 9`Á ¶·¶ Ä Â ½ ¶ Á µ=³%¶·¶-¾é6µ¡é ½ ·¶#µ=³*g e ³¯·ê Â Á Å nº ¼s nº ¹9ës ½ ¾¾À· Â ¸ Á ì ½ ¶ é ½ Å Á é ½ Â Â

·¶ Á ·aé `Á a£· Â Â ·h ½ ¶´¾é¯· e ú ·¶#ì ½ é ½ ·J¶¾ ½ ¾¾Àµ=é ½ ¾(c Á ì Æµsµ¡é6c¶ ½ é Á é ½ Å Á Å · Å Á ¶3é

t = τµ Å ½ ¶ ½ Å ² Å ¸µ Â ² Á ·a

h

½ ¾³ Á µ ú`Á ìà h `½ ú`ú ·³¯³ Á ¾ e ·J¶ì¾ÕéÀ·sµ¾é6µ=ê Â Á c Â Å ¾ ` µ e Á

limt→τ

h(x, t) = h(x), limt→τ

u(x, t) = u(x), limt→τ

T (x, t) = T (x), nTº ¹:åsê+sDµ=éDé ½ Å Á Å · Å Á ¶3é

t = τé `ÁÁzÁ:ú é ½ ¸ Áäú ³ ½ é ½ ú µ Â c Â Å ³À² e éÀ²³ Á é `½ ú ¿3¶ Á ¾¯¾ ½ ¾

³ Á µ ú`Á ìàµ¡é0h `½ ú` é `Á µ ú éÀ²µ Â c Â Å ³¯² e éÀ²³ Á · ú9ú ²³¯¾ôµ=é¾G· Å Á e · ½ ¶3éxr

h(xr, τ) ≈ 0,nTº ¹ s

ú s ú · Å)e ²é¯µ=é ½ ·¶¾ôµ³ Á ú ·J¶3é ½ ¶3² Á ì é ½ Â Â ú ·¶ì ½ é ½ ·J¶T (x, τ) ≈ 1

½ ¾ô¾¯µ¡é ½ ¾ c Á ì w Á µ¾¯¾G² Å Á é ` µ¡éñé `Á ¾À· Â ²é ½ ·¶§·a#¾/g¾é Á Å nTº ¼s3 nº ¹9ës e ·3¾À¾ Á ¾¯¾ Á ¾ é `Á ³ Áb ² ½ ³ Á ì

¾ Å ··=é ` ¶ Á ¾¯¾u, h, T ⊂ C

4(Ω)½ ¶

Ω = 0 ≤ x ≤ 1 a£·J³ 0 < t < τ

9U \ Y+_ ] 4]

9`Á c¶ ½ é Á 3¸· Â ² Å Á Å Á é ` ·ì ½ ¾²¾ Á ìmé¯· ú ·¶¾GéÀ³¯² ú é~é `Á ì ½ ¾ ú ³ Á é Á ¾ ú`Á Å Á Fh `½ ú`¾Àµ=é ½ ¾(c Á ¾é `Á Å µJ¾À¾A Å · Å Á ¶Jé¯² Å `Á µ=é j² êµ Â µ¶ úRÁ ¾+·J¶ Á µ ú` ú ·J¶Jé¯³À· Â ¸· Â ² Å Á µ=¶ìé `Á ³ Á a£·³ Á ·J¶ é `Á h ` · Â Á e ³¯·ê Â Á Å ì· Å µ ½ ¶

w Á ì Á c¶ Á µ~¶·¶²¶ ½ a£·J³ Å é ½ Å Á ´J³ ½ ì

Ωt = tj+1 = tj + 4tj , 4tj > 0, j = 0, . . . , Jτ , t0 = 0, tJτ= τ.

9`Á ¾ e µ úRÁ ´J³ ½ ì ½ ¾§µ Â ¾G·.¶·J¶3²¶ ½ a£·J³ Å àé `Á µJìµ e é6µ¡é ½ ·J¶ ·a´J³ ½ ì ¶·ì Á ¾ ½ ¾Ûì·¶ Áì=g¶µ Å ½ ú µ Â Â gì²³ ½ ¶´³ Á µ Â ½ ù µ¡é ½ ·¶b·aé `Á µ Â ´·³ ½ é ` Å 32 ·J³ Å ·³ Á µ ú9ú ²³6µ¡é Á µ eSe ³¯· Ä½ Å µ¡é ½ ·J¶ ·aé `Á ê·²¶#ìµ=³*g ú ·J¶ì ½ é ½ ·¶¾fh Á½ ¶Jé¯³À·ì² úRÁ é(h·b´J³ ½ ì¾sì ½ ¾ e Â µ úRÁ ì h ½ é `³ Á ¾ e ÁAú éé¯· Á µ ú` ·é `Á ³ 2 ²¶ ú é ½ ·J¶¾

uµ=¶#ì

Tµ=³ Á ì Á c¶ Á ìä·¶é `Á ´J³ ½ ì

û»=ë ±¯¡¤~¬¡Ù%®ôÚ# â6 J¡Þ¡ Ù1ã¯Ø¡Þ¡

Ωu,Tx = xi = i∆x, i = 0, . . . , N + 1, x0 = 0, xN+1 = 1,

h `½ Â Á a£²¶ ú é ½ ·¶h

½ ¾ì Á c#¶ Á ìä·¶é `Á ´³ ½ ì

Ωhx = xi−0.5 = (i − 0.5)∆x, i = 1, . . . , N + 1 .

u ¶é `½ ¾4hµgé `Á e ³¯·ê Â Á Å h ½ Â Â ê Á ¾G· Â ¸ Á ì'·J¶äµs¾Gé¯µ´´ Á ³ Á ì'¾ e µ ú9Á ´³ ½ ì é Á µ ú` é ½ Å Á ¾é Á e ¶·é¯µ=é ½ ·¶·a+ì ½ ¾ ú ³ Á é Á a£²¶ ú é ½ ·J¶¾ ½ ¾¾ ½ Åfe Â ½ c Á ìµ=¶ì ½ ¶ì Á Á ¾

µ=³ Á · Å ½ éÀé Á ì o år h·sé(g e Á ¾·a ú ·¶3é¯³À· Â ¸· Â ² Å Á ¾"µ=³ Á ²¾ Á ì Æ ·J¶ Á a£·³

uµ¶ì

Tµ=¶ìµ¶·=é `Á ³!a£·³

h

·¶3éÀ³¯· Â ¸· Â ² Å Á ¾xh ½ é ` úRÁ ¶3é Á ³6¾Dµ=éÛé `Á ´³ ½ ì ¶·ì Á ¾~µ=³ Á µ eSe Â ½ Á ì é¯·ì ½ ¾ ú ³ Á é ½ ù9ÁÁb ²µ¡é ½ ·J¶¾6nTº ¼sLnº ys 9`Á ì ½zÁ ³ Á ¶ úRÁ µ eSe ³¯· ½ Å µ=é ½ ·¶h·adnTº ¼s ½ ¾ ·êé6µ ½ ¶ Á ì ½ ¶3é Á ´³6µ¡é ½ ¶´ ½ é ·¶ é `Áú ·¶3é¯³À· Â ¸J· Â ² Å Á

[xi−1, xi]

ht +∆M − ∆M

−

∆x

= 0, i = 1, . . . , N + 1,n¦û ¹s

h `Á ³ Á∆M = ∆Mi =< h >i< u >i .

9`Á ¶·é¯µ=é ½ ·¶<>

1¾ ½ Å ½ Â µ=³ Â gµJ¾ ½ ¶ o rÅ Á µ=¶¾Ûé ` µ=éÛé `Á ¸¡µ Â ² Á ¾D·ah

µ=¶#ìu

µ=³ Á é¯µ¿ Á ¶b·J¶ é `Áú ²³¯³ Á ¶3é úRÁ9Â Â ê·³6ì Á ³ w Áµ Â ¾G·²¾ Á ¶·=é6µ¡é ½ ·¶

∆Mi = u+

i hi + u−

i hi+1, ui = u+

i + u−

i ,

u+

i = 0.5(ui + |ui|) ≥ 0, u−

i = 0.5(ui − |ui|) ≤ 0.

² ÅÅ ½ ¶´s² e nTû ¹sÕ·¡¸ Á ³é `Á ´J³ ½ ìΩ

hxh Á ·êé6µ ½ ¶ é `Á Áb ²µ Â ½ é(g

N∑

i=2

hi∆x = 1 .

u éµ eSe ³¯· ½ Å µ=é Á ¾é `Á Å µ¾¯¾ ú ·¶#¾ Á ³À¸¡µ¡é ½ ·J¶ Áb ²#µ¡é ½ ·¶ nTº ¹Aësh ½ é ` é `Á µ ú9ú ²³¯µ ú gO(∆x)

u aé `Á ê·²¶ìµ=³*g¸¡µ Â ² Á ¾h(0, tj+1)

µ=¶ìh(1, tj+1)

µ=³ Á a£·J²¶ì6a£³¯· Å ê·²¶ì Äµ=³*g ú ·J¶ì ½ é ½ ·¶¾6nº ¹:ºsR3é `Á ¶é `Á Áb ²µ Â ½ é(g

N+1∑

i=1

hi∆x = 1

´ ½ ¸ Á ¾O(∆x

2) µ eSe ³¯· ½ Å µ=é ½ ·¶ ·a nº ¹9ës 2 ·³1¾À· Å Á ¸¡µ Â ² Á ¾à·a e µ=³6µ Å Á é Á ³6¾A Á ´ a£·J³ Â µ³À´ Á

Re$h `½ ú` ú ·J³À³ Á ¾ e ·J¶ì"éÀ·¾GéÀ³¯·¶´ú ·¶¸ ÁAú é ½ ·J¶ ì· Å ½ ¶µ¶ úRÁ =h Á ·êé6µ ½ ¶¾ ½ ¶´² Â µ=³ e ³¯·ê Â Á Å ¾ h ½ é ` ¾ Å µ Â Â1ú · Á sú9½ Á ¶3é¯¾µ¡é

é `Á ì Á ³ ½ ¸¡µ¡é ½ ¸ Á ·a"é `ÁäÂ Á µJì ½ ¶´ ·J³¯ì Á ³ n£ê·²¶#ìµ=³*g Â µg Á ³ e ³¯·ê Â Á Å ¾s u ¶ e ³¯µ ú é ½ ú9Á ¾G² ú` e ³À·Jê Â Á Å ¾"µ³ ÁDú` µ=³6µ ú é Á ³ ½ ùAÁ ìäê.g.ê#·J²¶ìµ³/g Â µg Á ³6¾ASh `Á ³ Á ¾À· Â ²é ½ ·J¶¾ ú` µ¶´ Á¸ Á ³*g a¦µ¾Gé w Á ¶·é Á é ` µ=é§é `Á ¾ Á ³ Á ´ ½ ·J¶¾Dµ Â ¾À· ú` µ¶´ Á½ ¶ é ½ Å Á µ ú9ú ·J³¯ì ½ ¶´éÀ·é `Áe ³À·Jê Â Á Å ¾ e ÁAú9½ c ú ¾ u ¶ ·³6ì Á ³ÛéÀ·µ eSe ³¯· ½ Å µ¡é Á ¾G· Â ²é ½ ·J¶¾ ·a"¾G² ú` e ³À·Jê Â Á Å ¾9 ½ é ½ ¾¶ ÁAú9Á ¾¯¾Àµ³/gé¯·ì Á ¸ ÁAÂ · e ¾ e ÁAúR½ µ Â ì ½#z5Á ³ Á ¶ ú9Á ¾ ú`Á Å Á ¾

2 ·³µ eSe ³¯· ½ Å µ¡é ½ ·J¶ ·aÝé `Á ¸ ÁAÂ · úR½ é(g Áb ²#µ¡é ½ ·¶.¾À² ú` µ~¾ ú`Á Å Á6` µ¸ Á ê ÁAÁ ¶ e ³¯· Äe ·3¾ Á ì ½ ¶do ër u é ` µ¾Ýµ¾À² 'úR½ Á ¶3é Â g´J··ì µ ú9ú ²³¯µ ú g!a£·³ ¾ Å µ Â Â µ=¶#ì Â µ³À´ Á ¸ ÁAÂ · úR½ é ½ Á ¾ u ¶

£¢:¡© 3ø.D à±¯¯äÚ5¢+© ¤D û»Jå·²³ Å ·ì ÁAÂÁb ²µ=é ½ ·¶ nTº qs ` µ¾¾ ½ Å ½ Â µ³Fa Á µ=éÀ²³ Á ¾Ah ½ é ` é `Áú ·J¶3¸ ÁAú é ½ ·¶~·³+ì ½z ²#¾ ½ ·¶ì· Å ½ ¶µ=¶ ú9Á a£·J³ì ½#z5Á ³ Á ¶3é

Re

-2 ·³ ½ é¯¾¶² Å Á ³ ½ ú µ Â ¾G· Â ²é ½ ·J¶µ¶ ½ Åfe ³¯·¡¸ Á ìs¸¡µ=³ ½ µ¶3éÕ·aé `Á ¾ ú`Á Å Á a£³¯· Å o ër ½ ¾ e ³¯· e ·J¾ Á ì

ut +1

2(u+0.5u)x = (µ+0.5 ux)x +

ε

We

hxxx −A

3(h−3)x,

nTû ºs

h `Á ³ Áh = 1

2(h + h+1)

µ+0.5 = µ+0.5FB(R+0.5), µ+0.5 =4

Re

h+1

h

, R+0.5 =1

8Re ∆x u+0.5

h

h

µ=¶ì é `Á a£²¶ ú é ½ ·J¶FB

½ ¾ôµ e ½ Á:úRÁ h ½ ¾ ÁÂ ½ ¶ Á µ³µ eSe ³À· ½ Å µ=é ½ ·¶·a R

exp R − 1

9`Á é Á Åfe Á ³¯µ=éÀ²³ Á%Áb ²µ¡é ½ ·J¶Dµ Â ¾G· ` µ¾¾ ½ Å ½ Â µ=³ia Á µ=éÀ²³ Á ¾Ah `Á ¶ ½ é ½ ¾¾À· Â ¸ Á ì ½ ¶ é `Á

¾ g¾Gé Á Å ·a Áb ²µ=é ½ ·¶¾ih ½ é ` ì· Å ½ ¶µ=¶3é ú ·¶¸ ÁAú é ½ ·J¶ é Á ³ Å ¾ w Á e ³¯· e ·J¾ Á µì ½ ¾ ú ³ Á é ½ ù µ Äé ½ ·¶1ih `½ ú` ½ ¾ ú ·¶#¾é¯³À² ú é Á ì ²¾ ½ ¶´é `Á ¾¯µ Å Á e ³ ½ ¶ úR½ e Â ÁÝ ·³ Á ·¡¸ Á ³A1µ³ Á ¾é¯³ ½ ú é ½ ·J¶ú ·¶¶ ÁAú é Á ì h ½ é ` e ` g¾ ½ ú µ Â"ú ·³¯³ Á:ú éÀ¶ Á ¾¯¾·aé `Á ¾À· Â ²é ½ ·J¶ ½ ¾ ½ Åfe ·3¾ Á ìàÕµ=é Â Á µ¾Gé ½ ¶¾G· Å Á ¾ ½ Å)e Â ÁÛú µ¾ Á ¾A=h `½ ú` · ú9ú ²³"µJ¾ e µ=³Àé ½ ú ² Â µ³ ú µ¾ Á ¾·aÝé `ÁDú ·¶¾ ½ ì Á ³ Á ì e ³À· ú9Á ¾¯¾ 2 ·³ Á µ Å)e Â Á Â Á é²¾ ú ·J¶¾ ½ ì Á ³µ `Á µ¡ééÀ³6µ=¶¾ a Á ³ e ³¯·ê Â Á Å h ½ é ` ·J²é `Á µ¡é Á ú` µ¶´ Áh ½ é ` é `Á µ Å ê ½ Á ¶3é´Jµ¾·¶ê#·J²¶ìµ³ ½ Á ¾A ½ Á h Áú ·J¶¾ ½ ì Á ³"é `Á ³ Å µ Â Â g ½ ¾G· Â µ=é Á ìc Â Å¾G²³/a¦µ ú9Á ¾:né `Á ³ ÁD½ ¾"¶·¾G·J²³ ú9Á é Á ³ Å ½ ¶ nTº ysµ=é

Ra = 0 s 9`Á ¶é `Á é Á Åfe Á ³¯µ=éÀ²³ Á·a c Â Å ³ Á Å µ ½ ¶¾ ½ ¶ é `ÁÂ ½ Å ½ é6¾·a1é `Á½ ¶ ½ é ½ µ Â é Á Åfe Á ³¯µ=éÀ²³ Á µ¶ìé `Á a£³6µ Å Á é Á Å)e Á ³ Äµ¡éÀ²³ ÁJ "·h Á ¸ Á ³A ½ aÕé `Á ³6µì ½ µ¡é ½ ·J¶Lh ½ é ` µ ú · Â ì Á ³µ Å ê ½ Á ¶3éé¯µ¿ Á ¾ e Â µ ú9Á é `Á ¶é `Ác Â Å é Á Å)e Á ³6µ¡é¯²³ Áú µ=¶¶·=éôê Á Â Á ¾¯¾é ` µ=¶äé `Á µ Å ê ½ Á ¶3éé Á Åfe Á ³¯µ=éÀ²³ Á

² ú` é Á ¾Gé¯¾ ` µ¸ Á ê ÁAÁ ¶ e Á ³ a£·J³ Å Á ìkh ½ é ` µ a Á hlì ½ ¾ ú ³ Á é Á ¾ ú`Á Å Á ¾Aµ eSe ³¯· ½ ÄÅ µ=é ½ ¶´'é `Áú ·¶#¾ ½ ì Á ³ Á ì Áb ²µ¡é ½ ·J¶ 5 µ¿ ½ ¶´ ½ ¶3éÀ·äµ úAú ·²¶Jéé `Á ³ Áb ² ½ ³ Á Å Á ¶3é·aé `ÁÅ µ ½ Å ² Å e ³ ½ ¶ ú9½ e Â Á a£·J³"é `Á ¾G· Â ²é ½ ·J¶µ=¶#ì²¾ ½ ¶´ Á e Á ³ ½ Å Á ¶3é6µ Â ³ Á ¾À² Â é¯¾·êé6µ ½ ¶ Á ìh ½ é ` ì ½#z5Á ³ Á ¶3é"ì ½ ¾ ú ³ Á é Á ¾ ú`Á Å Á ¾9=h Á6` µ¸ Á ¾ Á9Â ÁAú é Á ì é `Á a£· Â Â ·h ½ ¶´sì ½ ¾ ú ³ Á é ½ ù µ=é ½ ·¶1h `½ ú`.½ ¾µ=¶µ¶µ Â ·´J² Á ·aé `ÁDú · Å ê ½ ¶ Á ìä¾ ú`Á Å Á µ=¶ì ½ ¾ ú ·J¶¾é¯³À² ú é Á ì.¾ ½ Å ½ Â µ=³ Â gµ¾½ ¶po ër Æ

(hT )t + (h+1u+0.5T )x = (ϕ+0.5 Tx)x +Ra

Pe

(

T4a − T

4

)

,nTû ûs

h `Á ³ Áϕ+0.5 =

h+1

Pe

FB(S+0.5), S+0.5 = ∆x u+0.5Pe .

u ¶Jé Á ´J³¯µ=é ½ ¶´mnº ysÕ·¡¸ Á ³Õé `ÁÛú ·J¶3éÀ³¯· Â ¸· Â ² Å Á[0, x0.5]

µ=¶ì é¯µ¿ ½ ¶´ ½ ¶3éÀ·sµ úAú ·²¶Jéé `Á ê·²¶#ìµ=³*g ú ·J¶ì ½ é ½ ·¶.µ¡é

x0 = 0 =h Á ´ Á éôé `Á ê·²¶#ìµ=³*g'ì ½ ¾ ú ³ Á é ÁÁb ²µ=é ½ ·¶ Æ

(hT )t,0 +2

∆x

h1u0.5T0 =2

∆x

ϕ0.5 Tx,1 +Ra

Pe

(

T4a − T

40

)

.nTû ës

9`Á½ é Á ³¯µ=é ½ ·¶ e ³À· úRÁ ¾À¾ a£·³h

µ=¶ìu

½ ¾ì Á ¾ ú ³ ½ ê Á ì ½ ¶ì Á é¯µ ½ Â ¾ ½ ¶ o ër3é `Á ³ Á a£·³ Á"½ éh ½ Â Â ¶·=éê Á e ³ Á ¾ Á ¶Jé Á ì `Á ³ Á9`Á ¶·J¶ Â ½ ¶ Á µ=³¾ g¾Gé Á Å nTû ¹sLn¦û ës ½ ¾¾À· Â ¸ Á ì'ê.g'µ=¶½ é Á ³6µ¡é ½ ¸ Á e ³¯· úRÁ ¾À¾A ú ·¶#¾ ½ ì Á ³ ½ ¶´

hs+1 µ¶ì

us+1 µ¾ôµ Â ³ Á µìSgfa£·²¶ì Æ

û» ±¯¡¤~¬¡Ù%®ôÚ# â6 J¡Þ¡ Ù1ã¯Ø¡Þ¡

(s+1

h

s+1

T )t,0 +2

∆

s+1

h 1

s+1u 0.5

s+1

T 0 =2

∆x

s+1

ϕ0.5

s+1

T x,1

+Ra

Pe

(

− 4s

T30

s+1

T 0 + T4a + 3

s

T40

)

,

(s+1

h

s+1

T )t + (s+1

h +1

s+1u +0.5

s+1

T )x = (s+1

ϕ+0.5

s+1

T x)x

+Ra

Pe

(

− 4s

T3

s+1

T + T4a + 3

s

T4)

, 1 ≤ i ≤ N ,

s+1

T N+1 = Tg .

nTû ås

w Á ¶·é Á é ` µ=éé `Á e ³ Á ¾ Á ¶3é Á ì Â ½ ¶ Á µ=³ ½ ù µ¡é ½ ·J¶'·aÝé `Á ¾À·²³ úRÁ é Á ³ Å ½ ¶ nTû ûsRQnTû ëse ³ Á ¾ Á ³À¸ Á ¾%é `Á e ` g¾ ½ ú µ Â1ú ·³¯³ Á:ú éÀ¶ Á ¾¯¾Õ·a é `Á ¾G· Â ²é ½ ·¶ 9`Á ¾G· Â ²é ½ ·J¶m·a¾/g¾é Á Å nTû ås ½ ¾ a£·²¶ìmê.g é `Á9` · Å µ¾sµ Â ´J·³ ½ é ` Å h `½ ú`

é¯µ=¿ Á ¾ ½ ¶3éÀ·sµ úAú ·²¶Jéé ` µ¡é Å µ¡é¯³ ½ ·aé `Á ¾ g¾Gé Á Å ½ ¾é ` ³ Á9Á Ä ì ½ µ=´J·¶µ Â

−C0

s+1

T 0 + B0

s+1

T 1 + F0 = 0 ,

Ai

s+1

T i−1 − Ci

s+1

T i + Bi

s+1

T i+1 + Fi = 0, i = 1, . . . , N,

s+1

T N+1 = Tg

h ½ é `

Ai =τ

(∆x)2

s+1

ϕ i−0.5 +τ

∆x

s+1

h i, Bi =τ

(∆x)2

s+1

ϕ i+0.5 ,

Ci =s+1

h i +τ

(∆x)2(s+1

ϕ i−0.5 +s+1

ϕ i+0.5

)

+τ

∆x

s+1u i+0.5

s+1

h i+1 +4τRa

Pe

s

T3,

Fi = hiTi +τRa

Pe

(

3s

T4i + T

4a

)

.

