Potential Function in Fredholm ــ Volterra Integral … · Potential Function in Fredholm ــ...
Transcript of Potential Function in Fredholm ــ Volterra Integral … · Potential Function in Fredholm ــ...
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Umm Alــ Qura University
Faculty of Applied Sciences
Department of Mathematical Sciences
Potential Function in
Fredholm ــ Volterra Integral Equation
A Thesis Submitted in Partial Fulfillment of the Requirements of
Master's Degree
In
Applied Mathematics
( Integral Equations )
Prepared by Researcher
Faizah Mohamed Hamdi Alــ Saedy
Supervised by
Prof. Mohamed Abdella Ahmed Abdou
1427 AH2006 ــG
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References
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their origins in integral equation theory. Arch. Hist. Exact. Sci. Vol. 3 (1966) 1- 96.
[2] C.D.Green, Integral Equation Methods, NewYork, 1969.
[3] H.Hochstadt, Integral Equations, Awiley Inter Science Publication, NewYork,
1971.
[4] M.A.Golberg.ed. Solution Methods for Integral Equations, NewYork, 1979.
[5] F.G.Tricomi, Integral Equations, Dover, NewYork, 1985.
[6] T.A.Burton, Volterra Integral and Differential Equations, London, NewYork,
1983.
[7] R.P.Kanwal, Linear Integral Equations Theory and Technique, Boston, 1996.
[8] P.Schiavone, C.Constanda and A.Mioduchowski, Integral Methods in Science
and Engineering, Birkhauser Boston, 2002.
[9] N.I.Muskhelishvili, Singular Integral Equations, Noordhoff, Groningen, The
Netherland, 1953.
[10] Peter Linz, Analytic and Numerical Methods for Volterra Equations, SIAM,
Philadelphia, 1985.
[11] K.E.Atkinson, A Survey of Numerical Method for the Solution of Fredholm
Integral Equation of the Second Kind, Philadelphia, 1976.
[12] K.E.Atkinson, The Numerical Solution of Integral Equation of the Second Kind,
Cambridge University, Combridge, 1997.
[13] Christopher T.H.Baker, Treatment of Integral Equations by Numerical Methods ,
Academic Press, 1982.
[14] L.M.Delves and J.L.Mohamed, Computational Methods for Integral Equations,
NewYork, London, 1985.
[15] M.A.Golberg.ed, Numerical Solution for Integral Equations, NewYork, 1990.
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[16] M.A.Abdou, FredholmــVolterra integral equation of the first kind and contact
problem, J. Appl. Math. Comput. 125 (2002) 177193ــ.
[17] M.A.Abdou, FredholmــVolterra integral equation and generalized potential
kernel, J. Appl. Math. Comput. 131 (2002) 8194ــ.
[18] M.A.Abdou, On asymptotic methods for FredholmــVolterra integral equation of
the second kind in contact problem, J. Comp. Appl. Math. 154 (2003) 431446ــ.
[19] M.A.Abdou, FredholmــVolterra integral equation with singular kernel, J. Appl.
Math. Comput. 137 (2003) 231243ــ.
[20] M.A.Abdou, F.A.Salama, VolterraــFredholm integral equation of the first kind
and spectral relationships, Appl. Math. Comput. 153 (2004) 141153ــ.
[21] M.A.Abdou, O.L.Moustafa, FredholmــVolterra integral equation in contact
problem, J. Appl. Math. Comput. 138 (2003) 199215ــ.
[22] M.A.Abdou, A.A.Nasr, On the numerical treatment of the singular integral
eqution of the second kind, J. Appl. Math. Comput. 146 (2003) 373380ــ.
[23] M.A.Abdou, Fredholm integral equation with potential kernel and its structure
resolvent, Appl. Math. Comput . 107 (2000) 169 180 ـ .
[24] I.S.Gradshteyn and I.M.Ryzhik, Table of Integrals, Series and Products,
Academic Press, New York, 1980 .
[25] H.Bateman, A.Erdely, Higher Transcendental Functions, Vol.2, Nauka Moscow
1973 .
[26] S.M.Mkhitarian, M.A.Abdou, On different methods for solving the integral
eqution of the first kind with logarithmic kernel, Dokl. Acad. Nauk. Armenia 90
.10ــ1 (1990)
[27] E.V.Kovolenko, Some approximate methods of solving integral equations of
mixed problem, Appl. Math. Mech. 53 (1) (1989) 8592 ـ .
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[28] M.A.Abdou, N.Y.Ezzeldin, Krein's method with certain singular kernel for
solving the integral equation of the first kind, Period. Math. Hung. 28 (2) (1994)
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[29] M.A.Abdou, K.I.Mohamed and A.S.Ismal, Toeplitz matrix and product Nystrom
methods for solving the singular integral equation, Le Mathematicle, Vol. LVII
.37ــFasc 1.pp. 21ـ(2002)
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ntial equation, Quart. Appl. Math. 56 (1996) 409424ــ.
[31] Orsi, A.Palamara, Product integration for Volterra integral equation of the
second kind with weakly singular kernel. Math. Comp. Vol. 65 No.215 (1996)
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[32] N.K.Artiunian, Plane contact problems of the theory of creel. Appl. Math.Mech.
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)مكة المكرمة(معة أم القرى جا
آـلـيـة الــعــــلوم الـتـطـبـيقـيـــة
قـســم الـعــلـوم الـريــاضـيـــــة
ـةــــي مــعــادلـــــــــد فــدالـــة جــهــ
فـردهـولـم ــ فـولـتـيـرا الـتـكـامـلـيــة
ث تكميلي مقدم لنيل درجة الماجستير ــحـب
فـــــــــي
الـتطبيـقـيـــــةاتـياضـريـال
)معادالت تكـامـلـيـــــــــة (
إعداد الباحثـــة
فايـــزة محمـد حـمـدي الـصـاعــــدي
تحت إشـراف
محمـد عبد الـاله أحمــد عبــده/ األســتاذ الـدآتــور
142٧ م٢٠٠٦ هـ ــ
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Contents
Introduction ………………………………………………………… i ـ v
Chapter 1 Basic Concepts
§ 1.1 Definitions and theorems . ………………………………………1
§ 1.2 Laplace transformation …………………………………………..12
§ 1.3 Classification of integral equations …………………………….. 16
§ 1.4 Fredholm theorems for continuous kernel ………………………..21
§ 1.5 Integral operator ………………………………………………….24
§ 1.6 Compact operator …………………………………………………28
Chapter 2 Volterra Integral Equation
§ 2.1 Existence and uniqueness solution of Volterra equation…………. 37
§ 2.2 The resolvent kernel method …………………………………….. 44
§ 2.3 Solution method using Laplace transformation …………………. 49
§ 2.4 Method of successive approximation …………………………… 51
§ 2.5 Volterra integral equation of the first kind ……………………… 53
Chapter 3 Fredholm ــ Volterra Integral Equation with Potential Kernel
§ 3.1 