9`Á ¾Gé¯µ=ê ½ Â ½ é(g ú ·¶ì ½ é ½ ·¶s·aàé `Á9` · Å µJ¾ Å Á é ` ·ìs³ Áb ² ½ ³ Á ¾é `Á e ·3¾ ½ é ½ ¸ Á ¶ Á ¾¯¾+·aé `ÁÛú · Á 'úR½ Á ¶3é¯¾

AiBi

µ=¶ìDi = Ci −Ai −Bi

u é4a£· Â Â ·h¾<a£³¯· Å a£·J³ Å ² Â µ¾´ ½ ¸ Á ¶µ=ê·¡¸ Á é ` µ¡é ½ ¶ Áb ²#µ Â ½ é ½ Á ¾

Ai > 0 µ=¶#ìBi > 0 µ=³ Á ¾Àµ=é ½ ¾(c Á ìä²¶ ú ·¶ì ½ é ½ ·¶#µ Â Â g 9`Á

e ·3¾ ½ é ½ ¸ Á ¶ Á ¾¯¾·a

Di =s+1

h +τ

∆x

(s+1u i+0.5

s+1

h i+1 −s+1u i−0.5

s+1

h i

)

+4τRa

Pe

s

T3

ì Á e Á ¶ì¾·J¶é `Á~ú` µ=¶´ Á ·a b ²µ¶3é ½ é ½ Á ¾ s+1

hµ=¶#ì s+1

uµ=¶#ì ½ é ú µ=¶ê Á µ ú`½ Á ¸ Á ìàa£·³Á µ Åfe Â Á ê.gDì ½ Å ½ ¶ ½ ¾ `½ ¶´é `Á é ½ Å Á ¾Gé Á e =9`Á ¸ ½ · Â µ=é ½ ·¶§·aé `Á ¾é6µ=ê ½ Â ½ é(g ú ·¶ì ½ é ½ ·¶Â Á µì¾~é¯· ¾À· Å Á ³ Á ¾GéÀ³ ½ ú é ½ ·¶¾~·¶mé ½ Å Á ¾é Á e ¾ +9`Á ³ Á a£·³ Á ½ ¶ ·J³¯ì Á ³~éÀ· ·êé6µ ½ ¶ é `Á

ú ·³¯³ ÁAú éô¾À· Â ²é ½ ·¶·a é `Á e ³¯·ê Â Á Å ½ é ½ ¾¶ Á:úRÁ ¾À¾¯µ=³*g~éÀ· ú`ÁAú ¿é `½ ¾ ú ·¶#ì ½ é ½ ·J¶äì²³ ½ ¶´ú · Åfe ²é¯µ¡é ½ ·J¶¾ u aé `Á½ é Á ³¯µ=é ½ ¸ Á e ³¯· úRÁ ¾À¾ ½ ¾Fc¶ ½ ¾ `Á ìdh `Á ¶'µ³ Á9Â µ¡é ½ ¸ Áú` µ=¶´ Á ·aàê·=é ` a£²¶ ú é ½ ·J¶¾

uµ=¶ì

Tµ=éôé(hÕ·¾À² ú9ú9Á ¾¯¾ ½ ¸ Á½ é Á ³6µ¡é ½ ·J¶¾ ½ ¾ Â Á ¾¯¾é ` µ=¶µ~´ ½ ¸ Á ¶.¾ Å µ Â Â ¸¡µ Â ² Á é `Á ¶h Á

` µ¸ Á ³ Á µ ú`Á ì'é `Á ¾Gé Á µJì=gsé `Á ³ Å µ Â µ=¶#ìì=g¶µ Å ½ ú ¾À· Â ²é ½ ·¶¾6nº ¹:ås

£¢:¡© 3ø.D à±¯¯äÚ5¢+© ¤D û»3¼ ·¶ì ½ é ½ ·¶dnº ¹ s ú µ¶¶·=éÝê Á a£² Â c Â Â Á ì Á µ ú é Â gµ¶ìé `Á ³ ÁÕÁ ½ ¾é6¾ µ ú ³ ½ é ½ ú µ Â é `½ ú ¿ Ä

¶ Á ¾¯¾hcri > 0 µ¡é4h `½ ú` é `Á µ ú éÀ²µ Â ³À² e éÀ²³ Á ·aÝé `Á c Â Å · úAú ²³6¾93é `Á ³ Á a£·J³ Á nTº ¹ s½ ¾ ú` µ=¶´ Á ìsé¯·

h(xr , τ) = hcri.nTû s

w Á ¶·=é Á é ` µ¡éé `Á ì ½zÁ ³ Á ¶3é ½ µ Â e ³À·Jê Â Á Å ê Á:ú · Å Á ¾¾ ½ ¶´² Â µ=³ ½ ¶é `Á ³ Á ´ ½ ·¶ h `Á ³ Áh

µ eSe ³À·3µ ú`Á ¾hcri

0U \ Y+_ ] .\(X9`Á c Â Å ì³¯µ ½ ¶µ´ Á~½ ¶ é ½ Å Á µ¶ì ½ é6¾ Â ·¶´ ½ é¯²ì ½ ¶µ Â ¸ Á9Â · úR½ é(g h Á ³ Á½ ¶¸ Á ¾é ½ ´3µ¡é Á ì ½ ¶o ër%ê.g µ¾ ½ Å ½ Â µ³ ¶3² Å Á ³ ½ ú µ Â ¾ ú`Á Å Á µ¾Ûì Á ¾ ú ³ ½ ê Á ì `Á ³ ÁJ u ¶ o ër+é `Á µ=¶µ Â g¾ ½ ¾ hµ¾ì·¶ Á a£·³¸¡µ=³ ½ ·²¾"¸µ Â ² Á ¾·a+é `ÁsôÁ g¶· Â ì¾¶² Å ê Á ³

Ren 1 ≤ Re ≤ 100 sµ¶ìì ½ Ä

Å Á ¶¾ ½ ·J¶ Â Á ¾À¾"µ Å µ=¿ Á ³ ú ·¶¾Gé¯µ¶JéAn 0.1 ≥ A ≥ 0 sR+h `Á ¶é `Áú µ e ½ Â Â µ³/gä¶² Å ê Á ³

Caµ=¶#ì

εh Á ³ Á c Á ìhéÀ·

Ca = ε = 0.01 nWe

ì Á e Á ¶ì Á ì ·¶ Â g ·J¶Re

%¾ ½ ¶ úRÁWe = 0.01Re

s u é ½ ¾ a£·J²¶ì ½ ¶ o ër5é ` µ=éé `Á ³À² e éÀ²³ Á e · ½ ¶3éxr

Å ·¡¸ Á ¾%éÀ·§é `Á ³ ½ ´ ` é¾ ½ ì Á a£³À· Å é `Ású9Á ¶3éÀ³6µ Â ¾/g ÅÅ Á é¯³/g e · ½ ¶3é

x = 0 éÀ·hµ=³6ì¾é `Á a£³¯µ Å Á e · ½ ¶3éx = 1µ=¶ìäµRì ½ Åfe Â Á ·a é `Á c Â Å ½ ¾ a£·J³ Å Á ì'é ` µ¡é Á ¶ Â µ³À´ Á ¾ ½ ¶.¾ ` µ e Á h ½ é ` é `Á ´³¯·h ½ ¶´

¸µ Â ² Á ¾·aRe

µ=¶ìA

à9`½ ¾ Å Á µ=¶¾é ` µ¡éé `Á~½ ¶ Á ³Àé ½ µ Â a£·³ úRÁ ¾ ` µ¸ Á µì Á ¾Gé¯µê ½ Â ½ ù9½ ¶´Á$z5ÁAú é·¶ é `Á c Â Å µ¶ì Â Á µì éÀ· ½ é¯¾~ì³6µ ½ ¶µ=´ Á µ=¶ì ³À² e éÀ²³ Á µ=éA > 0

9`Á ¸¡µ=¶ì Á ³fw µµ Â ¾xa£·³ úRÁ ¾~µ Â ¾À· ì Á ¾é6µ=ê ½ Â ½ ù9Á é `Á c Â Å ì=g¶µ Å ½ ú ¾§µ¶ì µ ú9úRÁAÂ Á ³¯µ=é Á é `Á c Â Å³À² e éÀ²³ Á

u ¶é `Á e ³ Á ¾ Á ¶3é!h·³¯¿mh Á a£· ú ²¾·¶é `Á§½ ¶=j#² Á ¶ ú9Á ·aé `Á§½ ¶ Á ³Gé ½ µ Â µ¶ì.¸¡µ=¶ì Á ³wbµJµ Â ¾>a£·³ úRÁ ¾·J¶é `Á `Á µ¡é éÀ³6µ=¶#¾(a Á ³Aàµ=¶#ì ¶#µ Å Á9Â g·¶ ú ·· Â ½ ¶´ 1"Â ¾À·h Á'ú ·¶¾ ½ ì Á ³¸µ³ ½ ·²#¾Õ³6µì ½ µ=é ½ ·¶ Å µ=´J¶ ½ éÀ²ì Á ¾ôµ=¶ìä¾GéÀ²ì=g'é `ÁA½ ³ ÁzÁ:ú éô·¶é `ÁÛú ·· Â ½ ¶´ e ³¯· úRÁ ¾¯¾µ¾µ h ` · Â Á9`Á e ³ Á ¾ Á ¶3é Á ì.¶² Å Á ³ ½ ú µ Â ³ Á ¾À² Â é¯¾µ=³ Á ·êé6µ ½ ¶ Á ì a£·J³ ú ·J¶¾Gé¯µ=¶3é"¸¡µ Â ² Á ¾·aú · Á'úR½ Á ¶3é6¾

Ca = 0.01, ε = 0.01, α = 1.37, P r = 1 .

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0

0,0

0,2

0,4

0,6

0,8

1,0

1,2

1,4

1,6

1,8

2,0

2,2

2,4

2,6

2,8

h

x

t=1.e-4 t=0.03 t=0.1 t=0.3 t=0.5 t=0.8 t=1. t=1.3 t=1.5 t=1.8 t=2.0 t=2.4

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

U

X

t=0.1e-3 t=0.03 t=0.1 t=0.3 t=0.5 t=0.8 t=1. t=1.3 t=1.5 t=2.4

M ? +N$I #I :HILH I" E =,3HJ?*I :H

h

Du

MA = 0.1

DRe = Pe = 1

DWe = 0.01

DT0 = 1.19

DTg = Ta = 1

Dα = 1, 37 M

h(x, t) ? u(x, t)-

û»q ±¯¡¤~¬¡Ù%®ôÚ# â6 J¡Þ¡ Ù1ã¯Ø¡Þ¡ u ¶ 2Ý½ ´ ¹é `ÁÁ ¸J· Â ²é ½ ·¶ ½ ¶ ¾ e µ úRÁ µ¶ìbé ½ Å Á ·aé `Á c Â Å é `½ ú ¿3¶ Á ¾¯¾

h Â ·¶´ ½ Ä

éÀ²ì ½ ¶µ Â ¸ Á9Â · úR½ é(gu

µ=¶ì é Á Å)e Á ³6µ¡é¯²³ ÁT

µ=³ Á ´ ½ ¸ Á ¶ a£·J³~é `Á a£· Â Â ·h ½ ¶´ ¸¡µ Â ² Á ¾·ae µ=³6µ Å Á é Á ³6¾

A = 0.1, Re = Pe = 1, We = 0.01, T0 = 1.19, Tg = Ta = 1.

h·Dì ½zÁ ³ Á ¶3é%³¯µJì ½ µ¡é ½ ·¶¶² Å ê Á ³¯¾Ra = 0 µ=¶#ì

Ra = 10 h Á ³ Á ²¾ Á ì 39`Á ³¯² e é¯²³ Á·aé `Á c Â Å ½ ¾·ê#¾ Á ³À¸ Á ì µ=é

x = 0 a£·³µ§é ½ Å Á Å · Å Á ¶3ét = 2.4 o ër

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0

1,00

1,05

1,10

1,15

1,20

T

x

t=1.e-4 t=0.03 t=0.1 t=0.3 t=0.5 t=0.8 t=1. t=1.28

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0

1,00

1,05

1,10

1,15

1,20

T

x

t=1.e-4 t=0.03 t=0.1 t=0.3 t=0.5

M ? +N$I I :H I HLI" E <F E E "61 E GM( E

TM

A = 0.1D

Re = Pe = 1D

We = 0.01DT0 = 1.19

DTg = Ta = 1

Dα = 1, 37 M

T (x, t)M

Ra = 0D ?

T (x, t)M$

Ra = 10-

u ¶ 2Ý½ ´ ºh Á e ³ Á ¾ Á ¶3éD³ Á ¾À² Â é¯¾D¾ ` ·h ½ ¶´ì=g¶µ Å ½ ú ¾Û·aôé `Á é Á Åfe Á ³6µ¡éÀ²³ ÁJ u é ½ ¾¾ ÁAÁ ¶ é ` µ¡éÛé `Á é Á Åfe Á ³¯µ=éÀ²³ Á ê ÁAú · Å Á ¾ ú ·¶#¾é6µ=¶3éDµ=¶ì Áb ²µ Â éÀ·é `Á ¾À· Â ½ ì ½ c ú µ¡é ½ ·J¶é Á Åfe Á ³6µ¡é¯²³ Á µ=éì ½#z5Á ³ Á ¶3éñé ½ Å Á Å · Å Á ¶3é¯¾A:é ` µ=é%µ=³ ÁÕÂ Á ¾¯¾é ` µ¶Dé `Á é ½ Å Á ·a#³¯² e é¯²³ Á9`Áú ·3· Â ½ ¶´ Å Á:ú` µ=¶ ½ ¾ Å ½ ¾~ì² Á é¯· ú ·¶¸ Á:ú é ½ ·¶ µ¶ì ú ·¶ì² ú é ½ ·¶ ½ ¶ é `Áú µ¾ Á ·aRa = 0 n 2½ ´ º# sh `½ Â Á µ=é

Ra = 10 n 2½ ´ ºâ*s ½ é ½ ¾%ì· Å ½ ¶µ¡é Á ì§êg§³¯µJì ½ µ¡é ½ ·¶µ=¶#ìé `ÁÂ µ³À´ Á ³Õ³¯µJì ½ µ=é ½ ·¶ ú µ=²¾ Á ¾ a¦µ¾Gé Á ³ ú ·3· Â ½ ¶´ 39`Á ¾ Áú ·J¶ úRÂ ²¾ ½ ·J¶¾%µ³ Áú ·J¶=c³ Å Á ìsê.gµìì ½ é ½ ·¶#µ Â ¶3² Å Á ³ ½ ú µ Â ¾ ½ Å ² Â µ¡é ½ ·¶#¾ e Á ³/a£·³ Å Á ì h ½ é ` ·=é `Á ³¸µ Â ² Á ¾·a

RaTg

µ=¶ìTa

2 ·³ Â µ=³¯´ Á ³

Re (Re > 1) é `Á ¸ Á9Â · úR½ é(gu

µ¶ì é `Á é `½ ú ¿¶ Á ¾¯¾h

` µ¸ ÁÂ µ³À´ Áµ Åfe Â ½ é¯²ì Á ¾ µ¶ìé `Á ³À² e éÀ²³ Á e · ½ ¶3é Å ·¡¸ Á ¾!a£³¯· Å é `ÁsúRÁ ¶3é Á ³

x = 0 éÀ·hµ=³6ì¾ôé `Áa£³¯µ Å Á e · ½ ¶Jé

x = 1 u ¶é `ÁÛú µ¾ Á ·a

Re = 100 We = 1 µ¶ì

A = 0.1 é `Á ³À² e éÀ²³ Á½ ¾Õµ ú`½ Á ¸ Á ì§é `Á é ½ Å Á Å · Å Á ¶3ét = 1.757

½ ¶sé `Á e · ½ ¶Jéx = 0.53 o ër 39`Á ¾ Á ³ Á ¾À² Â é¯¾

µ=³ Á ¾ ` ·hô¶ ½ ¶ 2½ ´ û u ¶ 2½ ´ ëµ¶ì 2½ ´ åÕé `Á+ú ·· Â ½ ¶´ô·a3é `Á c Â Å ½ ¾ e ³ Á ¾ Á ¶3é Á ìa£·J³5é `Á ¾¯µ Å Á ì=g¶µ Å ½ ú

e µ=³6µ Å Á é Á ³6¾ôµ=¶ìT0 = 1.19 ê²éôé(hÕ·'ì ½#z5Á ³ Á ¶3é4a£³6µ Å Á µ=¶ì.µ Å ê ½ Á ¶3éôé Á Åfe Á ³¯µ=éÀ²³ Á

³ Á ´ ½ Å Á ¾µ=³ Áú ·¶¾ ½ ì Á ³ Á ì Æµs

Tg = 1, Ta = 1 n¦¾ Á9Á 2½ ´ ë.sê+s

Tg = 1.15, Ta = 0.9 n¦¾ Á9Á 2½ ´ ås

£¢:¡© 3ø.D à±¯¯äÚ5¢+© ¤D û»y

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0

0,0

0,5

1,0

1,5

2,0

2,5

h

x

t=1.e-4 t=0.03 t=0.1 t=0.3 t=0.5 t=0.8 t=1. t=1.3 t=1.4 t=1.4203

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0

-2,5

-2,0

-1,5

-1,0

-0,5

0,0

0,5

1,0

1,5

2,0

U

X

t=0.1e-3 t=0.03 t=0.1 t=0.3 t=0.5 t=0.8 t=1. t=1.3 t=1.4203

M ? +N$I I# AHILHI" E ,3HJ?*I :H

h

Du

MA = 0.1

Dα = 1, 37

DRe = Pe = 100

DWe = 1.

DT0 = 1.19

DTg = Ta = 1 M

h(x, t) ? u(x, t)-

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 1,00

1,02

1,04

1,06

1,08

1,10

1,12

1,14

1,16

1,18

1,20

T

x

t=1.e-4 t=0.03 t=0.1 t=0.3 t=0.5 t=0.8 t=1. t=1.3 t=1.37

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 1,00

1,02

1,04

1,06

1,08

1,10

1,12

1,14

1,16

1,18

1,20

t=1.e-4 t=0.03 t=0.1 t=0.3 t=0.5 t=0.8 t=1. t=1.3 t=1.3668

T

X

M ? +N$I #I :HILH)I" E

TM$

A = 0.1Dα = 1, 37

DRe = Pe = 100

DWe = 1.

DT0 = 1.19

DTg = Ta = 1 M

T (x, t)M

Ra = 0 ? T (x, t)M$

Ra = 10-

· Åfe ²é6µ¡é ½ ·¶#¾Dµ=³ Á ì·J¶ Á a£·J³§³6µì ½ µ¡é ½ ·J¶ ¶3² Å ê Á ³6¾Ra = 0 µ=¶ì

Ra = 10 u a

é `Á ³ Á'½ ¾Û¶·³¯µJì ½ µ¡é ½ ·¶kn 2½ ´ ë5 s é `Á ¶ é `Áú ·J¶¸ ÁAú é ½ ·¶ ì· Å ½ ¶µ=é Á ¾§µ=¶ìbé `Á é Á Å~Äe Á ³¯µ=éÀ²³ Á e ³¯·c Â Á e ³À·éÀ³¯²ì Á ¾+é¯·hµ³¯ì¾+é `Á e · ½ ¶3é·a ³À² e éÀ²³ Á

x = 0.53 ³ Á µ ú`½ ¶´ é `Á¾G· Â ½ ì ½ c ú µ=é ½ ·¶'é Á Åfe Á ³6µ¡éÀ²³ Á

T = 1 a£·J³ôµÛé ½ Å Á Å · Å Á ¶3é¾ Å µ Â Â Á ³Õé ` µ¶ é `Á ³À² e éÀ²³ Áé ½ Å Á

t = 1.37

u aàé `Á ³¯µJì ½ µ¡é ½ ·¶ ½ ¾ ½ ¶ ú9Â ²#ì Á ìà3é `Á ¶äµ ú ·· Â ½ ¶´Û·a1é `Á c Â Å é ½ Â Â é `Á ¾À· Â ½ ì ½ c ú µ¡é ½ ·J¶é Á Åfe Á ³6µ¡é¯²³ Áä½ ¾~µ ú9ú9Á9Â Á ³6µ¡é Á ìà ê²éé `Á é Á Åfe Á ³¯µ=éÀ²³ Á e ³¯·=é¯³À²#¾ ½ ·¶ ½ ¾~¸ ½ ¾ ½ ê Â Á µ=´3µ ½ ¶n¦¾ Á9Á 2½ ´ ë5â§µ¶ì 2Ý½ ´ åsÛéÀ·hµ=³6ì¾Dé `Á e · ½ ¶3é

x

ú ·³¯³ Á ¾ e ·¶ì Á ¶3éDé¯· é `Á Å ½ ¶ ½ Å ² Å·a0c Â Å é `½ ú ¿¶ Á ¾À¾~µ¡é~é `Á.ú ²³À³ Á ¶3éDé ½ Å Á Å · Å Á ¶3éA ½ ÁJ

x = 0.55 µ=ét = 1.3668a£·³ é `Á c³6¾é§³ Á ´ ½ Å Á µ¶ì

x = 0.56 µ=ét = 1.3245 a£·J³ é `Á ¾ Á:ú ·J¶ìb³ Á ´ ½ Å ÁJ 9`Á

µ Å ê ½ Á ¶Jé%µ¶ì:a£³¯µ Å Á é Á Åfe Á ³6µ¡é¯²³ Á ³ Á ´ ½ Å Á ¾ì· ¶·=é ` µ¸ Á µ ¾ ½ ´¶ ½ c ú µ=¶3é ½ ¶Sj² Á ¶ úRÁ ·¶é `½ ¾·ê¾ Á ³¯¸¡µ¡é ½ ·¶µ¶ìé `Á ·¶ Â gì ½#z5Á ³ Á ¶ úRÁD½ ¾é ` µ=éa£·J³

Tg = 1.15, Ta = 0.9 é `Á

û¹A» ±¯¡¤~¬¡Ù%®ôÚ# â6 J¡Þ¡ Ù1ã¯Ø¡Þ¡

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 1,00

1,02

1,04

1,06

1,08

1,10

1,12

1,14

1,16

1,18

1,20

T

X

t=1.e-4 t=0.03 t=0.1 t=0.3 t=0.5 t=0.8 t=1. t=1.3 t=1.3245

+N$I I# AHILH0I" E C MA = 0.1

Dα = 1, 37

DRe = Pe = 100

DWe = 1.

DT0 = 1.19

DTg = 1.15

DTa = 0.9

DRa = 10

-

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9`Á é Á Åfe Á ³6µ¡éÀ²³ Á e ³À·c Â Á e ³¯·=é¯³À²#¾ ½ ·¶é Á ¶ì ½ ¶´~éÀ·§ê³ Á µ=¿² e µ¡éé `Á é `½ ¶¶ Á ³ô³ Á Ä´ ½ ·¶ ·aé `Á c Â Å ½ ¾ ú ·J¶=c³ Å Á ìê.gµJìì ½ é ½ ·¶µ Â ¶² Å Á ³ ½ ú µ Âú µ Â ú ² Â µ¡é ½ ·J¶¾ h ½ é ` ì ½#z5Á ³ Á ¶3éa£³¯µ Å Á µ Å ê ½ Á ¶3éé Á Åfe Á ³6µ¡é¯²³ Á ³ Á ´ ½ Å Á ¾µ¶ì~³¯µJì ½ µ¡é ½ ·¶§¶² Å ê Á ³6¾ =9`Á ¾ Á ³ Á ¾G² Â é¯¾µ³ Á¶·=é´ ½ ¸ Á ¶ ½ ¶äé `Á e ³ Á ¾ Á ¶Jé4h·³¯¿

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Ren 1 ≤ Re ≤ 100 sÕì ½ Å Á ¶#¾ ½ ·¶ Â Á ¾¯¾ µ Å µ=¿ Á ³ ú ·¶¾Gé¯µ¶JéA

n 0 ≤ A ≤ 0.1 ssµ¶ì³¯µJì ½ µ=é ½ ·¶ ¶² Å ê Á ³6¾

Ran 1 ≤ Ra ≤ 10 s 9`Á a£· Â Â ·h ½ ¶´¶² Å ê Á ³¯¾Dµ=³ Á µ¾¯¾G² Å Á ì

c Á ì Æ

• é `Áú µ e ½ Â Â µ³/g¶² Å ê Á ³Ca = ε = 0.01 ½ ÁJ Aé `Á w Á ê Á ³Ý¶² Å ê Á ³

We = 0.01Re

• é `Á ³6µ=¶#ìé Â ¶² Å ê Á ³Pr = 1 ½T Á é `Á ñÁAúRÂ Á é¶² Å ê Á ³

Pe = Re

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∂u

∂t

+∂

4u

∂x4

=∂

2φ(u)

∂x2

, x ∈ Ω =]0, 1[, t ∈ (0, T ], 0+1 1324 ½ ß ·dµ ±² ¶º¸ Ý Ù ×>± ¶º½ ß ½ ± ¶ ³

∂u

∂x

∣

∣

∣

x=0,1= 0,

∂3u

∂x3

∣

∣

∣

x=0,1= 0, t ∈ (0, T ], 0+1 Ú 2

¸:¶ºd¸:¶.½ ¶½ ß ½ ¸¹ ×T± ¶º½ ß ½ ± ¶

u(x, 0) = u0(x), x ∈ Ω, 051 ® 24 · ´>Ý´

φ(u) = γ(u3 − β2u)

¸:¶ºγ

»β

¸ Ý´ø×T± ¶ ³ïß ¸:¶ ß³ 4 ½ ß ·γ > 0

¼£· ´ , ¸:·¶À.-8½ ¹ ¹ ½¾¸ Ý º ´/3² ¸ ß ½ ± ¶ 0+1 132 ¸ Ý ½ ³©Á³ ¸ ³ ¸ Þ · ´ ¶ ± Á ´ ¶ ± ¹ ±6 ½ × ¸:¹äÁ ± º ´87 ±Ý

Þ ·¸ ³©´+³©´>Þ ¸ Ý ¸ ß ½ ± ¶½ ¶ ×T±± ¹ ½ ¶ 6 µ½ ¶¸ Ý Ù ³©± ¹ ²ß ½ ± ¶ ³Ï³G²#× ·¸ ³ ¸:¹ ¹ ± Ù ³ » 6 ¹¾¸ ³³©Á³ ¸:¶#º Þ± ¹ ÙÁ ´>Ý

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u(x, t)½ ³ ¸ Þ#ÁÝGß²Ý µ#¸ ß ½ ± ¶ ± 71ß · ´×>± ¶ ×>´ ¶ ß©Ý ¸ ß ½ ± ¶ ± 7± ¶ ´± 71ß · ´

Þ ·¸ ³©Á³ ¹ ± µ¸:¹ ´ ½ ³Gß©´ ¶ ×T´ ¸:¶º ² ¶½ /3²´ ¶ Ń³©³ ± 7Ïß · ´³G± ¹ ²ß ½ ± ¶ 7 ±Ý 0+1 132 ·¸Ø ´ µ ´>´ ¶ ³ · ± 4 ¶

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º ´ ¶ ±:ßÁ³ 6J´ ¶ ÁÝ ½ ××T± ¶ ³ïß ¸:¶ ß »b¶ ±:ß ¶ Á×TŃ³©³ ¸ Ý ½ ¹ Ù ß · ´³ ¸Á ´ ¸ ß º½.' ´>Ý´ ¶ ß8±×>×>²Ý©Ý´ ¶ ×T´

/$10 %3254 )5'7698;:<4 >=! #? ´Tß

r

¸¶ºl

µ ´ ½ ¶ ß´6J´>Ý6³ 4 ½ ß ·r ≥ 4

¸:¶#º1 ≤ l ≤ r − 2

A@ ´×T± ¶ ³ ½¾º ´>Ý ¸ 7 ¸:Á½ ¹ Ù± 7Þ ¸ Ý©ß ½ ß ½ ± ¶ ³

0 = x0 < x1 < x2 < ... < xI = 1±7

[0, 1]½ ¶ ß©± ³©² µ½ ¶ ß©´>Ý Ø§¸:¹ ³

Ji = (xi−1, xi)¸¶º ³G´>ß

h = max1≤i≤I

(xi − xi−1) .

¼£· Ý©±J²*6 · ±²ß ß · ´Þ ¸ Þ´>Ý » 4 ´²³©´D

ß± º ´ ¶ ±ß©´ ∂

∂x

¼£· ´ ¶ ±Ý Á ³ ± 7L

2(Ω)»L∞(Ω)¸:¶º

Hs(Ω)

¸ Ý´ º ´ ¶ ±:ß´ ºµÙ‖.‖

»‖.‖

∞

¸:¶º‖.‖s

¼£· ´ ³©´ Á½ûÀÔ¶ ±JÝ Á‖Ds

v‖½ ³ º ´ À

¶ ±:ß´ º.µÙ|v|s, (v, w) =

∫

Ω

vw dx

º ´ ¶ ±ß©Ń³£ß · ´ ½ ¶¶ ´>Ý"ÞÝ©± º ²×Tß8± 7L

2(Ω)7? ´Tß

Shµ ´ ¸ ³©Þ ¸ ×>´±7ß · ´Þ ½ Á×T´ 4 ½ ³©´øÞ± ¹ Ù3¶ ± Á½ ¸¹ ³GÞ ¹ ½ ¶ Á³ Â

Sh =

χ ∈ Cl(Ω), χ

∣

∣

Ji

∈ Pr−1(Ji), i = 1, ..., I ; Dχ(0) = Dχ(1) = 0

,

4 · ´>Ý´Pr−1(Ji)

º ´ ¶ ±:ß©Ń³ß · ´³©´TßÖ± 71Þ± ¹ Ù3¶ ± Á½ ¸¹ ³%± ¶Ji

± 7 º ´6Ý´>´ ¹ Ń³©³%±Ý%´/3² ¸:¹ ß©±r − 1.

? ´TßSh ⊂ H

2(Ω)» 4 · ÁÝ©´

H2(Ω) =

u ∈ H2(Ω),

∂u

∂x

= 0, x ∈ ∂Ω

,

¸:¶ºdº ´ ¶ ±:ß©´Sh = Sh ∩ χ, (χ, 1) = 0 .

z¶¸ ß²Ý ¸¹ ¸:¹ ´>Ý ½ ¶d¸ ÞÞÝ± ½ Áª¸ ß ½ ± ¶ ß± 0+1 12 ½ ³ ÂB¶ºuh ∈ Sh

³ ¸ ß ½ ³+7 Ù½ ¶ 6(

∂uh

∂t

, χ

)

+ (D2uh, D

2χ) = (φ(uh), D2

χ), ∀χ ∈ Sh, 0 Ú 12

:¡ >§ J©¦ §¦§T¦§ø ¥ A¥ ø¡¡L :© ¢>%§¦§ ® 1 à4 ½ ß ·

uh(0) = u0h, 0 Ú Ú 24 · ´>Ý´

u0h ∈ Sh

½ ³ ¸:¶¸ ÞÞÝ©±JÞÝ ½¾¸ ß´ ¸ ÞÞÝ± ½ Áª¸ ß ½ ± ¶ ß©±u0.@ ´ ½ ¶ ß©Ý± º ²×T´8ß · ´³G±«× ¸:¹ ¹ ´ º ´ ¹ ¹ ½ Þß ½ ×8ÞÝ©±ïÁ×ß ½ ± ¶

Ph : H2(Ω) → Sh

º ´ ¶ ´ ºªµÙß · ´ 7 ± ¹ ¹ ± 4 ½ ¶ 6°ÞÝ± µ¹ ´ Á 0 ³GÁ´ » ´ 6 ¿%¹ ¹ ½ ±ßGß ´Tß ¸¹U7 11 2 Â 7 ±JÝ

v ∈ H2(Ω)

» ß · ´ 7 ² ¶ ×Tß ½ ± ¶Phv

½ ³£ß · ´ø² ¶½ /3²´ø³©± ¹ ²ß ½ ± ¶ ± 7

(D2(Phv − v), D2χ) = 0, ∀χ ∈ Sh,

(Phv − v, 1) = 0.

0 Ú ® 2¼£· ´«´ ½ ³ïß´ ¶ ×>´ø± 7 ¸ ² ¶½ /3²´

Phv

³ ¸ ß ½ ³57 Ù3½ ¶ 6 0 Ú ® 2 7 ± ¹ ¹ ± 4 ³$7 Ý©± Á ß · ´ ?¸ÀÔ½ ¹ 6Ý ¸:Á¼£· ´>±JÝ©´ Á ¸:¶º ß · ´ + Ý ½ ´ º Ý ½ × · ³ À ± ½ ¶ × ¸ Ý ½ ¶ ´ /3² ¸¹ ½ ß Ù

‖w‖2 ≤ C(|w|2 + |(w, 1)|), ∀w ∈ H2(Ω).

@ ´ µ ´6 ½ ¶ 4 ½ ß · ß · ´ 7 ± ¹ ¹ ± 4 ½ ¶ 6 ÝÁ³©² ¹ ß³ º ²´ß©± ¿+¹ ¹ ½ ±:ß©ß ¸:¶º · ´ ¶ 6 1 1 æãå#é ¥

Ph

J !%6"$# 0 Ú ® 2&%(' ¥%*)§+T¦§v ∈ H

r(Ω) ∩ H2(Ω),

2∑

j=0

hj |v − Phv|j ≤ Ch

r‖v‖r, 0 Ú ¬ 2§'

v ∈ H2(Ω),

¥>

‖v − Phv‖∞

≤ Chr‖v‖W r

∞(Ω) . 0 Ú à 2

,.-0/ #æ /21 #é #N¦> ¥ :¥T¦:¡L¦§u(t)

¦3 0+1 132 ¾854 >T#U¡6# ©Ô¨¡§7T¦§D¨8)>

T > 0: ¥%§ ¥ø>¦§¡ûU¦:¦3 0 Ú 132 §¾9! T

‖uh(t)‖∞

≤ CT , 0 ≤ t ≤ T. 0 Ú ) 29

u0h = Phu0,¥T

‖u(t) − uh(t)‖∞

≤ CT (u)hr, ∀t ∈ [0, T ]. 0 Ú 2

:" :; ¼£· ´ ¸ ³©³©² Á Þß ½ ± ¶ 0 Ú ) 2 ½ ³ ¶ ±:ß ¸ Ý©Ń³ïßÝ ½ ×ß ½ ± ¶1=<ÖÙ¸ ³Gß ¸:¶#º¸ Ý ºª¸ Ý 6² Á ´ ¶ ß¸:¶º ²³ ½ ¶ 6´>ÝÝ±Ý Á³Gß ½ Á¸ ß©Ń³ 0 Ú 2 4 ´ Áª¸Ù ï²³Gß ½ 7 Ù 0 Ú ) 2 ¸ Þ#±3³ïß´>Ý ½ ±JÝ ½ 7 ±Ý ¸¶3Ù

T > 0

0 ¯Á´ ¼£· ± Á A´ 1 » ÞÞ ÖÚ 1 ®?> Ú 1 ¬ 2 @ ´³ ·¸¹ ¹¶ ´>´ º ß · ´ ³ ¸Á ´ ¸ ³³G² Á Þß ½ ± ¶ ½ ¶³GŃ×ß ½ ± ¶ ³ ® » ¬ ¸:¶ºà

¼£· ´ , Ý ¸:¶ À8½ ×T± ¹ ³©± ¶ 7 ² ¹ ¹ Ùøº½ ³©×>Ý©´>ß©´ ¸ ÞÞÝ± ½ Áª¸ ß ½ ± ¶½ ³ º ´ ¶ ´ ºø½ ¶ ß · ´ 7 ± ¹ ¹ ± 4 ½ ¶ 64 ¸Ù5Â ¶º.¸ ³©´ /3²´ ¶ ×T´

UnN

n=0⊂ Sh

³ ¸ ß ½ ³+7 Ù½ ¶ 6

(∂tUn, χ) + (D2

Un− 1

2 , D2χ) = (φ(Un− 1

2 ), D2χ), ∀χ ∈ Sh,

U0 = u0h.

0 Ú 24 · ´>Ý´

u0h ∈ Sh

½ ³ ¸ ³G² ½ ß ¸µ¹ ´ ¸ ÞÞÝ± ½ Áª¸ ß ½ ± ¶ ß©±u0

»U

n½ ³ß · ´ ¸ ÞÞÝ± ½ Áª¸ ß ½ ± ¶

½ ¶Sh

± 7u(t)

¸ ßt = t

n = nk

¸:¶#ºk =

T

N

º ´ ¶ ±:ßÁ³Ûß · ´]³ ½- ´ ±7Dß · ´ß ½ Á ´º½ ³×TÝ´Tß ½-N¸ ß ½ ± ¶1¶ 0 Ú 2 4 ´ ·¸Ø ´²³©´ º ß · ´ ¶ ±:ß ¸ ß ½ ± ¶

® 1 ) :

∂tUn =

1

k

(Un − Un−1), U

n− 1

2 =1

2(Un + U

n−1).

+ ±Ý8×>± ¶ ß ½ ¶ ²±²#³ 7 ² ¶ ×ß ½ ± ¶u(t)

» 4 ´ 4 Ý ½ ß©ú

n− 1

2 = u(tn−1

2 )

¶ ß ·½ ³ ¸ ÝGß ½ × ¹ ´ » ß · ´ Áª¸:½ ¶ 6J± ¸¹°½ ³ß©± ¸ ÞÞÝ± ½ Áª¸ ß´ ß · ´ ³©± ¹ ²ß ½ ± ¶ ³± 7 ß · ´, ¸:·¶À.-8½ ¹ ¹ ½¾¸ Ý º ´/3² ¸ ß ½ ± ¶ÛµÙ 7 ² ¹ ¹ ÙÛº½ ³×TÝ´Tß©´ ¶½ ß´«´ ¹ ´ Á ´ ¶ ß8³× · ´ Á ´ #¼£· ´«³ïß ¸:¶º¸ Ý º ¸:¹ ´>Ý ½ ¶Á ´Tß · ± º'½ ³%²³G´ º 7 ±Ý ¸ ÞÞÝ± ½ Áª¸ ß ½ ± ¶½ ¶ ³©Þ ¸ ×>´ ¸:¶º , Ý ¸:¶ À8½ ×T± ¹ ³©± ¶À ß Ù Þ´³GŃ×T± ¶#º ±Ý º ´>Ý ¸ ×>×>²Ý ¸ ß©´ º½ ³×TÝ´Tß ½-N¸ ß ½ ± ¶ª½ ³ ²³©´ º 7 ±JÝ ¸ ÞÞÝ± ½ Áª¸ ß ½ ± ¶ ±7Ïß ½ Á ´ º ÁÝ ½ Ø§¸§Àß ½ Ø Á³ >@ ´ 4 ½ ¹ ¹ ÞÝ± Ø ´ß ·¸ ß ß · ´«³× · ´ Á ´ ½ ³ ×T± ¶3Ø ´>Ý 6´ ¶ ß

$ ) >4 &'74 6# &)$% 4 4 é#é Uã>å / è /¶ ß ·½ ³8³©Á×ß ½ ± ¶ 4 ´«³ ·¸:¹ ¹ ÞÝ± Ø ´ß · ´«´ ½ ³ïß´ ¶ ×>´ø± 7 ¸ ³G´/3²´ ¶ ×>´

UnN

n=0

³ ¸ ß ½ ³57 Ù3½ ¶ 6ÞÝ©± µ¹ ´ Á 0 Ú 2 7+ ±Ýß ·½ ³ » 4 ´°³ ·¸¹ ¹ ²³©´Dß · ´7 ± ¹ ¹ ± 4 ½ ¶ 6 Ø§¸ Ý ½¾¸:¶ ß±7ß · ´ 4 ´ ¹ ¹ À ¶ ± 4 ¶ ´ º Þ#± ½ ¶ ß£ß · ´>±Ý´ Á ± 7 < Ý±² 4 ´>Ý Ú» ® / 1 1 ç é T

H

"6$!+": >¦:A§¡ J'

¥5T 5©¦N§>(., .)H %

:¦§

‖.‖H .

>ä¥ 6g : H → H

"66¦:¦§ N¦>¥TJ¾α > 066¥'¥%§

(g(Z), Z)H ≥ 0T¦:$§¡¡

Z ' ¥‖Z‖H = α

¤5¥Td¥TJ¾Z

∗ ∈ H,66¥. ¥ :g(Z∗) = 0

§5‖Z∗‖ ≤ α.

@ ´«³ ·¸:¹ ¹Ï¶ Á´ º ß · ´ ¸ ² ½ ¹ ½¾¸ Ý Ù Á³Gß ½ Á¸ ß©Ń³ Â / 1 1 ç1é ¦§

v ∈ Sh %('¥%*)§

|v|21 ≤1

γβ2|v|22 +

γβ2

4‖v‖2

. 0 ® 12Ö©¦N¦3 + ±JÝ

v ∈ Sh

» 4 ´ÞÝ± Ø ´ ½ ÁÁ ´ º½ ¸ ß©´ ¹ Ù

−(D2v, v) = |v|21,

7 Ý©± Á 4 ·½ × ·1» 4 ´ ·#¸Ø ´|v|21 ≤ |v|2‖v‖. 0 ® Ú 2

³ ½ ¶ 6°ß · ´ ½ ¶ ´ /3² ¸:¹ ½ ß Ùab ≤

a2

γβ2

+γβ

2

4b2» 4 ´ÞÝ± Ø ´ß · ´ ¹ ´ ÁÁª¸

/ 1 1 ç é ¦§v ∈ Sh

¥T¥¦§¡ §

(φ(v), D2v) ≤ γβ

2|v|21. 0 ® ® 2Ö©¦N¦3 + ±JÝ

v ∈ Sh

» 4 ´ ·¸Ø ´

(φ(v), D2v) = −(Dφ(v), Dv) = −(φ′(v)Dv, Dv),

²³ ½ ¶ 6ß · ´ º ´ ¶½ ß ½ ± ¶ ± 7φ4 ´ #¶º

φ′(v) = γ(3u

2−β2) ≥ −γβ

2,4 ·½ × · ×>± Á Þ ¹ ´TßÁ³

ß · ´øÞÝ±± 7

:¡ >§ J©¦ §¦§T¦§ø ¥ A¥ ø¡¡L :© ¢>%§¦§ ® 1 ¼£· ´%ÞÝ±± 7±7´ ½ ³ïß´ ¶ ×>´ ±7

UnN

n=0

ÞÝ©±×>´>´ º ³ ½ ¶ ¸¶«½ ¶#º ²×ß ½ Ø ´ 4 ¸Ù:µØ½ ±J²³ ¹ ÙU

0´ ½ ³ïß6³ % ±Ý´>± Ø ÁÝ »ä¸ ³³©² Á ´.ß ·¸ ß

Ujn−1

j=0

´ ½ ³ïß6³ + ±ÝZ ∈ Sh

»%º ´ ¶ ´g :

Sh −→ Sh

µ3Ù

(g(Z), χ) := (Z−Un−1

, χ)+k

2

(

(D2Z, D

2χ)−(φ(Z), D2

χ))

, ∀χ ∈ Sh. 0 ® ¬ 2¯²× · ¸DÁ¸ Þª´ ½ ³ïß6³ µÙ ß · ´ ½ Á³ - Ý´>ÞÝÁ³©´ ¶ ß ¸ ß ½ ± ¶ ß · ´>±Ý´ Á.»

g

½ ³%± µØ3½ ±²#³ ¹ Ù ×>± ¶ ß ½ ¶À²±²³ ¼ ¸ ½ ¶ 6

χ = Z

½ ¶ 0 ® ¬ 2 ¸:¶º ²³ ½ ¶ 6 0 ® ® 2 » 4 ´± µ ß ¸½ ¶

(g(Z), Z) ≥ ‖Z‖2 − (Un−1, Z) +

k

2|Z|22 −

kγβ2

2|Z|21.

< Ù ? ´ ÁÁª¸'Ú» 4 ´ ¶º

(g(Z), Z) ≥ ‖Z‖2 − ‖Un−1‖‖Z‖ −kγ

2β

4

8‖Z‖2

.

¼£· ´>Ý´ 7 ±JÝ©´ »

(g(Z), Z) ≥ ‖Z‖(

(1 −kγ

2β

4

8)‖Z‖ − ‖Un−1‖

)

.

- ´ ¶ ×T´87 ±Ýk <

8

γ2β

4

»‖Z‖ =

8

8 − kγ2β

4‖Un−1‖ + 1

» 4 ´ ·¸Ø ´(g(Z), Z) > 0

¼£· ´ ¶ª½ ß 7 ± ¹ ¹ ± 4 ³ 7 Ý± Á ? ´ ÁÁª¸ 1 ß ·¸ ß ß · ÁÝ©´"´ ½ ³Gß³

Z∗ ∈ Sh

³G²#× · ß ·¸ ßg(Z∗) = 0

ß ½ ³£´ ¸ ³ ½ ¹ Ù ³©´>´ ¶ ß ·#¸ ß

Un = 2Z

∗ − Un−1

³ ¸ ß ½ ³ Á³ 0 Ú 2

é Ïé 7 / / ãNã ³³G² Á ´Öß ·#¸ ßäß · ´ ³©± ¹ ²ß ½ ± ¶

u

± 7 0+1 12 ½ ³ä³G²'× ½ ´ ¶ ß ¹ Ù Ý©´ 6² ¹¾¸ Ý ¸:¶º ß ·#¸ ßäß · ´ ³©± ¹ ²ß ½ ± ¶± 7 0 Ú 2 ³ ¸ ß ½ ³ Á³

‖Un‖∞

≤ c0, n = 0, 1, . . . , N. 0 ® à 2+ ±Ý ² ¶½ /3²´ ¶ Á³³ » ³G²ÞÞ#±3³G´ß ·#¸ ß

Vn ∈ Sh

¸:¶#ºV

0 = u0h

³ ¸ ß ½ ³+7 Ù(∂tV

n, χ) + (D2

Vn− 1

2 , D2χ) = (φ(V n− 1

2 ), D2χ), ∀χ ∈ Sh, 0 ® ) 2

¸:¶º‖V n‖

∞≤ c0, n = 0, 1, . . . , N.

# ´ ¶ ±:ß ½ ¶ 6E

i = Ui − V

i» 4 ½ ß ·

E0 = 0

» 7 Ý± Á 0 Ú 2 ¸:¶º 0 ® ) 2 » 4 ´ ·#¸Ø ´ 7 ±JÝχ ∈ Sh

(∂tEn, χ) + (D2

En− 1

2 , D2χ) = (φ(Un− 1

2 ) − φ(V n− 1

2 ), D2χ). 0 ® 2

± 4 » ³©²ÞÞ±J³ ½ ¶ 6E

n−1 = 0¸¶º × · ±±J³ ½ ¶ 6

χ = En− 1

2

½ ¶ 0 ® 2 » 4 ´± µ ß ¸:½ ¶

1

2k

(‖En‖2−‖En−1‖2)+|En− 1

2 |22 ≤1

4‖φ(Un− 1

2 )−φ(V n− 1

2 )‖2+|En− 1

2 |22. 0 ® 2 ³ ½ ¶ 6°ß · ´«×T± ¶ ß ½ ¶ ²±J²³ º½' ÁÝ©´ ¶ ß ½¾¸:µ½ ¹ ½ ß Ù ±7

φ(.)» 4 ´ 6´>ß

® 1 :

‖φ(Un− 1

2 ) − φ(V n− 1

2 )‖ ≤ C‖En− 1

2 ‖, 0 ® ! 24 · ´>Ý´

C

½ ³ ¸ ×T± ¶ ³ïß ¸:¶ ß º ´>Þ´ ¶º ´ ¶ ß"± ¶c0

<ÖÙ 0 ® 2 ¸¶º 0 ® ! 2 » 4 ´ #¶º

1

2k

(‖En‖2 − ‖En−1‖2) ≤C

2

4‖En− 1

2 ‖2 ≤C

2

8(‖En‖2 + ‖En−1‖2),

7 Ý©± Á 4 ·½ × ·1» 7 ±Ýk

³©²'× ½ ´ ¶ ß ¹ Ù ³ Áª¸:¹ ¹U» 4 ´ 6´Tß

‖En‖ ≤ λ‖En−1‖, λ =

4 + kC2

4 − kC2

1

2

.

@ ´«³©´>´ß ·#¸ ßE

n = 0¸:¶#º ß ·½ ³ ×T± Á Þ ¹ ´TßÁ³£ß · ´øÞÝ±± 7b±7ß · ´ø² ¶½ /3²´ ¶ Ń³©³

*) 256 *)+! ! = 4 ! 4 4 '74 6 >4¶ ß ·½ ³d³GŃ×ß ½ ± ¶1» 4 ´Ń³ïß ½ Áª¸ ß©´ ß · ´´>ÝÝ©±JÝ'± 7øß · ´]³G± ¹ ²ß ½ ± ¶ ± 7øß · ´ 7 ² ¹ ¹ Ùpº½ ³©×>Ý©´>ß©´ÞÝ©± µ¹ ´ Á 0 Ú 2 >@ ´ø²³©´ß · ´«³Gß ¸¶º¸ Ý º ´>ÝÝ©±JÝ º Á×T± Á Þ#±3³ ½ ß ½ ± ¶ 4 ½ ß ·

un = u(tn)

Â

Un − u

n = (Un − Phun) + (Phu

n − un) = θ

n + ρn. 0 ¬ 12

,.-0/ #æ /21 Ïé ># §66 Ö ¥ §5©Ô¨¡§#"§66 ¦§ 0 ® à 2 ¾+ §¾9! 6 % ' ¥>©U

n§5

u

:© ¥ >¦§¡L¦§ ¦3 0 Ú 2 : 0+1 12&% ©T6>U8)§T¡6# N¦> ¥%§£¥T¦:¡L¦§

u

¾654 >T#U¡ #ª¨¡: 9ø ¥« §¡ *§ ª :U¾9 #' ¥° :Ô

‖U0 − u0‖ ≤ Chr, 0 ¬ Ú 2

¥T ' ¥%*)§ T¦:54 >T#U¡6# §¡¡k

¥%§

‖Un − u(tn)‖ ≤ C(hr + k2), 0 ¬ ® 2

' ¥>©C

¾;d6¦§ §#%JÔ>> ¦3h

§5k.

Ö©¦N¦3 ¯ ½ ¶ ×T´Dß · ´°Á³Gß ½ Áª¸ ß´ ±7ρ

n 7 ± ¹ ¹ ± 4 ³7 Ý©± Á 0 Ú ¬ 2 »½ ß ½ ³´ ¶ ±J²*6 · ß©±dÁ³Gß ½ Á¸ ß©´θ

n ³ ½ ¶ 6ß · ´ º ´ ¶½ ß ½ ± ¶ ± 7 ß · ´«´ ¹ ¹ ½ Þß ½ ×ÞÝ± ïÁ×Tß ½ ± ¶

Phv

½ ¶ 0 Ú ® 2 4 ½ ß · ´ /3² ¸ ß ½ ± ¶ ³(1.1)

¸¶º(2.8)

» 4 ´8± µ ß ¸½ ¶ ß · ´ 7 ± ¹ ¹ ± 4 ½ ¶ 6ø´ /3² ¸ ß ½ ± ¶1» 4 ·½ × · ½ ³ Ø¸¹ ½¾º 7 ±JÝ ¸:¹ ¹χ ∈ Sh³G²× · ß ·#¸ ß

(χ, 1) = 0Â

(∂tθn, χ) + (D2

θn− 1

2 , D2χ) = (φ(Un− 1

2 ) − φ(un− 1

2 ), D2χ)

− (∂tPhun − u

n− 1

2

t , χ) − (D2(u

n + un−1

2− u

n− 1

2 ), D2χ) . 0 ¬ ¬ 2

< Ù ß ¸ ½ ¶ 6χ = θ

n− 1

2

» 4 ´± µ ß ¸½ ¶

(∂tθn, θ

n− 1

2 ) + |θn− 1

2 |22 ≤ I |θn− 1

2 |2 + J ‖θn− 1

2 ‖ + K |θn− 1

2 |2,

4 · ´>ÝÍ = ‖φ(Un− 1

2 ) − φ(un− 1

2 )‖, J = ‖∂tPhun − u

n− 1

2

t ‖,

:¡ >§ J©¦ §¦§T¦§ø ¥ A¥ ø¡¡L :© ¢>%§¦§ ® 1 !

K = ‖D2

(

un + u

n−1

2− u

n− 1

2

)

‖.

¼£·½ ³ Ù3½ ´ ¹¾º ³

1

2k

(‖θn‖2 − ‖θn−1‖2) + |θn− 1

2 |22 ≤1

2I2 +

1

2|θn− 1

2 |22 +1

2J

2 +1

2‖θn− 1

2 ‖2

+1

2K

2 +1

2|θn− 1

2 |22.

¼£· ´ ¸:µ ± Ø ´ ½ ¶ ´ /3² ¸:¹ ½ ß Ù'µ Ń×T± Á Ń³

1

k

(‖θn‖2 − ‖θn−1‖2) ≤ ‖θn− 1

2 ‖2 + I2 + J

2 + K2. 0 ¬ à 2

@ ´ ·¸Ø ´"ß · ´ 7 ± ¹ ¹ ± 4 ½ ¶ 6Ń³ïß ½ Áª¸ ß´ 0 ³©´>´ ¸:¹ ³©± Á Ý ¸:¶½9 1 ¬ » 1 à 2I = ‖φ(Un− 1

2 ) − φ(un− 1

2 )‖ ≤ C‖Un− 1

2 − un− 1

2 ‖,

· ´>ÝĆ

½ ³ ¸ ×T± ¶ ³ïß ¸:¶ ß º ÁÞ#´ ¶º ´ ¶ ß£± ¶c0

¸¶º‖un−1

2 ‖∞

¼£· ´×>± ¶ ß ½ ¶ ²±²#³ º½' ÁÝ©´ ¶Àß ½¾¸:µ½ ¹ ½ ß Ù ±7

φ(·)·¸Ø ´ µ ´>´ ¶ ²#³G´ º ß± º ´>Ý ½ Ø ´ß ·½ ³ ½ ¶ ´ /3² ¸¹ ½ ß Ù

¼£· ´>Ý´ 7 ±JÝ©´ 4 ´ 6J´Tß£ß · ´øŃ³ïß ½ Áª¸ ß©Á³

‖Un− 1

2 − un− 1

2 ‖ ≤ ‖Un− 1

2 − Ph(u

n + un−1

2)‖ + ‖Ph(

un + u

n−1

2) − u

n− 1

2 ‖

≤ ‖θn− 1

2 ‖+ I1.

I1 = ‖Ph

(

un + u

n−1

2

)

− un− 1

2 ‖ ≤‖ρn‖ + ‖ρn−1‖

2+ ‖

un + u

n−1

2− u

n− 1

2 ‖

=‖ρn‖ + ‖ρn−1‖

2+

1

2‖

∫ tn−

1

2

tn−1

(s − tn−1)utt(s)ds −

∫ tn

tn−

1

2

(s − tn)utt(s)ds‖

≤‖ρn‖ + ‖ρn−1‖

2+ Ck

3

2

(

∫ tn

tn−1

‖utt(s)‖2ds

)1

2

.

¯ ½ Á½ ¹ ¸ Ý ¹ Ù 4 ´ 6´Tß ß · ´øÁ³Gß ½ Áª¸ ßÁ³

J = ‖∂tPhun − u

n− 1

2

t ‖ ≤ ‖∂tPhun − ∂tu

n‖ + ‖∂tun − u

n− 1

2

t ‖

=1

k

‖ρn − ρn−1‖+

1

2k

‖

∫ tn−

1

2

tn−1

(s − tn−1)2uttt(s)ds +

∫ tn

tn−

1

2

(s − tn)2uttt(s)ds‖

≤ k−

1

2

(

∫ tn

tn−1

‖ρt(s)‖2ds

)1

2

+ Ck

3

2

(

∫ tn

tn−1

‖uttt(s)‖2ds

)1

2

,

K = ‖D2

(

un + u

n−1

2− u

n− 1

2

)

‖ ≤ Ck

3

2

(

∫ tn

tn−1

‖D2utt(s)‖

2ds

)1

2

.

® Ú : ³ ½ ¶ 6 0 ¬ à 2 4 ½ ß · ß · ´øŃ³ïß ½ Áª¸ ß©Á³£±7

I, J

¸:¶ºK

» 4 ´± µ ß ¸:½ ¶

1

k

(‖θn‖2 − ‖θn−1‖2) ≤ C[ ‖θn− 1

2 ‖2 + ‖ρn‖2 + ‖ρn−1‖2

+1

k

∫ tn

tn−1

‖ρt(s)‖2ds + k

3

∫ tn

tn−1

(‖uttt(s)‖2 + ‖utt(s)‖

2 + ‖D2utt(s)‖

2)ds ],

¸:¶ºÛ· ´ ¶ ×T´‖θn‖2 − ‖θn−1‖2 ≤ Ck

(

‖θn− 1

2 ‖2 + Rn

)

,

4 · ´>Ý´ß · ´ ¹¾¸ ß©ß©´>Ý ´/3² ¸:¹ ½ ß ÙÛº ´ ¶ Á³Rn

¯± 4 ´ÞÝ©± Ø ´ º ß ·¸ ß

(1 − Ck)‖θn‖2 ≤ (1 + Ck)‖θn−1‖2 + CkRn,

¸:¶º 7 ±Ý8³ Áª¸:¹ ¹k ≤ k0

ß · ´«³Gß ¸µ½ ¹ ½ ß Ù Á³Gß ½ Áª¸ ß´ ½ ³ Ø§¸:¹ ½ º1Â

‖θn‖2 ≤(1 + Ck

1 − Ck

)

‖θn−1‖2 + CkRn.

7ß©ÁÝ8Ý´>Þ´ ¸ ß´ ºd¸ ÞÞ ¹ ½ × ¸ ß ½ ± ¶1» ß ·½ ³ Ù3½ ´ ¹¾º ³

‖θn‖2 ≤(1 + Ck

1 − Ck

)n

‖θ0‖2 + Ck

n∑

j=1

(1 + Ck

1 − Ck

)n−j

Rj ,

±Ý

‖θn‖2 ≤ C‖θ0‖2 + Ck

n∑

j=1

Rj . 0 ¬ ) 2 ±:ß ½ ¶ 6 ß ·¸ ß

‖θ0‖2 ≤ ‖ρ0‖2 + ‖U0 − u0‖2, 0 ¬ 2

¸:¶ºÛµÙ 0 Ú ¬ 2 » 4 ´± µ ß ¸:½ ¶

‖ρn‖ ≤ Chr(‖u0‖r +

∫ tn

0

‖ut(s)‖rds). 0 ¬ 2 ³ ½ ¶ 6 0 ¬ ) 2 » 0 ¬ 2 ¸¶º 0 ¬ 2 » 4 ´ ¶#º

‖θn‖2 ≤ C

(

‖U0 − u0‖2 + h

2r(

‖u0‖2r +

∫ T

0

‖ut(s)‖2r ds

)

+ k4

∫ T

0


2 + ‖D2utt(s)‖

2) ds

)

.

ß 7 ± ¹ ¹ ± 4 ³7 Ý± Á 0 ¬ Ú 2 ß ·#¸ ß8Á³Gß ½ Áª¸ ß´ 0 ¬ ® 2 · ± ¹ º ³

´ ß 4 ´ 4 ½ ¹ ¹ ²³©´ß · ´ "½ Ý´ ¶µ ´>Ý 6 ½ ¶ ´ /3² ¸¹ ½ ß Ù 1 / 1 1 çé ¦§ j

m

≤ a ≤ 1 % 1

p

=j

n′

+ a(1

r

−m

n′

) + (1 − a)1

q

% ¥>©¥¦§¡

‖Djv‖Lp(Ω) ≤ C

(

‖Dmv‖a

Lr(Ω)‖v‖1−aLq(Ω)

+ ‖v‖Lq(Ω)

)

,

' ¥>©Ω

¾ "6¦:6Û¦ §Û : n′

:¡ >§ J©¦ §¦§T¦§ø ¥ A¥ ø¡¡L :© ¢>%§¦§ ® Ú 1,.-0/ #æ /21 é >

Un"6° ¥ >¦§¡L¦§ ¦ 0 Ú 2 N¦>° ¥ § ¥ >¦:¡LU¦:]¦3

0+1 12 ¾54 T>¡ # ¦A¦:¥§ ¥D §¡*§ '¾D!+56

u0h = Phu0. 0 ¬ ! 2¤¥T

%T¦§

k

654 >T#U¡ # §¡¡%

¥ TU §

max0≤n≤N

‖Un − u(tn)‖ ≤ C(hr + k2)

¥¦§¡ % '

¥T©C

¾; 6¦§ :%5>JT#£¦ h

:k.

Ö©¦N¦3 ¯´>ßGß ½ ¶ 6χ = ∂tθ

n½ ¶ 0 ¬ ¬ 2 » 4 ´± µ ß ¸½ ¶

‖∂tθn‖2 + |∂tθ

n|22 ≤ I |∂tθn|2 + J ‖∂tθ

n‖ + K |∂tθn|2

≤ I2 + 1

4|∂tθ

n|22 + 1

4J

2 + ‖∂tθn‖2 + K

2 + 1

4|∂tθ

n|22 .

¼£· ´ ¸:µ ± Ø ´ ½ ¶ ´ /3² ¸:¹ ½ ß Ù 6 ½ Ø Ń³%ß · ´øÁ³Gß ½ Áª¸ ß´

1

k

(|θn|22 − |θn−1|22) ≤ C(I2 + J2 + K

2). 0 ¬ 1 2 ³ ½ ¶ 6°ß · ´øÁ³Gß ½ Áª¸ ß©Ń³ ± 7

I, J

¸¶ºK

4 ½ ß · 0 ¬ ® 2 » 4 ´± µ ß ¸:½ ¶

|θn|22 − |θn−1|22k

≤ C

(

C(u)(hr + k2)2 + ‖ρn‖2 + ‖ρn−1‖2 +

1

k

∫ tn

tn−1

‖ρt(s)‖2ds

+ k3

∫ tn

tn−1

(

‖uttt(s)‖2 + ‖utt(s)‖

2 + ‖D2utt(s)‖

2)

ds

)

.

- ´ ¶ ×T´ß · ´ 7 ± ¹ ¹ ± 4 ½ ¶ 6³ïß ¸:µ½ ¹ ½ ß Ù'½ ¶ ´ /3² ¸¹ ½ ß Ù

|θn|22 ≤ |θn−1|22 + C(u)k(hr + k2)2 + CkRn

½ ³ Ø§¸:¹ ½¾ºÏ» 4 · ´>Ý´ß · ´ ½ ¶ ´ /3² ¸¹ ½ ß Ù'¸:¹ ³G± º ´ ¶ Á³Rn

*- ´ ¶ ×T´ »µÙ Ý©ÁÞ#´ ¸ ß©´ º ¸ ÞÞ ¹ ½ × ¸ ß ½ ± ¶± 7 ß · ´ø± µ ß ¸½ ¶ ´ º Ń³ïß ½ Áª¸ ß©´ ¸:¶º 0 ¬ ! 2 4 ´ 6J´Tß

|θn|22 ≤ T C(u)(hr + k2)2 + Ck

n∑

j=1

Rj .

¼£· ´>Ý´ 7 ±JÝ©´ »

|θn|22 ≤ C

(

C(u)(hr + k2)2 + k

n∑

j=1

‖ρj‖2 +

∫ T

0

‖ρt(s)‖2ds

+ k4

∫ T

0


2 + ‖D2utt(s)‖

2) ds

)

.

± 4 » ²³ ½ ¶ 6 0 Ú ¬ 2 » 4 ´ø×>± ¶ × ¹ ² º ´ß ·¸ ß » 7 ±Ý8³©± Á ´ø×T± ¶ ³Gß ¸:¶ ßC = C(u, T )

® ÚÚ :

|θn|2 ≤ C(hr + k2), 0 ≤ n ≤ N, 0 ¬ 1132

¸:¶ºÏ»· ´ ¶ ×>´ø²³ ½ ¶ 6 0 ¬ ® 2 » 0 ¬ 1132 ¸¶º 0 ® Ú 2 » 4 ´± µ ß ¸½ ¶

|θn|1 ≤ C(hr + k2), 0 ≤ n ≤ N. 0 ¬ 1 Ú 2

ÞÞ ¹ Ù½ ¶ 6 0 ¬ ® 2 ¸¶º 0 ¬ 1 Ú 2 »½ ß 7 ± ¹ ¹ ± 4 ³ 7 Ý± Á ? ´ ÁÁª¸ ¬ 4 ½ ß ·p = ∞

ß ·¸ ß

‖θn‖∞

≤ C(hr + k2), 0 ≤ n ≤ N. 0 ¬ 1 ® 2

¼ ± 6´Tß · ´>ÝÖß · ´øÁ³Gß ½ Áª¸ ß©Ń³ 0 Ú à 2 ¸:¶º 0 ¬ 1 ® 2 ³ · ± 4 ß · ´ ¼£· ´>±JÝ©´ Ád

$ ) 4 6 ") 4 # 698 4 )5 :<4 = ! #¼£· ´ ¸:µ ± Ø ´ Á ´Tß · ± º]·¸ ³øß · ´ º½ ³ ¸ºØ§¸¶ ß ¸ 6J´ ß ·¸ ß ¸¶ ± ¶¹ ½ ¶ ´ ¸ Ý«³ Ù ³Gß©´ Ár·¸ ³ß©± µ ´³G± ¹ Ø ´ ºd¸ ß ´ ¸ × · ß ½ Á ´«³Gß©´>Þ %+ ±JÝ£ß ·½ ³ Ý´ ¸ ³©± ¶ 4 ´«³ ·¸¹ ¹ ×>± ¶ ³ ½ º ´>Ý ¸°¹ ½ ¶ ´ ¸ Ý ½- ´ ºÛÁ ± º½ À × ¸ ß ½ ± ¶ ± 7ß · ´ Á ´>ß · ± ºd½ ¶ 4 ·½ × · ß · ´ ¸ Ý 6² Á ´ ¶ ß

f

½ ³ ± µ ß ¸½ ¶ ´ ºdµ3Ù ´ ß©Ý ¸ Þ± ¹¾¸ ß ½ ± ¶7 Ý©± Á

Un−1

¸:¶ºU

n−2»½ ´

Un =

3

2U

n−1 −1

2U

n−2» 7 ±Ý

n ≥ 2,

(∂tUn, χ) + (D2

Un− 1

2 , D2χ) = (φ(Un), D2

χ), ∀χ ∈ Sh. 0 à 12¼£·½ ³ Á ´Tß · ± º 4 ½ ¹ ¹ Ý´ /3² ½ Ý©´ ¸ ³©´>Þ ¸ Ý ¸ ß©´øÞÝ©Ń³©×>Ý ½ Þß ½ ± ¶ 7 ±JÝ"× ¸¹ ×>² ¹ ¸ ß ½ ¶ 6

U1 0 ³GÁ´ » ´ 6

¼£· ± Á >´ 1 » ÞÞ Ú 1 > ÚÚJÚ 2 @ ´ ¸:¶#¸:¹ Ù ³©´ ¸ ÞÝ©´ º½ ×ß©±JÝ ×>±ÝÝ©Ń×ß©±JÝ Á ´Tß · ± º 7 ±Ý%ß ·½ ³Þ²ÝÞ#±3³G´ » 4 ·½ × ·.½ ³ 7 ±Ý Á ² ¹¾¸ ß´ ºd¸ ³ 7 ± ¹ ¹ ± 4 ³ Â

U0 = u0h,

(

U1,0 − U

0

k

, χ

)

+(

D2(U

1,0 + U0

2

)

, D2χ

)

= (φ(U0), D2χ),

(∂tU1, χ) + (D2

U

1

2 , D2χ) = (φ(U1,0

+U0

2), D2

χ), ∀χ ∈ Sh.

0 à Ú 2

:" : + ±Ýu(t)

³©²'× ½ ´ ¶ ß ¹ Ù ³ Á ±±:ß ·b» 4 ´ ·¸Ø ´

un =

3

2u

n−1 −1

2u

n−2 = un− 1

2 + O(k2) as k → 0. 0 à ® 2 ± 4 4 ´ 4 ½ ¹ ¹ ÞÝ± Ø ´ß ·¸ ßß · ´°ÞÝ±Þ±J³©´ º¸ ÞÞÝ± ½ Áª¸ ß ½ ± ¶ 4 ½ ¹ ¹ 6 ½ Ø ´øß · ´°³©Á×>± ¶º

±Ý º ´>Ý ¸ ×A×T²Ý ¸ × Ù,.-0/ #æ /21 bé >

Un"6£ ¥£T¦:¡L¦§ ¦3 0 à 12&% ' ¥

U0§5

U1

!+56 "5# 0 à Ú 2 #N¦>"¥%§¥ >¦§¡ûU¦:u

¦ 051 12 ¾£54 >>¡ #ø©Ô¨:¡§ % §5‖Un‖

∞, ‖un‖

∞§©"6¦§56 9D¥°6¦:§¦§ 0 ¬ ! 2 ¾ )§¡L % ¥T % T¦§k

54 >>¡ #ª :¡¡ % ' ¥ *)§ ¥T¦:¡¡ ¦ ' ¨ :Ô

max0≤n≤N

‖Un − u(tn)‖ ≤ C(hr + k2) 0 à ¬ 2

' ¥C = C(u, T )

:¡ >§ J©¦ §¦§T¦§ø ¥ A¥ ø¡¡L :© ¢>%§¦§ ® Ú ®Ö©¦N¦3 ³ ½ ¶ 6 0 Ú ¬ 2 »½ ß ½ ³%³©²'× ½ ´ ¶ ßß©±DÁ³Gß ½ Á¸ ß©´

θn

*+ Ý± Á 0 Ú ® 2 » 051 12 ¸¶º 0 à 132 »4 ´± µ ß ¸:½ ¶ 7 ±Ýn ≥ 2

ß · ´ø´/3² ¸ ß ½ ± ¶ 7 ±JÝθ

n

(∂tθn, χ) + (D2

θn− 1

2 , D2χ) = (φ(Un) − φ(un− 1

2 ), D2χ)

− (∂tPhun − u

n− 1

2

t , χ) −(

D2(

un + u

n−1

2− u

n− 1

2 ), D2χ

)

. 0 à à 2¯´Tß©ß ½ ¶ 6

χ = θn− 1

2

» 4 ´ ¶#º

1

2k

(

‖θn‖2 − ‖θn−1‖2)

+ |θn− 1

2 |22 ≤1

2I′2 +

1

2|θn− 1

2 |22 +1

2J

2

+1

2‖θn− 1

2 ‖2 +1

2K

2 +1

2|θn− 1

2 |22,

4 · ´>ÝÍ′ = ‖φ(Un) − φ(un− 1

2 )‖.¼£· ²³

1

k

(‖θn‖2 − ‖θn−1‖2) ≤ ‖θn− 1

2 ‖2 + I′2 + J

2 + K2. 0 à ) 2

³ ½ ¶ 6ß · ´ º½' ´>Ý´ ¶ ß ½¾¸:µ½ ¹ ½ ß Ù ± 7φ(·)

» 4 ´ ¶º 7 ±JÝ"³G± Á ´«×T± ¶ ³ïß ¸:¶ ßC

º ´>Þ´ ¶º ´ ¶ ß"± ¶‖Un‖

∞

¸:¶#º‖un− 1

2 ‖∞

I′ ≤ C‖Un − u

n− 1

2 ‖ ≤ C

(

‖θn‖ + ‖ρn‖ + ‖un − un− 1

2 ‖)

.

< Ù 0 Ú ¬ 2 ¸¶º 0 à ® 2 » 4 ´ ·#¸Ø ´

I′ ≤ C(‖θn−1‖ + ‖θn−2‖) + C(u)

(

hr + k

2)

.

³ ½ ¶ 6 0 Ú ¬ 2 » 0 à ) 2 ¸¶º ß · ´øÁ³Gß ½ Áª¸ ßÁ³ ±7I′

»J

»K

» 4 ´± µ ß ¸½ ¶

‖θn‖2 ≤ (1 + Ck)‖θn−1‖2 + Ck‖θn−2‖2 + C(u)k(hr + k2)2.

¼£·½ ³ Ù3½ ´ ¹¾º ³

‖θn‖2 + Ck‖θn−1‖2 ≤ (1 + 2Ck)(

‖θn−1‖2 + Ck‖θn−2‖2)

+ C(u) k(hr + k2)2.

¼£· ´>Ý´ 7 ±JÝ©´ » 7 ±JÝnk ≤ T

¸¶ºn ≥ 2

‖θn‖2 ≤ C

(

‖θ1‖2 + k‖θ0‖2 + (hr + k2)2

)

. 0 à 2 ´ ß 4 ´«³ ·¸¹ ¹ Ń³ïß ½ Áª¸ ß©´

‖θ1‖%+ Ý©± Á ´ /3² ¸ ß ½ ± ¶ ³ 0 à Ú 2 4 ´ 6J´Tß£ß · ´øÁ³Gß ½ Á¸ ß©´

1

k

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(

‖U0 − u

1

2 ‖2 + (hr + k2)2

)

,

· ´>Ý´θ1,0 = U

1,0 − Phu1»θ0,0 = θ

0 ß 7 ± ¹ ¹ ± 4 ³ 7 Ý©± Á 0 ¬ 2 ß ·¸ ß

‖U0 − u

1

2 ‖ ≤ ‖θ0‖ + ‖ρ0‖ + ‖u0 − u

1

2 ‖ ≤ ‖θ0‖ + C(hr + k),

4 ·½ × ·.Ù½ ´ ¹¾º ³

® Ú ¬ :

1

k

(‖θ1,0‖2 − ‖θ0‖2) ≤ C

(

‖θ0‖2 + h2r + k

2)

.

¼£· ²#³ »

‖θ1,0‖2 ≤ (1 + Ck) ‖θ0‖2 + Ck(h2r + k2) ≤ C

(

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3)

. 0 à 2¶ ß · ´«³ ¸:Á ´ 4 ¸Ù'¸ ³ ¸:µ ± Ø ´ 4 ´± µ ß ¸:½ ¶ 7 Ý©± Á 0 à Ú 2

1

k

(‖θ1‖2 − ‖θ0‖2) ≤ C

(

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2− u

1

2 ‖2 + (hr + k2)2

)

. 0 à ! 2¼£· ´>Ý´ 7 ±JÝ©´ » ²#³ ½ ¶ 6 0 à 2 » 4 ´Á³Gß ½ Á¸ ß©´ß · ´ø´>ÝÝ©±JÝ

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2− u

1

2 ‖ ≤ ‖θ1,0 + θ

0

2‖ + ‖Phu

1

2 − u

1

2 ‖

≤1

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(

hr + k

2)

≤ C‖θ0‖ + C(u)(

hr + k

3

2

)

.

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‖θ1‖2 ≤ (1 + Ck)‖θ0‖2 + Ck(h2r + k3) ≤ C

(

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)

. 0 à 1 2 ß 7 ± ¹ ¹ ± 4 ³7 Ý± Á 0 ¬ ! 2 » 0 à ) 2 ¸¶º 0 à 1 2 » ß ·¸ ß£ß · ´øÁ³Gß ½ Á¸ ß©´

‖θn‖ ≤ C(hr + k2), 0 ≤ n ≤ N 0 à 1132

½ ³ Ø§¸¹ ½¾º 7 ±Ý%³©± Á ´ ×T± ¶ ³ïß ¸:¶ ßC = C(u, T )

, · ±±J³ ½ ¶ 6χ = ∂tθ

n½ ¶ 0 à à 2 » 4 ´ ± µ ß ¸½ ¶

7 ±Ýn ≥ 2

‖∂tθn‖2 + |∂tθ

n|22 ≤ I′ |∂tθ

n|2 + J ‖∂tθn‖ + K |∂tθ

n|2

≤ I′2 +

1

4|∂tθ

n|22 +1

4J

2 + ‖∂tθn‖2 + K

2 +1

4|∂tθ

n|22.

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¸:¶#º 0 Ú ¬ 2 » 4 ´± µ ß ¸:½ ¶

1

k

(|θn|22 − |θn−1|22) ≤ C(‖θn−1‖2 + ‖θn−2‖2) + C(u)(hr + k2)2.

³ ½ ¶ 6 0 à 112 » 4 ´ 6J´Tß

1

k

(|θn|22 − |θn−1|22) ≤ C(u)(hr + k2)2.

, ± ¶ ³©´ /3²´ ¶ ß ¹ ÙJ» 7 ±Ýn ≥ 2

¸:¶#ºnk ≤ T,

4 ´ ·¸Ø ´

|θn|22 ≤ C(u, T )(

|θ1|22 + (hr + k2)2

)

.

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:¡ >§ J©¦ §¦§T¦§ø ¥ A¥ ø¡¡L :© ¢>%§¦§ ® Úà

|θ1|22 ≤ C(hr + k2)2.

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2), 0 ≤ n ≤ N. 0 à 1 Ú 2¼£· ´ ¶ ²³ ½ ¶ 6 0 ® Ú 2 » 0 à 1132 ¸:¶º 0 à 1 Ú 2 » 4 ´ø± µ ß ¸:½ ¶

|θn|1 ≤ C(u, T )(hr + k2), 0 ≤ n ≤ N. 0 à 1 ® 2

+ Ý± Á 0 à 112 » 0 à 1 ® 2 ¸:¶º ? ´ ÁÁ¸ ¬ 4 ½ ß ·p = ∞

» 4 ´ 6J´Tß

‖θn‖∞

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ää ¾ #¯:¥°³ ± ¤ © ¥ ² ¯¦¤ ² © ¿ ¹$LÂ à =¿Í Æ?à ÉÂÁçFÍçÅÂ Z>Ë Æ àTÆ ¸ºÉ Æ Ã¹ÃË¿Á¦¸ÊÂ ¹I Æ çºÂÈ¸IJ'PJÍÂ à çWÝÉ Æ Ã ã ¹â¿ Æ ÍçÂ ËçºÂ HLÍCË¿È¸ÊÂ Ã¿ Æ8ãÊÆ L ãºÆ çºçºÂ ¹I K ¹ ã Ã¹ÃË¿Á¦¸ÊÂ ¹I' Æ #çFÂ ¸IJ Ã ãºÆ Â à ¸ºÂ ¹=¿Áçº¹ò¹ à?à Ë ã QÉ Æ àTÆWã ¸ÊÂ Â I¸ ÆWã à ¸ºÂ ¹çÅâ Æ ¸IQ ÆWÆ .Â Æ Ã Æ Æ 3¸8á¦ ã ÂÁ=â¿ Æ ç ÆT ÂÁç¸:

C9 G< E G AY@ >PAY@ ;=< EF> A D >HGIA?<"! A B# G >$&%'(

) ð+*#ð-,~îì/.1032 îì54 ïî6 î7®î6981ë/4Vë;:<=2>:?@036ÝÉ Æ ã ç¸YQÉ¹ ãºÆ Q|=¸F¸ Æ 3¸ºÂ ¹q¸º¹CÂ ÍÃ¹ ã ¸Ê= àTÆ ¹MK1 Æ ¿ J K ¹ ã Æ çFÂ ¸IJPJ=ÍÂ à ç:QÝç¶8Ë¸ à ÉÂ #çF¹I: , ¶ Æ çºËHL$L Æ çF¸ Æ Â8 ÆWãÊÆ 3¸ºÂÁ=¿ Ì Â ÆWãÊÆ àWÆÀÆ O Ë¦¸ÊÂ ¹I1Î

dN(t)

dt

= r

(

1 −

N(t − h)

K

)

N(t),#µå è %

QÉ ÆWãÊÆr

ÂÁç5¸ÊÉ Æ%à ¹ Æ Z à Â Æ 3¸1¹MK¿ Â Æ ã L ã ¹WQÝ¸ºÉh

¸ºÉ Æ á ÆWã ML Æ L Æ ¹MKË¿È¸xÂ Â áÂÁË=¿ÁçÂ .Ã¹ÃË¿Á¦¸ÊÂ ¹I1

K

¸ºÉ Æ Í T Â ÍCËÍçºËç¸6=Â =â¿ Æ Ã#¹IÃË¿Á¦¸ÊÂ ¹I1ÝÉ Æ ÍÂ K Æ ¦¸ºË ãºÆ çÝ¹MK Æ O Ë¦¸ÊÂ ¹I:#µå è %Å ãÊÆ è:ä µÎ

è 8¿ ¿çº¹¿ Ë¸ºÂ ¹çCQÂ ¸ºÉ.Ã¹IçºÂÈ¸ÊÂ á Æ Â Â ¸ºÂÁ=¿ à ¹#ÂÈ¸ÊÂ ¹Iç ãÊÆ Ã#¹3çFÂ ¸ºÂ á Æ â¹Ë Æ æå=É Æ

0 < rh ≤

1

e

Å¸ºÉ Æ çº¹¿ Ë¸ºÂ ¹N(t)

Íq¹I¹=¸Ê¹Â à ¿ ¿VJ à ¹I3á ÆWã L Æ ç¸º¹K#J7®ÂVL è ¸ºÉ Æ IçFÉ Æ ¿ Â Æ % æ

ä =É Æ 1

e

< rh ≤

π

2

1¸ÊÉ Æ çº¹¿ Ë¸ºÂ ¹N(t) à ¹á ÆWã L Æ ç¸Ê¹

K

QÂÈ¸ÊÉ ¹Iç à Â ¿ ¿Á¦¸ÊÂ ¹Iç Æ ãÆ O ËÂ ¿ Â â ã Â ËÍ #J7®ÂVL è ¸ºÉ Æ çº¹¿ Â ¿ Â Æ % æ

¼=É Æ rh >

π

2

:¸ÊÉ ÆWãÊÆ ÂÁç¹I¿VJ¹I Æ çF¸Êâ¿ Æ Ã ÆWã Â ¹Â à çº¹¿ Ë¸ºÂ ¹ #@7®ÂSL# è W¸ÊÉ Æ ¹=¸F¸ Æ ¿ Â Æ %

A BDCFE1+HG3I N=ø ñ õIúHKø?G þ øWù D õLTúHKø:G 9 KP KJ +E D Gh = 1

r = 0.1

K = 2

þ ø ñ õúHKø:GN(t)öòø?G3øWúø:G3HKAL ñKñKû AÊø:GM D óô D6þ úø

K

ý L þ E Dý'ñ HKG DKL +E D Gh = 1

r = 0.7

K = 2

þ ø ñ õúHKø:GN(t)

AÊø:GM D óô D6þ úøK

+HüúE ø þ A6H ñKñ LWúHKø:G þ G D LWó D õIH ñ HK÷IóHKõIö °þ ø ñ H ý ñ HKG D KL +E D Gh = 5.3

r = 0.3

K = 2

þ ø ñ õIúHKø:GN(t)

H þ+ÿD óHKø ý HKA"L?G ý þ úFL?÷ ñKD ý ø?úú D6ý ñ HKG D

D8§Y³ ©W±E:ª© £°£K¥° FR N ²W© LW³ ²T §Y£ ¤¦³µ¥ © ² ¥°³V´'C´¦¤Dq¥JL ² ääèÝÉ Æ ÃÃ ã ¹ T Â Í¦¸ Æ çF¹I¿ Ë¸ºÂ ¹ QÝç à ¹#ç¸ ã Ë à ¸ Æ 'WK°¸ ÆWã =¿SJçºÂ ç%¹MK Æ O Ë=¸ºÂ ¹E#µå è %

â=J Í Æ =çÝ¹K®âÂ K Ë ã6à =¸ºÂ ¹ ¸ºÉ Æ ¹ ã J è?ä Î

N(t) ≈ K

[

1 + 2.32√

rh − 1.57 cos(π

2τ

)

+ 0.54(rh − 1.57)#Våüå-%

×

(

sin(πτ) + 2 cos(πτ))]

,

QÉ ÆWãÊÆhτ [1 + 0.17(rh − 1.57)] = t.

) ð ) ð-, :(?106+694+7 í 0ë@6 ì?ëÝÉ ÆWãÊÆ Q Æ?ãºÆÆ ¹ ã ¸6ç¸Ê¹ ÆT Ã¿Á=Â q¸ºÉ Æà ç Æ ç+¹MK à J à ¿ Â àà É=HL Æ çÂ Â ç Æ:à ¸6ç4[IÃ#¹IÃË¿Á¦¸ºÂ ¹ Æ çºÂ ¸IJâ2J Í Æ =çò¹K Æ O Ë¦¸ÊÂ ¹I #Vå è % è?ä è »®å=»µ· JÂ ÉÂ çCÍ¹¹$L ã =ÃÉ? è »à ¹ÍÃ ãºÆ ¸ÊÉ ÆÀãºÆ çFË¿ ¸ÊçÝ¹KxËÍ ÆWã Â à =¿r=¿VJçFÂÁçÅ¹MK Æ O Ë¦¸ÊÂ ¹I1#Vå è %?QÂÈ¸ÊÉ Æ4T Ã ÆWã Â Í Æ Ì¸Ê=¿ ¦¸6 K ã ¹IÍ)¸ºÉ Æ à ¿ÁçÊçFÂ à ¿:Q ¹ ã Xâ=Jj8Â à É¹¿Áçº¹ å=¼%¹Ij¹3ç à Â ¿ ¿Á¦¸ºÂ ¹#çÀ¹MK8 Æ #çFÂ ¸IJ¿ â#¹ ã =¸º¹ ã JÃ¹ÃË¿Á¦¸ÊÂ ¹I ¹KÝ"ËçF¸ ã =¿ Â jçºÉ ÆWÆ ÃÌ â#¹WQ>)J?#CLW¥°£K¥ ¤:L :§ ± ¥°¤ % LI¹=¸çº=¸ºÂÁçIKJÂ HL ãºÆ çFË¿È¸6çW®ÝÉ Æ =ÃÃ ã ¹ T Â Í¦¸ Æ çº¹¿ Ë¸ºÂ ¹ #µåüå-%ò¿ çº¹R¸Êç O ËÂÈ¸ Æ Q Æ ¿ ¿+¸º¹.¸ÊÉ Æ8Â à É¹¿ÁçF¹I\[ ç Æ4T Ã ÆWã Â Í Æ 3¸Ê=¿=¸Êr¶"¹WQ Æ á ÆWã Â ¸çFÉ¹Ë¿Áâ Æ ¹=¸ Æ ¸ÊÉ¦¸ à ¹ ãºãÊÆ ¿Á¦¸ÊÂ ¹Iâ Æ ¸IQ Æ?Æ ¸ºÉ Æ ¹ ãÊÆ ¸ÊÂ à =¿r= Æ4T Ã ÆWã Â Í Æ 3¸Ê=¿5¹3ç à Â ¿ ¿ =¸ºÂ ¹ç à ¹Ë¿Á â Æ â Æ ¸º¸ ÆWã #J7®ÂSL##å-%T

0

2

4

6

8

10

0 50 100 150 200 250 300 350

t, days

N(t

),th

ou

san

ds

0

2

4

6

8

10

0 2 4 6 8 10

Experimental data

N(t

),th

ou

san

ds

L ÷

A BDCFE1+I L ø?ö ÿ LTóH þ ø?GøWùOÅHKAFE3ø ñKþ ø?G þ 9T>D('ÿD óHKö D G¦úFL ñ=ý LWúFL ø?G ñ L?÷øWóFLWúøWó ûÿ ø ÿ õ Sñ LTúHÈø?GqøWù õ þ úóFL ñ HÈL?G þ E DÊD6ÿIS ÷ø û M +HKúE L ÿIÿ óø ' HKöLWú D%þ ø ñ õIúHKø:G

N(t) 9 9 ÷ A6ø?óó DÊñ LTúHKø:GA6ø D A6H D G¦úr = 0.67

CO ã =¸F¸[ çr¿Á=â¹ ã ¦¸Ê¹ ã Â ¿ ãºÆ ç Æ ã6à É Æ çüå ¹KÃÉÂÁÝÃ#¹IÃË¿Á¦¸ÊÂ ¹IÂ èO, ¼ äÅà â Æà ¹#çFÂÁ ÆWãÊÆ Cç® à ¿ IçºçºÂ à =¿ Æ?à ¹¿ ¹LÂ à ¿ Æ4T Ã ÆWã Â Í Æ 3¸?¦ÃÉÂÁ¹3¸º¹$L Æ Æ çºÂ ç ãºÆ ç Æ ÍCâ¿ Æ ç¸ºÉ Æ Æ á Æ ¿ ¹IÃÍ Æ 3¸CçF¸ÊML Æ çÀ¹KÝâ ÆWÆ ¸º¿ Æ Ã¹ÃË¿Á¦¸ÊÂ ¹I1=ÃÉÂ Æ çºÂ ¸IJ PJÍqÂ à ç K ¹ ãÂ ÆWãÊÆ 3¸¸ Æ ÍÃ Æ?ã =¸ºË ãÊÆ çÝÂÁççºÉ¹WQ Â )7®ÂSL# ä

Ç Æ ¸Ëç à ¹#çFÂÁ ÆWã K ¹IË ã Í Æ ¸ºÉ¹ç K ¹ ã Æ Â ¸ºÂ ¹¹MKÃ ã =Í Æ ¸ ÆWã çr

h

¹MKrÃÉÌÂ 8Ã#¹IÃË¿Á¦¸ºÂ ¹O ã =Í Æ ¸ ÆWã

r

ÂÁç¸ºÉ Æ ·=¿ ¸ºËçºÂÁ= à ¹ Æ Z à Â Æ 3¸¹MK Æ4T Ã¹ Æ 3¸ºÂÁ=¿$L ã ¹WQÝ¸ºÉ1QÉÂ à É à =Àâ Æ ¹âç ÆWã á Æ ¹¿VJ=¸1¸ºÉ Æ â Æ LÂ Â HL¹MKÃ¹ÃË¿Á¦¸ÊÂ ¹I L ã ¹WQÝ¸ºÉË Æ?ã9(Wà ¹ÍqÌÃ Æ ¸ºÂ ¸ºÂ ¹ á¦ à ËËÍ ( à ¹Â ¸ºÂ ¹#çWÇ Æ ¸

[t1, t1 + ∆t]â Æ çºË à É ¸ºÂ Í Æ Â 3¸ Æ?ã á¦=¿CQÂ ¸ºÉÂ

QÉÂ à ÉN(t)

N(t − h)

á¦=¿ Ë Æ ç ãºÆ çºÍ=¿ ¿VÝÉ Æ N

′(t) ≈ rN(t)=#

r ≈

lnN(t1 + ∆t)

∆t

.

#µå ä %

ää å #¯:¥°³ ± ¤ © ¥ ² ¯¦¤ ² ©

L ÷

A BDCFE1+I L ÿ E3G3HÈL ýID G þ HKú û ýû GLWöòHÈA þ J L B P 0 ? ÷ B9 0

Æ ¿ Jh

Â ç Ã ÆWã Â ¹ ¹K¸ºÂ Í Æ QÉÂ à É Æ O Ë=¿Áç =ÃÉÂÁ Æ á Æ ¿ ¹IÃÍ Æ 3¸ à J à ¿ Æ Ã Æ?ã Â ¹rÃÉÂÁ Æ á Æ ¿ ¹ÃÍ Æ I¸ à J à ¿ Æà ¹çºÂ çF¸ÊçÅ¹Kx¸IQÅ¹Ã ã ¸Êç"# ç ÆWÆ 7®ÂVL¼%Î

è % Æ?ã Â ¹'¹K ä Ì½ Jç?Ë ã Â HL"QÉÂ à É ,J¹IËHL =ÃÉÂÁ à = â Æ É=¸ à É Æ '¹Ë¸ æå-% Æ?ã Â ¹¹KÅ¦¸À¿ Æ çF¸åÌ ä Jç?QÉÂ à Éh ãÊÆ Æ?Æ Æ K ¹ ã NJI¹ËHLÃÉÂÁ ¸º¹

àWà ËÍCË¿ =¸ Æ K ¦¸"çºËâçF¸Ê àTÆ ç%¸Ê¹â ÆòãÊÆ IPJK ¹ ã ¿Á JÂ )L Æ LL3çW

h

3-5 days 2-3 days

t egg young Daphnia egg young Daphnia

A BDCFE1+ I Hüù D A û A ñKD øWù L ÿ E3G3HÈLM

ÝÉ Æ á¦=¿ Ë Æ ¹Kh

Æ Ã Æ #ç"¹I.¸ Æ ÍqÃ ÆWã ¦¸ÊË ãÊÆ C=É Æ ¸ÊÉ Æ ¸ Æ ÍÃ Æ?ã =¸ºË ãÊÆCà ÉHL Æ çW¸ºÉÂÁç"Ã# ã =Í Æ ¸ ÆWã8à =¿ÁçF¹á¦ ã J K ã ¹IÍ¼'¸ºÂ ¿ ¿®»' Jç" Æ á Æ .Í¹ ãÊÆ 7 ã ¹IÍP ã =¸F¸ [ ç¦¸6 à É ã ¸6ç ¹IdâÂ ã ¸ÊÉ =#Q Æ ¦¸ºÉ ã =¸ Æ PJ=ÍÂ à ç ÂÈ¸ K ¹¿ ¿ ¹WQ8ç ¸ºÉ#¦¸¦¸ è »

0S

Æ ¿Á Jh ≈ 12

Jç=|¦¸' ÀåI½0S Æ ¿Á J

h ≈ 9 JçW+¹#:QÉ Æ

rh = 1.2# 1e

< 1.2 ≤

π

2

%TK = 135

=r = 0.1

Æ O Ë#¦¸ºÂ ¹ #Vå è %C Æ ç àTã Â â Æ ç O ËÂ ¸ Æ Q Æ ¿ ¿ÃÉÂÁ. Æ çºÂ ¸IJPJ=ÍÂ à çÀ¦¸ è »

0Y=É Æ

rh = 1.62#1.62 >

π

2

%TK = 30=

r = 0.18 Æ O Ë¦¸ÊÂ ¹I?#Vå è % Æ ç àTã Â â Æ ç O ËÂ ¸ Æ Q Æ ¿ ¿=ÃÉÂ . Æ çºÂÈ¸IJPJÍÂ à ç

¹. Àå½0#Vç Æ?Æ 7®ÂVL#½q 7®ÂVL 5 %

C9 G< E G AY@ >PAY@ ;=< EF> A D >HGIA?<"! A B# G >$&% G$>P@)GD >F B&%' =

ÝÉ ãºÆ?Æ çF¸ÊML Æ ç"¹MK% Æ á Æ ¿ ¹ÃÍ Æ 3¸8Ì Æ LL#¿Á ã á =IË¿ ¸çºÃ Æ:à Â Æ ç ãÊÆCà ¹IçºÂ Æ?ãºÆ Â Ë [ ç8Í=¸ºÉ Æ Í=¸ºÂ à ¿1Í¹ Æ ¿¹IÂ #ç Æ:à ¸8Ã¹ÃË¿Á¦¸ÊÂ ¹IPJ=ÍÂ à ç üå , µ"ÂÁç¸ ã Â âË¸ Æ Æ ¿Á Jç ãÊÆ Â 3¸ ã ¹Ë àWÆ qÂ 3¸Ê¹òçJçF¸ Æ Í¹MK¸ºÉ ãºÆ?Æ Â ÆWãÊÆ 3¸ÊÂ ¿ Æ O Ë=¸ºÂ ¹ç?Ë à ÉÂ 3¸ ã ¹=ÌË à ¸ÊÂ ¹I ¹KÂÁçF¸ ã Â âË¸ Æ Æ ¿ JçxÂ 3¸Ê¹ÀÍ¹ Æ ¿ÂÁç à Ëç Æ â2Jò¸ÊÉ Æ Â #=âÂ ¿ Â ¸IJ¹KÂÁç àWãºÆ ¸ Æ Æ ¿Á JÍq¹ Æ ¿I Æ O Ë=¸ Æ ¿SJ Æ ç àTã Â â Æ çF¹IÍ ÆÆT ¸ ã =¹ ã Â ã J K à ¸º¹ ã ç? Æ L 3ÂÈ¸ÝÂ ç Â ÍqÃ¹IçFÌçFÂ â¿ Æ ¸Ê¹çFÂ ÍCË¿ =¸ Æ ËçºL Æ ¹MKÃ Æ ç¸ÊÂ à Â Æ çÂ à ¿ ËÂ HLçº¹Í Æ IÂ ¸ºÂ ¹¿2K Ë à ¸ºÂ ¹çÂ 3¸º¹

D8§Y³ ©W±E:ª© £°£K¥° FR N ²W© LW³ ²T §Y£ ¤¦³µ¥ © ² ¥°³V´'C´¦¤Dq¥JL ² äää

0

50

100

150

200

250

0 50 100 150 200 250

Experimental data

Mo

dell

ing

resu

lts

L ÷

A BDCFE1+I !qø ýD6ñKñ HÈGIô8ø?ù L ÿ E3GIHÈL ýID G þ HKú ûýû GLWöòHÈA þ +E D G B P 0 J L rþ HKöõ ñ LWúHKø:GÀó DºSþ õ ñ ú þ +E D Gr = 0.1

h = 12

LWG ýK = 135

÷ A6ø?ö ÿ LWóH þ ø?GÀø?ùróFLWúú þ 9:;4D('ÿD óHKö D G¦úFL ñý LWúFL ÿ ó D6þD G¦ú Dý HÈG5HÈô 7 +HüúEöòø ýID6ñKñ HKG3ôó DÊþ õ ñ ú þ øWóó D6ñ LWúHKø:GA6ø D A6H D G¦ú

r = 0.83

0

20

40

60

80

100

0 10 20 30 40 50 60 70 80 90

Experimental data

Mo

de

llin

gre

su

lts

L ÷

A BDCFE1+I !qø ýD6ñKñ HÈGIô8ø?ù L ÿ E3GIHÈL ýID G þ HKú ûýû GLWöòHÈA þ +E D G B9 0 J L rþ HKöõ ñ LWúHKø:GÀó DºSþ õ ñ ú þ +E D Gr = 0.18

h = 9

LWG ýK = 30

÷ ø?ö ÿ LTóH þ ø?Gòø?ùróFLWúú þ 9:;D('ÿD óHKö D G¦úFL ñý LWúFL ÿ ó D6þD G¦ú Dý HÈG5HÈô 7 +HüúE öòø ýID6ñKñ HKG3ôó D6þ õ ñ ú þ ø?óó DÊñ LTúHKø:GA6ø D A6H D G¦ú

r = 0.73

'Í¹ Æ ¿µÝÉ Æ Í¹ Æ ¿®Éçâ Æ?Æ ¸ Æ çF¸ Æ â2J Æ4T Ã Æ?ã Â Í Æ 3¸Ê¿x=¸Ê¹IÂ #ç Æ:à ¸Êç [Ã#¹IÃËÌ¿ =¸ºÂ ¹¹K#³V¥JL6¤ ±6² ¥ ¤ F3© DD¤¦³¤¦£K¥ ² CIW ©W± #ÝÉ Æqà ¹ ãºãÊÆ ¿Á¦¸ÊÂ ¹Iâ Æ ¸IQ ÆWÆ ¸ºÉ Æ ¹ ãÊÆ ¸ÊÂ à =¿= Æ4T Ã Æ?ã Â Í Æ 3¸Ê¿ à Ë ã á Æ çCÃÃ Æ ãÊÆ j¸Ê¹â Æ O ËÂ ¸ Æ L¹¹rx¶"¹WQ Æ á ÆWã 1Â ¸çºÉ¹IË¿ hâ ÆÍ Æ 3¸ºÂ ¹ Æ rr¸ºÉ#¦¸Ã ã à ¸ºÂ à =¿ËçÊML Æ ¹MK Í¹ Æ ¿% Æ ÍçÀá¦çF¸C=Í¹ËI¸ò¹KÅÂ PK ¹ ã Í¦Ì¸ºÂ ¹1= Æ çFÉ¹IË¿ÁX3¹WQ ¸ÊÉ Æ âÂ ã ¸ºÉ'=# Æ =¸ºÉ ã ¦¸ Æ K Ë à ¸ÊÂ ¹Iç%=¸ Æ à É Æ á Æ ¿ ¹ÃÍ Æ 3¸ç¸6ML Æ 3'¿ çº¹ K Ë à ¸ºÂ ¹çÅ¹KML Æ ÂÁç¸ ã Â âË¸ÊÂ ¹I ¦¸Å¸ºÉ Æ Â ÂÈ¸ÊÂ ¿#¸ÊÂ Í Æ Í¹Í Æ 3¸

t = 0

QÅ¹ ã X »/jÍ Æ ¸ºÉ¹¹I¿ ¹$LJ ¹K à ¹çF¸ ã Ë à ¸ºÂ ¹|¹MKçºÂ HL¿ Æ ÌçFÃ Æ?à Â Æ ç¿ â#¹ ã =¸º¹ ã JÃ#¹IÃË¿Á¦¸ºÂ ¹ PJ=ÍÂ à çÍ¹ Æ ¿ÁçQÂ ¸ºÉ ãºÆ çFÃ Æ?à ¸¸Ê¹ ML Æ çF¸ ã Ë à ¸ÊË ãÊÆ ÂÁç Æ ç àWã Â â Æ rrÝÉ ÆLÂ á Æ Í Æ ¸ºÉ¹¹I¿ ¹$LJ à =â Æ Ëç Æ K ¹ ã ¸ÊÉ¹Iç Æ Â ç Æ?à ¸6ç+¸ÊÉ Æ ¿ ÂSK Æà J à ¿ Æ ¹K\QÉÂ à É à =â ÆÂ áÂÁ Æ hÂ 3¸º¹ç Æ Ã# ã ¦¸ Æ Æ á Æ ¿ ¹IÃÍ Æ 3¸CçF¸ÊML Æ ç?ÝÉ Æ?ãºÆ ÉIçâ ÆWÆ à ¹#ç¸ ã Ë à ¸ Æ ¸ÊÉ ÆçJçF¸ Æ Ía¹MKÝÂ ÆWãÊÆ 3¸ÊÂ ¿ Æ O Ë=¸ºÂ ¹ç QÂ ¸ºÉ ç Æ á Æ?ã ¿ Æ ¿Á JçW QÉÂ à É à â Æ Ëç Æ hçòÍq¹ Æ ¿5¹MK1¸ÊÉ Æ â#¹¦á Æ Â ç Æ?à ¸ÝÃ¹ÃË¿Á¦¸ÊÂ ¹I1ÝÉ Æ Íq¹ Æ ¿5Éç â Æ?Æ =ÃÃ¿ Â Æ ¸Ê¹ Æ4T Ã¿ Â ¿ â#¹ ã =¸º¹ ã JÃ¹ÃË¿ =¸ºÂ ¹'PJÍqÂ à ç:K ã ¹IÍ à ¿ IçºçºÂ à =¿"Â à É¹¿ÁçF¹I\[ ç ÆT Ã Æ?ã Â Í Æ 3¸ üå¦¼ ÝÉ Æ ËÍ Æ?ã Â à ¿5=¿VJçFÂÁç+¹K1¸ºÉ Æ Í¹ Æ ¿É#çÝçFÉ¹WQ O ËÂÈ¸ Æ LI¹3¹ à ¹ ãÊãºÆ ¿ =¸ºÂ ¹ QÂ ¸ºÉ8Â à É¹¿ÁçF¹I\[ ç Æ4T Ã Æ?ã Â Í Æ 3¸Ê¿1¦¸6

ÝÉ Æ çÊ=Í Æ Í Æ ¸ÊÉ¹ QÅIç8Ëç Æ ¸Ê¹ à ¹Iç¸ ã Ë à ¸"Í¦¸ÊÉ Æ Í=¸ºÂ à =¿xÍ¹ Æ ¿Áç8¹MK¿Á=â¹ ã ¦Ì¸º¹ ã JCÃ¹ÃË¿ =¸ºÂ ¹PJÍqÂ à ç QÂ ¸ºÉ à ¹çºÂÁ ÆWã ¦¸ÊÂ ¹I ¹Ká¦ ã Â ¹ËçÃ¹IçÊçFÂ â¿ ÆÆ Æ?à ¸Êç+Ë Æ ¸º¹¿ Â ÍÂÈ¸ Æ K ¹¹ K ¹ ã ¿ ã á¦ åI½H

ää ¼ #¯:¥°³ ± ¤ © ¥ ² ¯¦¤ ² © 9 KE GI>3 B G<= E G AY@ >PAY@F ;$< EF> AD >PGA <&! A B=K

ð+*#ð-, :(?106 ï :7xë?ìWí ®ï=ì549:7î7@?'2 îì/.1032 îì/4Vïî6 î7®î6 81ë54 ë)âIçFÂ à ¿ ¿VJ¬ Æ Q)=ÃÃ ã ¹3 à É¬¸º¹ ¸ºÉ Æ Ã ã ¹Iâ¿ Æ Í ¹MKÂ ç Æ:à ¸6ç4[ÅÃ#¹IÃË¿Á¦¸ÊÂ ¹IdPJ=ÍÂ à çÍq¹ Æ ¿ ¿ Â HL É#çqâ Æ?Æ |ç¸6 ã ¸ Æ Â Q ¹ ã XçCâ=J ¡¹¿ Æ çº¹¦á è ¼ è ½H +¶ Æ çÊçFËÍ Æ h¸ºÉ#¦¸¸ºÉ Æ Í=Â à =Ë#ç Æ K ¹ ã Æ çºÂÈ¸IJ ¹3ç à Â ¿ ¿Á¦¸ºÂ ¹#ç ÂÁç à ¹IÍqÃ Æ ¸ÊÂÈ¸ÊÂ ¹I QÂ ¸ºÉÂ Ã¹ÃË¿Á¦¸ÊÂ ¹I1ÝÉ3Ë#çW:Â ÆWãÊÆ 3¸ÊÂ ¿ Æ O Ë=¸ºÂ ¹çx Æ ç àTã Â âÂ )LÃ#¹IÃË¿Á¦¸ºÂ ¹HJ3#=ÍÂ à ççºÉ¹Ë¿ ÀÉá Æ =Ë¸º¹ ¹Iç à Â ¿ ¿Á¦¸ÊÂ HLÀçº¹¿ Ë¸ºÂ ¹ç?=¶8Ë¸ à ÉÂ çº¹\[ ç Æ O Ë=¸ºÂ ¹R#Vå è %xÂÁç= ÆT =ÍÃ¿ Æ ¹MK5çFË à ÉqÍ¹ Æ ¿µ ¸+Â ç+çºËÃÃ¹Iç Æ Â è ½¸ºÉ=¸+Â #ç Æ:à ¸Êç [= Æ çFÂ ¸IJ HJ3#=ÍÂ à ç à â Æ Æ ¸ ÆWã ÍÂ Æ qâ=J¸IQ ¹ à ¸ÊÂ á Æ ÃÉIç Æ çÌx¿Á ã á¦ò=Â ÍML¹#=ÝÉ ÆWãÊÆ K ¹ ãºÆ ¦¸ºÉ Æ K ¹¿ ¿ ¹WQÂ HLòÍ¦¸ÊÉ Æ Í=¸ºÂ à =¿#Í¹ Æ ¿Â ç8çºËHL$L Æ çF¸ Æ K ¹ ã Æ ç àTã Â Ã¸ºÂ ¹¹MK¸IQÅ¹Â #ç Æ:à ¸Êç [ Æ á Æ ¿ ¹IÃÍ Æ 3¸8ç¸6ML Æ çWÎ

N′

1(t) = r1

[

1 − a(1 −

N2(t − h1)

K2

) −N1(t − h2)

K1

]

N1(t) ,

N′

2(t) = r2

[N1(t − (1 − h1 − h2))

K1

−

N2(t)

K2

]

N2(t) .

# ¼ è %

qçJç¸ Æ Í # ¼ è %N1(t)

Â ç®¸ÊÉ Æ 3ËÍCâ ÆWã ¹MKÂ ÍL¹N2(t)

¸ÊÉ Æ ËÍCâ ÆWã ¹MK¿ ã á¦h1(t)¸ºÂ Í Æ Ã Æ?ã Â ¹â Æ ¸IQ Æ?Æ ¿Á ã á¦=Â ÍMLI¹=ÃÃ Æ ã àTÆ

1−h1

¸ºÂ Í Æ Ã ÆWã Â ¹â Æ ¸IQ ÆWÆ Â ÍMLI¹=# ¿Á ã á=ÃÃ Æ ã àTÆ

h2

á ÆWã ML Æ Â ÍML¹[ çCÃ#¹IÃË¿Á¦¸ÊÂ ¹I ¿ ÂSK Æ Ã Æ?ã Â ¹ Ã ÆWãJ Æ ã

K1

K2

á ÆWã ML Æ ËÍ â Æ?ã ¹KÂ ÍMLI¹ =#¿ ã á¦a < 1 à É ã à ¸ ÆWã Â<; Æ çÝÃ¹WQ ÆWã

¹MK ãÊÆ ¿Á¦¸ºÂ ¹Câ Æ ¸IQ Æ?Æ Â ÍML¹[ çx=¿ ã á¦H[ çxÃ#¹IÃË¿Á¦¸ºÂ ¹#çWr01 = r1(1−a)

·=¿ ¸ºËçºÂ \[ çà ¹ Æ Z à Â Æ 3¸¹MK¿ Â Æ ã L ã ¹WQÝ¸ÊÉ1

r2

¿Á ã á)[ çÝÃ¹ÃË¿ =¸ºÂ ¹¿ Â Æ ã L ã ¹WQÝ¸ºÉ à ¹ Æ Z à Â Æ I¸:=É Æ

r2 −→ ∞

=À¸ÊÉ Æ ¹¸ºÉ Æ?ã Ã ã Í Æ ¸ Æ?ã ç ãÊÆ+à ¹#ç¸6=3¸?:çF¹I¿ Ë¸ÊÂ ¹Iç1¹KçJçF¸ Æ Í# ¼ è %Å ãºÆ çºÂ ÍÂ ¿ ã ¸º¹çF¹I¿ Ë¸ºÂ ¹çÝ¹K Æ O Ë=¸ºÂ ¹

N′(t) = r

[

1 − a

(

1 − N(t − (1 − h)))

− N(t − h)]

N,

# ¼üå-%QÉ ÆWãÊÆ

N(t) =N1(t)

K1

, r = r1, h = h2,N2(t)

K2

= N

(

t − (1 − h1 − h2))

.

"Â8 Æ?ãºÆ 3¸ºÂÁ=¿ Ì Â8 ÆWãÊÆ àWÆÆ O Ë#¦¸ºÂ ¹1# ¼üå-%%ÂÁçÅçºÂ ÍqÃ¿ Æ?ã ¸ÊÉ= ¸ºÉ Æ çJç¸ Æ Í # ¼# è %% ÃÌÃ ã ¹ T Â Í¦¸ Æ çÅÂÈ¸"çÊ¦¸ÊÂ çK à ¸º¹ ã Â ¿SJI

8ËÍ ÆWã Â à =¿I=¿SJçºÂ ç5¹KçJç¸ Æ Í # ¼# è %r= Æ O Ë#¦¸ºÂ ¹ # ¼üå-%5ÉIçrâ ÆWÆ ¹ Æ Â è åH ÝÉ Æ âÂ ¹I¿ ¹$LÂ à ¿Ã ÆWã Â ¹ ÂÁç à ¿ ¹Iç Æ ¸Ê¹q¹I Æ É¹WQ Æ á ÆWã I¸ºÉ Æ?ãºÆ ãÊÆ Íq¹ Æ çCQÂÈ¸ÊÉ Ã ÆWã Â ¹çFÂVLÂ8 à =3¸º¿VJ¿ Æ çºç%¸ºÉ ¹I Æ ÝÉ Æ ç¸6¦¸ºÂ ¹# ã J Í¹ Æ ç ¹MK Æ O Ë¦¸ÊÂ ¹IE# ¼# å%¹K°¸ Æ Éá Æ à ¹IÍÃ¿ Â à =¸ Æ K ¹ ã Í. Æà '¹â#ç Æ?ã á Æ L ã Ë#=¿Â àWãºÆ ç Æ ¹MK¹3ç à Â ¿ ¿Á¦¸ºÂ ¹#çYQÂ ¸ºÉ'¸ÊÉ ÆK ¹¿ ¿ ¹WQÂ HLh Æ Ã ãÊÆ çÊçºÂ ¹I¬çF¸ÊML Æ QÉÂ à É¬ÂÁç L¹¹ ÆT ÍqÃ¿ Æ ¹MKQ Æ ¿ ¿ Ì X3¹WQ =¸ºË ã =¿ÃÉ Æ ¹Í Æ ¹K Æ çºÂÈ¸IJ ¹Ë¸ÊâË ã ç¸:#¶"¹WQ Æ á ÆWã 3¸ºÉ Æ ç¸6¦¸ºÂ ¹# ã J'Í¹ Æ ç è åH® ãÊÆ á ÆWã Jç Æ çFÂ ¸ºÂ á Æ ¸º¹ Æ á Æ ÍÂ ¹ ã+à É=)L Æ çx¹MKÍ¹ Æ ¿Ã ã Í Æ ¸ Æ?ã ç®=C¸ºÉ ÆÅã =)L Æ çx¹MKÃ ã =ÍqÌÆ ¸ Æ?ã ç8á¿ Ë Æ çCK ¹ ã QÉÂ à É ¸ÊÉ Æ Í¹ Æ Ã ÆWã Â ¹.ÂÁç à ¿ ¹Iç Æ ¸º¹¹I Æ ãÊÆ á ÆWã J # ãÊã ¹WQòÝÉËççFË à É à ¹IÍqÃ# ã Â çº¹¹K1¸ºÉ Æ ¹ ãÊÆ ¸ÊÂ à =¿5= ÆT Ã Æ?ã Â Í Æ 3¸Ê¿¦¸Ê,LÂ á Æ ç ã ÂÁç Æ ¸º¹Â Z à Ë¿È¸ÊÂ Æ çâ Æ:à =Ë#ç Æ çº¹Í Æ Ã ã Í Æ ¸ Æ?ã çÝ¹K®Í¹ Æ ¿ ç# ¼ è %8= # ¼# å% ãÊÆ ¹¸X¹WQ1 ¸çºÉ¹IË¿

D8§Y³ ©W±E:ª© £°£K¥° FR N ²W© LW³ ²T §Y£ ¤¦³µ¥ © ² ¥°³V´'C´¦¤Dq¥JL ² ää ½â Æ =¿ÁçF¹¹¸ Æ .¸ºÉ=¸8¸ºÉ Æ çF¸Ê¦¸ÊÂ ¹I ã J Í¹ Æ ç8¹MK Æ O Ë#¦¸ºÂ ¹ # ¼üå-% ãºÆ Íq¹ ãºÆ ç Æ #çFÂ ¸ºÂ á Æ¸º¹ à É=)L Æ çÝ¹KxÃ ã Í Æ ¸ Æ?ã ç ¸ÊÉ=¸ºÉ Æ çºÍ Æ Í¹ Æ çCK ¹ ã çJçF¸ Æ Í # ¼# è %

¹ ã ÆWã ¸Ê¹ ÆT Ã¿Á=Â j¸ºÉ Æ ¿Á=â¹ ã ¦¸Ê¹ ã J=¸Ê.¹MKÂ ç Æ?à ¸6ç4[xÃ#¹IÃË¿Á¦¸ÊÂ ¹Ih Æ çºÂÈ¸IJIxÍq¹Â8 à ¦¸ºÂ ¹¹MKÍ¹ Æ ¿ K ã ¹Í è ½HÉIçÝâ Æ?Æ .Ã ã ¹Ã¹Iç Æ 'Â µÎ

N′

1(t) = r1

[

1 − a(1 −

N2(t − h1)

K2

) + c

(

1 −

N1(t)

K1

)

−

N1(t − h2)

K1

]

N1(t),

N′

2(t) = r2

[

N1

(

t − (αT − h1 − h2))

K1

+ b

(

1 −

N2(t − h3)

K2

)

−

N2(t)

K2

]

N2(t),# ¼ ä %QÉ ÆWãÊÆ

T

Â ç.¹I Æ L Æ Æ?ã =¸ºÂ ¹QË ã =¸ºÂ ¹1ÅÂµ Æ Ý¸ºÂ Í Æ Ã ÆWã Â ¹ K ã ¹IÍ Æ LL¬¸ºÂ ¿ ¿òÂ ÍMLI¹h3

ÂÁçò¿Á ã á¦ =¿ Æ á Æ ¿ ¹ÃÍ Æ 3¸CçF¸ÊL Æ Ë ã ¦¸ÊÂ ¹I1α ≥ 1 ãºÆ > Æ?à ¸ÊçÀ¸ÊÉ Æ L Æ ÆWã ¦¸ÊÂ ¹Iç

¹¦á ÆWã ¿Á=ÃÃÂ HL K à ¸:·¹ Æ ¿# ¼# ä %+ÉIç%â ÆWÆ à ¹çF¸ ã Ë à ¸ Æ ¹¸ÊÉ Æ â#ç Æ ¹KçFÃ Æ?à Â ¿¿Á=â¹ ã ¦¸Ê¹ ã J Æ4T Ã ÆWã Ì

Â Í Æ 3¸8¹IR ±A² §ZB¥°£ ¤+D © £ ¤¦ AF ¤ ² ³ ©T± Ã#¹IÃË¿Á¦¸ºÂ ¹Z=É Æ

b = 0, c = 0, α = 1, T = 1

Q Æ L Æ ¸¸ºÉ Æ çJç¸ Æ Íf¹K Æ O Ë#¦¸ºÂ ¹#ç # ¼ è %T ¸ Æ Í

c(1 −

N1(t)

K1

)Â ¸ÊÉ Æ ã ç¸ Æ O Ë=¸ºÂ ¹¹MK'# ¼# ä % çºÉ¹WQ8ç ãÊÆ Ë à ¸ºÂ ¹ ¹MK K ÆWã ¸ºÂ ¿ Â ¸IJ

¹MK IË¿ ¸òçºÃ Æ?à Â Æ çÂ à ç Æ ¸ÊÉ Æ Â ã Æ çºÂ ¸IJâ Æ:à ¹Í Æ çÍ¹ ãºÆ ¸ºÉ á ÆWã ML Æ #IçQ Æ ¿ ¿çÂ àWãºÆ ç Æ ¹K1¸ºÉ Æ Â ã Æ =¸ºÉ ã =¸ Æ QÂÈ¸ÊÉÂ çºË à É Ã ÆWã Â ¹çÝ¹MK1¸ÊÂ Í Æ C ã Í Æ ¸ Æ?ã 0 ≤ c < 1à É ã à ¸ ÆWã Â<; Æ ç8¸ºÉ Æ Ã¹WQ Æ?ã ¹K+¸ºÉÂÁçÀÂÈ¸ÊÂ ¹I=¿Â ÆWã K Æ?Æ â à X¦¸¸ºÉ Æ Â ÍL¹ç¸6ML Æ¹MK8Â ç Æ:à ¸6ç4[®Ã#¹IÃË¿Á¦¸ÊÂ ¹I1®ÝÉ Æ ÂÈ¸ÊÂ ¹I=¿ ¸ Æ?ã Í Â ¸ÊÉ Æ ç Æ?à ¹ Æ O Ë¦¸ÊÂ ¹I ¹MK # ¼ ä %ãºÆ > Æ?à ¸Êç ¸ºÉ Æ Â ÍÃ à ¸.¹MK à ¹ÍÃ Æ ¸ºÂ ¸ºÂ ¹ QÂÈ¸ÊÉÂ QÃ¹ÃË¿Á¦¸ÊÂ ¹Id¹t Æ #çFÂ ¸IJPJ=ÍÂ à ç¹MK¿Á ã á¦ Æ ¿Á J

h3Æ O Ë#=¿ÁçC¸ÊÉ Æ Ã Æ?ã Â ¹ ¹K"¸ÊÂ Í Æ QÂÈ¸ÊÉÂ QÉÂ à É¬¸ºÉ Æà ¹ÍÃ Æ ¸ºÂ ¸ºÂ ¹

â Æ ¸IQ Æ?Æ ¿Á ã á Â ç'¸ºÉ Æ ç¸ ã ¹IHL Æ ç¸:6 ã =Í Æ ¸ ÆWãb

à É ã à ¸ ÆWã Â8; Æ ç ¸ºÉ Æ à Ë¸ Æ Æ çÊç'¹K¸ºÉÂÁç à ¹ÍÃ Æ ¸ÊÂÈ¸ÊÂ ¹I137 à ¸º¹ ã

α ≥ 1=ÃÃ Æ ã çË Æ ¸Ê¹ ¹¦á Æ?ã ¿Á=ÃÃÂ )L¹MK L Æ Æ?ã =¸ºÂ ¹çÂ

¿ â#¹ ã =¸º¹ ã JÃ#¹IÃË¿Á¦¸ÊÂ ¹I #ç ÆWÆ 7®ÂVL %T

egg larva imago

T

h1

h2

T T T

A BDCFE1+FI Hüù D A û A ñKD øWù óø þ ø ÿ E3H ñ L

ÝÉ Æ ¿ Â Æ ã =¿SJçºÂ çÅÉIççFÉ¹WQ ¸ÊÉ¦¸QÉ Æ

cr1h2 >

π

2, c < 1, br2h3 >

π

2, b > 1

=a

Â ç çFÍ¿ ¿µIçJçF¸ Æ Í # ¼ ä %É#ç¸IQÅ¹ =K ãºÆ O Ë Æ à JCç¸ Æ IPJ Í¹ Æ Æ ã ç¸ Æ HJqçF¸Ê=¸ ÆN1(t) ≡ K1

N2(t) ≡ K2

ää-5 #¯:¥°³ ± ¤ © ¥ ² ¯¦¤ ² © Ç Æ ¸ à ¹çºÂ ÆWã ¸ºÉ Æ K ¹¿ ¿ ¹WQÂ HLqÍq¹ Æ ¿VÎ

N′(t) =

αN(t − h)

1 +(

N(t−h)

K

)n − βN(t),# ¼ ¼ %

QÉ ÆWãÊÆα > 0

ÂÁç à ¹ Æ Z à Â Æ 3¸¹MKÃ ã ¹Ë à ¸ÊÂ ¹I1β > 0 à ¹ Æ Z à Â Æ 3¸Ý¹K ãÊÆ Ë à ¸ÊÂ ¹I1

n > 0¸ºÉ Æ ¹ ã ÆWã ¹MK Æ á3Â ã ¹Í Æ 3¸Å¹Ì ¿ Â Æ ã ÂÈ¸IJIK > 0

á Æ?ã L Æ ËÍ â Æ?ã ¹KÂ #Â áÂÁË¿ ç?ÝÉ Æ Â ÆWãÊÆ 3¸ÊÂ ¿ Â ÆWãÊÆ àTÆÆ O Ë¦¸ÊÂ ¹I1# ¼ ¼ % Éç Í¹ ãÊÆòà ¹ÍÃ¿ Â à ¦¸ Æ PJ=ÍqÌ

Â à çÝ¹MK®çº¹¿ Ë¸ÊÂ ¹Iç ¸ºÉ Æ O Ë#¦¸ºÂ ¹ #Vå è %Å= çºÍ=¿ ¿ Æ?ã ç Æ çFÂ ¸ºÂ áÂÈ¸IJ K ¹ ã8à ÉHLÂ HL¸ÊÉ ÆÍq¹ Æ ¿ [ çÃ# ã =Í Æ ¸ ÆWã ç à ¹ÍÃ ãºÆ QÂÈ¸ÊÉ Æ O Ë=¸ºÂ ¹ # ¼# å% ã ¹Ë à ¸ºÂ ¹ ã ¦¸ Æ Â # ¼ ¼ %Â ç Æ ç àTã Â â Æ .â=J ¸ºÉ Æ ·Â à É Æ ¿ ÂÁç · Æ I¸ Æ .¿Á Q K ¹ ã K Æ?ã Í Æ 3¸6¦¸ºÂ á Æ X3Â Æ ¸ºÂ à ç å¾µ5ÝÉ Æ Æ ¿Á J É#ç+â ÆWÆ 'Â I¸ ã ¹Ë àTÆ QÂÈ¸ÊÉ ãºÆ çFÃ Æ?à ¸+¸º¹C¸ºÉ Æ · à X Æ J 0¿ÁçÊç QÅ¹ ã X è I%¿ Æ:à Ì¸ ã ¹Â à =¿ ¹$L ¹K # ¼ ¼ % ÉIçÝâ Æ?Æ .Ã ãÊÆ ç Æ 3¸ Æ Â üå ä

ÝÉ Æ ÍÂ .Ã ã ¹Ã ÆWã ¸ºÂ Æ çÅ¹K Æ O Ë#¦¸ºÂ ¹ # ¼# ¼%Å ãÊÆ ¸ÊÉ Æ K ¹¿ ¿ ¹WQÂ HL ä å ää Î

è ÝÉ Æòà ¹I3á ÆWã L Æ àTÆ ¹KN(t)

¸Ê¹C¸ÊÉ Æ ç¸ Æ HJçF¸Ê=¸ ÆN(t) ≡ K n

√

αβ− 1

K ¹I¿ ¿ ¹WQ8ç%Â ¸IQ ¹QÝ JçWÎ%ÅÍ¹¹=¸º¹IÂ à =¿ ¿VJÂ K

0 <

(α − β)βn

α

− β ≤

1

he1+βh

;

â %Å¹Iç à Â ¿ ¿ =¸ºÂ HL ã ¹Ë'¸ºÉ Æ ¹3¸ ã Â áÂÁ=¿rçF¸ Æ IPJ çF¸Ê=¸ Æ Â K

1

he1+βh

<

(α − β)βn

α

− β ≤

√

(ω∗

h

)2 + β2,

QÉ ÆWãÊÆω∗

ÂÁçÅ¸ºÉ Æ çF¹I¿ Ë¸ÊÂ ¹I ¹Kx¸ÊÉ ÆòÆ O Ë=¸ºÂ ¹

−

1

βh

ω = tan ω, ω∗∈ (0, π) .

å"Â8 ÆWãÊÆ 3¸ºÂÁ=¿ Ì Â ÆWãÊÆ àWÆÀÆ O Ë¦¸ÊÂ ¹I:# ¼ ¼ % ÉIçÝ¹Ì à ¹IçF¸Ê=3¸Ã ÆWã Â ¹Â à çF¹I¿ Ë¸ÊÂ ¹Iç?ä 8¹¿ Ë¸ÊÂ ¹IçÝ¹MK Æ O Ë=¸ºÂ ¹ # ¼ ¼ % à =â Æ Éá Æòà É¹=¸ºÂ à ¿ ¿VJ

ÝÉ Æ çºÍ Æ Ã ã ¹Ã ÆWã ¸ºÂ Æ ç ãºÆ =¿Áçº¹qá¦¿ ÂÁ K ¹ ã ¸ºÉ ÆòÆ O Ë¦¸ÊÂ ¹I1Î

N′(t) =

αN(t − h)

1 + (αβ− 1)(N(t−h)

K)n

− βN(t) ,

# ¼ü½-%

QÉÂ à É.ÉIçç¸ Æ HJ ç¸6¦¸ Æ çº¹¿ Ë¸ÊÂ ¹IN(t) ≡ K

=#NK ¹ ã ¸ºÉ ÆòÆ O Ë¦¸ÊÂ ¹I1Î

N′(t) = α

N(t − τ(N))

1 + (N(t − τ(N))

K

)n

− βN(t),

τ(N) = h exp(

γ

(

1 −

N

Kn

√

α/β − 1

)

)

.

# ¼ 5 %

QÉ ÆWãÊÆ ¸ºÉ Æ Æ ¿ Jτ = τ(N)

Æ Ã Æ ç¹IN(t)

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350; K2 = 700

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α = 10; β = 1; γ = 0.5; h = 4; n = 7.7; K = 2800

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(<, Pn) , í I H¦ÝHPn

N ÝH ÞMKMJ H N Ý å àBæ Jç H ç à ß H¦Ý NTâ à@Ýáè Ýà ìïîÖN@J3N@C>I á ß3N@C HX

KMJâ àX,

N ÝHß ÝàTÜH çð F I H6áHñÝH6áæ Þãâ á N ÝH N@ß<ß3ÞMK H ç8â à+á â æ çòÇC à J ÜH¦ÝéBH J3C H N@C6C H Þ H¨Ý N:âK à J æ3á KMJ é%éBH J H¦Ý N@ÞMKMê H çë%K H6á êÇì H âI à çN:Jç é@H J H¦Ý N:ÞMKóê H ç'ôò é ì æ J3çõì H âI à çðö÷øpù"úû:ü5ýBþ ÞMKóJ H N Ýáæ ìõìäN å KMÞóKãâVòì H âI à ç á6ÿ C à J ÜH¦Ýé@H J3C H N@C¦C H Þ H¦Ý N:âK à J ÿ F N æ å H¦Ý KóN@JÝH ìäN@KMJç H¦Ý âI H6à:ÝH ì á

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$1ÆT´X, Y

ÍÆ&%Ä©?¬©<È(' ³¨Ì©È:Æ@³©?¬ÊL (X, Y )

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X

« ¬<´¦µY.

³¨Æ.-+Æ@¬ÈTÆx = (ξk) (ξk ∈ X)

«¶³"È@©?ª ª ÆBÊλ

·ÅÍµ*+¬ÊÆ@Ê«/)

∃ lim ξk = ξ ∧ βk = λk (ξk − ξ) ∧ βk = O (1) ,

0 'Æ:Î¦Æ@©<³λ = (λk) 0 «ó´1'

0 < λk .$1ÆT´m

λX

Í#Æ´1'Æä³¨ÆT´Çµ2)Ö©?ª ªλ

·ÅÍ#µ3+¬ÊÆBÊ.³>Æ4-3+Æ:¬È:Æ@³@|5)λk = O (1) ,

´6'Æ@¬m

λX =

cX ,0 '« ª Æ

cX

« ³© ³>Æ:´µ*) ÈTµ<¬ÉÆ:Î17Æ@¬3´³>Æ4-+Æ:¬È:Æ@³ 0 « ´6'ξk ∈ X.

³>Æ4-3+Æ:¬È:Æx =

(ξk)« ³È@©?ª ª ÆBÊy98 3:T &;³¨Æ:Æ=< >8®4>@?BAÍC ©D7Æ:¬Æ:Î6©?ª «FE@Æ@Ê ¸Æ:´6'µÊ

A = (Ank),Ank ∈ L (X, Y )

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ηn =∞∑

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0 < µk .

~'Æ´¨Î6©?¬#³)µ<Î¨¸©´¦« µ<¬A

«¶³8È:©ª ª Æ@Ê©È:È:Æ:ª Æ:Î6©´¦« ¬7

λ

·¤Íµ*+¬#ÊÆ@Ê¬Æ@³¦³Ä«/)Am

λX ⊂ m

µY0 « ´6'

lim µk/λk = ∞.y¸ÆT´1'µÊA = (Ank) 0 « ´6'

(Ank ∈ L (X, X))«¶³ È@©?ª ª ÆBÊ¨Å§*8 «F)

AcX ⊂ cX©?¬Êlimn

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ξk ,

>?Ð3Ï 8# :¨?

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λ

·ÅÍ#µ3+¬ÊÆBÊ¬ÆB³¨³%« ¬'©ÌÌª « ÆBÊ¸©´6'Æ:¸©´¦« È@³ª « ¬ÆB©?Î¸ Æ:´6'µÊ³¡©?Î¦Æ +³¨Æ@Êõ´¨µ©<È:È:Æ:ª Æ:Î6©´¨ÆÖ´1'ÆÄÈTµ¬É<Æ:Î17Æ:¬#ÈTÆ ;³¨Æ:Æ< ®B°@?BAT?Ë+È('G©¬G©È@ÈTÆ:ª Æ:Î6©´¦« µ<¬« ³%Ì#µ3³¨³¨« Íª Æ8«F) 0 Æ +³¨Æ"³>µ<¸Æ"³ +Í³>Æ:´¦³%µ2)

mλX .

8ª¶³>µ )µ<Î ©<È:ÈTÆ@ª Æ@Î¦©?´¨« µ¬« ´ «¶³+Ìµ<³¦³>« Íª ÆÇ´¨µ+³>Æ¬µ<¬Î¦Æ 7*+ª ©Î¸Æ:´6'µÊ³Çµ<ÎÄ« ¬³¨µ¸ÆõÈ:©³¨Æ@³´1'ÆõÌ³¨Æ +Êµ<³6+¸¸©Í« ª « Ć ;³¨Æ:Æ < ®@°@? A

|¬G< ®BÏ@?"µ<Î¨¬)Æ:ª¶Ê ³ÖÎ¦Æ@³6+ª ´« ³7<Æ:¬Æ@Î¦©ª « E:ÆBÊ )µ<Î+³>Æ@ÉÆ@Î¦©ª3¸Æ:´6'µÊ³A = (Ank) 0 « ´6'

(

Ank ∈ L (X, X))

.

|¬ < ¯ ?ÇÈTÆ@Î>´6©?« ¬ ¬+¸ Æ@Î¨«¶È:©ª ³6+¸¸©?Í« ª « Ć ¸ÆT´6'µÊ³©Î¨Æ¥È:µ¸ ·Ì©?Î¦Æ@ÊÍCäÎ¦©?´¨ÆB³dµ*)È:µ¬ÉÆ@Î67<Æ:¬È:ÆB|¬ <ãÏ@?©Î¨ÆÌÎ¦Æ@³¨Æ:¬3´¨ÆBÊG¸©?« ¬ Î¨ÆB³ +ª ´¦³Öµ*)È:µ¬ÉÆ@Î67<Æ:¬È:Æ©È:È:Æ:ª Æ:Î6©´¦« µ<¬ +³¨« ¬ 7 ¬µ¬ª « ¬Æ@©?ÎÄ¸ÆT´1'µÊ³@$1ÆT´ +³ ÊÆ:¬µ´¨ÆiÍC

(<, Pn)µ<Î³6'µÎ¨´¨ª CbÍC

<

´6'ÆG7Æ@¬Æ:Î6©?ª « E:Æ@Ê Ç« Æ@³6E¥¸Æ:´6'µÊ;³¨Æ:Æ < ®?.? AÊÆ ¬ÆBÊ ÍC

Ank = Rnk =

RnPk (k = 0, 1, . . . , n) ,

θ (k > n) ,

0 'Æ:Î¦ÆPk, Rn ∈ L (X, X) ,

0 '« ª ÆRn

« ³8ÊÆT´¨Æ@Î¨¸« ¬Æ@Ê ÍC

Rn

n∑

k=0

Pkζ = ζ (ζ ∈ X, n ∈ N0) .

;>®ãÏ*A

$1ÆT´Bn

Í#ÆG©¥³>Æ4-3+Æ:¬È:Æäµ2)ÖµÌÆ:Î6©´¦µÎ6³Bn ∈ L (X, X)

³¦©´¨«¶³ ) C3« ¬7´1'ÆGÈTµ<¬Ê« ·´¨« µ¬³

(n + 1) ‖Bn+1 − Bn‖ = O (‖Bn‖) (n ∈ N);®< >A

©?¬ÊB0 = θ, Bn 6= θ (n ∈ N) .

$1ÆT´ +³ÊÆ@¬µ?´¦ÆÍC(Z , Bn)

µ<Î³6'µ<Î>´¦ªFC.ÍCZ

´1'Æ7<Æ:¬Æ@Î¦©ª « E:ÆBÊC7¸+¬Ê¸Æ:´6'µÊ©³Ç© 7Æ@¬Æ:Î6©?ª «FE@Æ@ÊiÇ« ÆB³ Eõ¸Æ:´6'µÊ

(<, Pn) 0 « ´6'Pn = Bn+1 − Bn.

ÙÛ ; <.? A¦??V?

lim ‖Rn‖ = 0;>® ÐA

‖Rn‖

n∑

k=0

‖Pk‖ = O(1);®< 3A

85M£'G:<

¶õ¨Å§28 ? Ù1Û

B−1n ∈ L (X, X) (n ∈ N) ,

lim∥

∥

B−1n

∥

∥ = 0;>® ²3A

∥

∥

B−1

n+1

∥

∥

n∑

k=0

‖Bk‖

k + 1= O(1),

;®<*A

T(Z , Bn)

¶õ¨Å§28 ?

::¨?MB äT +¢:V§<:6"66: T¦¤ >?Ð3>¨B "³¨« ¬7õ´6'Æ"ÊÆ #¬«ó´¦« µ<¬'µ2)5´1'Æ8¸ÆT´1'µÊ

(Z , Bn)©?¬#Ê $1Æ:¸¸©® 0 Æ 7Æ:´+´6'#©´

;® ²3A « ¸Ìª « Æ@³;® ÐAT "³¨« ¬ 7 ;®< >A ©¬ÊD;®<*A 0 Æ 7Æ:´

‖Rn‖

n∑

k=0

‖Pk‖ =∥

∥

B−1

n+1

∥

∥

n∑

k=0

‖Bk+1 − Bk‖

=∥

∥

B−1n+1

∥

∥

n∑

k=0

O(1)‖Bk‖

k + 1= O(1).

~'+#³Ç´6'ÆÈTµ<¬Ê« ´¨« µ¬ ;>®*AÄ«¶³³¦©´¦« ³ ÆBÊ©?¬Ê.ÍC $1Æ:¸¸©®õ´1'ÆG¸Æ:´6'µÊ(Z , Bn)

«¶³Î¨Æ.7*+ª¶©?ÎB

ÙñÛ ; < ®Ï4?BAA = (Ank)

(Ank ∈ L (X, X))

¶ ¨Å§*8 V?3§*8 ?ViT :¤¶6£3§i ¦??V?

n∑

k=0

Ank = I (n ∈ N0) ,

TA

65Ç66T :¨?ÅG6#¢:V§3T56"³¨« ¬7 $ñÆ:¸¸©³8Ï ©?¬ÊG> 0 Æ 7<ÆT´Ä´6'Æõ¬Æ ´"ÈTµ<Î¨µ<ª ª¶©?Î1C

?¨ £ ~'Æ¸ Æ:´6'µÊZ ,

³¦©´¦« ³ ) C« ¬ 7ä´6'ÆÈTµ<¬Ê« ´¨« µ¬³ ;®< ²A©?¬#Ê ;®<*AÈ:©?¬¥¬µ?´©È:È:Æ:ª Æ:Î6©´¦Æ"´1'ÆäÈTµ<¬3É<Æ:Î17Æ@¬ÈTÆ<

|¬¥´1'Æä³>Æ4-+Æ:ªλk ∞

µk ∞

©?¬Êτk ∞

ÙdÛ ; < ®<®9?BA856?V? ;® ÐA ;®*A

µn ‖Rn‖

n∑

k=0

‖Pk‖

λk

= O(1)

¨ä98 :T#T?õä5T/86?<m

λX ⊂ m

µX .

?¨ £|5)B

−1n ∈ L (X, X) (n ∈ N) ,

´6'Æ@¬¥´1'ÆõÈTµ¬#Ê«ó´¦« µ<¬³ ;®< ²A ;®<*A ©¬Ê

µn

∥

∥

B−1n+1

∥

∥

n∑

k=0

‖Bk‖

(k + 1)λk

= O(1)

©?Î¦Æ³6+'ÈT« Æ:¬3´)µÎÄ´1'Æõ« ¬ÈTª +³¨« µ<¬Zm

λX ⊂ m

µX .

~ËÎ¦¸+³,;³¨Æ:Æ < ±2?BA¡³>´6+Ê« ÆBÊG©?¬« ¬É<Æ:Î6³>Æ ÌÎ¨µ<Íª Æ:¸p©?¬ÊGÌÎ¦µÉÆ@ÊG³>µ·ÅÈ@©?ª ª ÆBÊõ~d©*+Í#Æ@Î¨«¶©?¬´6'Æ@µÎ¦Æ:¸ )µÎ 7Æ@¬Æ:Î6©?ª « E:Æ@ÊGÇ« ÆB³ E¸ÆT´1'µÊ

<

« ¬´6'ÆÄÈ:©<³>Æλn = O(1)

©?¬Êµn = O (1)©?¬Ê 7µ´´1'Æõ¬Æ3´8Î¦Æ@³6+ª ´@

Ù1Û TP

−1

k , R−1n ∈ L (X, X)

G§<::¨ó@ !TT <

:6¨Å§#"8

x = (ξk)¶

(<, Pn) $98*:: GÅ

η

?

P−1

k R−1

k ∆ξk = o(1),

:¨6∆ξk = ξk − ξk−1,

:ξk → η.

>?ÐÐ 8# :¨?~'Æ)µ<ª ª µ 0 « ¬7³¨µ?·È:©ª ª Æ@Ê'~¡©2+Í#Æ@Î¨«¶©?¬'Î¦Æ:¸©« ¬ÊÆ:Î ´6'Æ@µÎ¦Æ:¸ )µÎ7<Æ:¬Æ@Î¦©ª « E:ÆBÊ¸ Æ:´6'µÊµ2)³ +¸¸©?Í« ª «ó´C

A

«¶³ÄÌÎ¦µÉÆBÊ'« ¬ < ®BÏ4?Å ÙñÛ T

A = (Ank)

Ank ∈ L (X, X):6 i¨Å§*8 V?3§*8 §<: "

T¨?MB 'T ¥>98*::M¤£:¤¶6£3§iG6 ¤

λn

n∑

k=0

λ−1

k ‖Ank‖ = O(1),;®< ±A

λk

∥

∥

∥

∥

∥

k∑

ν=0

Anν

∥

∥

∥

∥

∥

‖∆ξk‖ = O(‖Ank‖) (k ≤ n, n ∈ N0) ,

;>® ¯3A

λn

∥

∥

∥

∥

∥

n∑

k=0

Ank − I

∥

∥

∥

∥

∥

‖ξn‖ = O (1) .

;® ®@°A

TA x ∈ m

λX

85MTx ∈ m

λX .

Ù1Û ; < 4? A 8T6¨Å§*8 "TT (<, Pk)

Pk > 0

R−1

k = O

(

R−1

k−1

)

¶ 5¨TT:¢B3§λ

" :628 ?T¦G?

τkR−1

k ∆ξk = O (Pk) ,

TRx ∈ m

λR

85Mx ∈ m

λR

, T¦¦

1 ≤

λk

τk

↑, λkτk ↑, µk =√

λkτk.

#6 " "

Ø Õ× #Û ¦??V? ;>® ÐA6 ;>®*Aä?

λn ‖Rn‖

n∑

k=0

‖Pk‖

λk

= O(1);VÏ ®@A

85M£λn ‖Rn‖

λν ‖Rν‖= O (1) (ν ≤ n) ,

;VÏ Ï3A

λn ‖Rn‖ = o (1) .

;VÏ >A

¨B "³

n∑

k=0

‖Pk‖

λk

≥

ν∑

k=0

‖Pk‖

λk

≥

1

λν

ν∑

k=0

‖Pk‖ ≥

1

λν ‖Rν‖‖Rν‖

ν∑

k=0

‖Pk‖

::¨?MB äT +¢:V§<:6"66: T¦¤ >?Ð ©?¬Ê ;®< Ï3A« ¸Ìª « ÆB³

‖Rν‖

ν∑

k=0

‖Pk‖ ≥ 1,

;¤Ï ÐA

´6'Æ@¬n

∑

k=0

‖Pk‖

λk

≥

1

λν ‖Rν‖.

;¤Ï*A

"³¨« ¬7 ;VÏ ®4A©¬Ê ;¤Ï*A 0 Æ 7ÆT´ ;¤ÏãÏ*A~'ÆäÈTµ¬#Ê«ó´¦« µ<¬ ;¤ÏãÏ*A« ¸Ìª « Æ@³

λn ‖Rn‖ = O (1) .

~'Æ:Î¦Æ )µ<Î¨Æ´6'Æ@Î¨ÆõÆ« ³>´¦³Ç³6+È('m > 0,

´6'©?´

1

λn

≥ m ‖Rn‖ .

;¤Ï ²3A

Î¦µ¸ ;¤Ï ²3A©?¬ÊD;¤Ï ÐA «ó´ )µ<ª ª µ 0 ³´6'#©´

‖Pn‖

λn

≥ m ‖Rn‖ ‖Pn‖ ≥ m

‖Pn‖

n∑

k=0

‖Pk‖

.

Î¦µ¸ ;>® ÐA©?¬ÊD;¤Ï ÐA 0 Æ 7<ÆT´

n∑

k=0

‖Pk‖ → ∞.

~'Æ:Î¦Æ )µ<Î¨Æ´6'Æä³¨Æ:Î¦« Æ@³ ∑

∞

k=0‖Pk‖ /λk

«¶³ÇÊ« ÉÆ:Î17Æ@¬3´@"³¨« ¬ 7 ;VÏ ®@A 0 Æ 7Æ:´;VÏ >3AT

?¨ £ ~'ÆäÈ:µ¬Ê« ´¨« µ¬#³R

−1n ∈ L (X, X) ,

;®< ÐA ;®< 3A;¤Ï ®4A©?¬#Ê∥

∥

R−1n

∥

∥ = O

(∥

∥

R−1

n−1

∥

∥

)

« ¸Ìª Cλn = O (λn−1) .

":? 3|¬È@©³¨ÆGµ2)¬+¸GÍÆ:Î6³©¥Î¦Æ@³6+ª ´@ ³>« ¸« ª¶©?Î´¨µ+Î¦µÌµ<³¨«ó´¦« µ<¬ ®< 0 ©³ÌÎ¦µÉÆBÊÍC õ©?¬7<Î¨µ ;³¨Æ:Æ < Ð2? A Ø Õ× Û

λn ‖Rn‖

n∑

k=0

‖Pk‖

λk

= o(1),

T<

¶G<¦6: :¨¤3§λ

" :¦28< ?T6

¨B $1Æ:´x ∈ m

λX

©¬Êηn

« ³"ÊÆ ¬Æ@Ê ÍC ;®< ®@A ;®ãÏ*ATË« ¬ÈTÆ

n∑

k=0

RnPk = I

>?Ð<² 8# :¨?©?¬Ê +³¨« ¬7 ;VÏ*A 0 Æ 7Æ:´

λn ‖ηn − η‖ = λn

∥

∥

∥

∥

∥

n∑

k=0

RnPkξk −

n∑

k=0

RnPkξ

∥

∥

∥

∥

∥

= λn

∥

∥

∥

∥

∥

n∑

k=0

RnPk (ξk − ξ)

∥

∥

∥

∥

∥

≤ λn

n∑

k=0

‖Rn‖ ‖Pk‖ ‖ξk − ξ‖

≤ λn ‖Rn‖

n∑

k=0

‖Pk‖

λk

λk ‖ξk − ξ‖ ≤ λn ‖Rn‖

n∑

k=0

‖Pk‖

λk

O (1)

≤ o (1) O (1) = o(1),´6'Æ@¬i´1'Æõ¸ÆT´1'µÊ

(<, Pn)«¶³Ç©È:È:Æ:ª Æ:Î6©´¦« ¬7

λ

·¤Íµ*+¬#ÊÆ@Ê¬Æ@³¦³:

?¨ £|5)B

−1n ∈ L (X, X) (n ∈ N)

©¬Ê

λn

∥

∥

B−1

n+1

∥

∥

n∑

k=0

‖Bk‖

(k + 1) λk

= o(1),

´6'Æ@¬i´1'Æõ¸ÆT´1'µÊZ

«¶³Ç©È@ÈTÆ@ª Æ@Î¦©?´¨« ¬7λ

·ÅÍ#µ3+¬ÊÆBÊ¬Æ@³¦³@Ò8Æ ´ 0 Æ )µÎ¦¸+ª¶©´¦Æ"´1'Æ )µª ª µ 0 « ¬7ÌÎ¦µÍª Æ:¸.¹

¨*:: : 3|³«ó´Ì#µ3³¨³¨« Íª Æõ´¦µ ¬Ê©'³>« ¸Ìª ÆG¬µ<¬Î¨Æ.7*+ª¶©?Î7Æ:¬Æ:Î6©?ª «FE@Æ@Ê ¸ÆT´6'µÊ1³¦©·´¨«¶³) C« ¬7´6'ÆäÈ:µ¬Ê«ó´¦« µ<¬³Äµ2) +Î¨µ<Ì#µ3³>« ´¨« µ¬ÏµÎ µ<Î¨µ<ª ª¶©?Î1CÐ|¬b´1'Æ'³¨Æ.-+Æ:ª 0 Æ 7Æ:´G~d©*+ÍÆ:Î¦« ©¬bÎ¦Æ:¸©?« ¬ÊÆ@Î´6'Æ@µÎ¦Æ:¸³ )µÎ´1'Æ'¸ÆT´1'µÊ³

<

©?¬ÊZ .

Ø Õ× 1Û : 6?V? ;® ÐA ;®< 3A6 ;VÏ ®@AG

λk

∥

∥

RnR−1

k

∥

∥

‖∆ξk‖ = O (‖RnPk‖) (k ≤ n, n ∈ N0);¤Ï 2A

:¦ä:¤¶ 85:<x ∈ m

λX

"óx ∈ m

λX .

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<x ∈ mλX

« ¸Ìª « ÆB³x ∈ m

λX .

?¨ £ $1Æ:´B

−1n ∈ L (X, X) (n ∈ N)

©¬Ê ´6'ÆbÈTµ<¬Ê« ´¨« µ¬³ ;®< ÐA;>® 2AT;® ±3AT

λn

∥

∥

B−1n+1

∥

∥

n∑

k=0

‖Bk‖

(k + 1) λk

= O(1)

©?¬Ê

λk

∥

∥

B−1

n+1Bk+1

∥

∥

‖∆ξk‖ = O

(∥

∥

B−1

n+1 (Bk+1 − Bk)∥

∥

)

(k ≤ n, n ∈ N0)Í#Æä³¦©´¦« ³ Æ@Ê1~'Æ:¬Zx ∈ m

λX

« ¸ Ìª « Æ@³x ∈ m

λX .

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W $XIYBZM<\[X<QT%,0 <CFH$&%I< ÿItG_ÿ D:E ;Ä= D ; E ÿ#;R@R U ð AäF ð@P,m Ý ì æ<á ð F N æ å H¨Ý K N:JÇâI H6à:ÝH ì á#è à@Ý5é@H J H¨Ý N@ÞMKMê H ç áæ ìõìäN å KóÞMKãâ¤ò ì H âI à ç á KóJ"îÖN:JN:C¨Iá ß3N@C H6á ðDT> UV< W $X'YB;M<l[@X<QT^%,0 <lFg$&%I< ÿ'tGuÿ;VN E =;A D ÿ D:EBE@E ð R 5"ð Pâ>NBçâìv<ÞMÞ H¨Ý N@J3çf"ð F N:ÞóKð 8¡à ìõß3N Ý K áà J à:è C H¦Ý â>N:KóJ áæ ìõìäN å KMÞóKãâVò'ì H âI à ç á å òá ß H6H ç á+à@è C à J ÜH¦Ýé@H J3C H ðwB;@ 0 "& 4FH$&%I!R$/ ÿK@uÿ DBD N"= D 9 D ÿ D:EBE 7 ð ; E O>ð F N:ìõì H¦Ý N@Kóçð 8¡à J ÜH¦ÝéBH J3C HÖÝ NTâ H%à:è Kãâ H¦Ý N:âK ÜH ß Ýà C H6áá N@J3ç í H K é I?â H çì H N@J á ðGT> UV<

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MATHEMATICAL MODELLING AND ANALYSIS MODELLING/vo… · During the reign of Kestutis Naujieji Trakai...

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Transcript of MATHEMATICAL MODELLING AND ANALYSIS MODELLING/vo… · During the reign of Kestutis Naujieji Trakai...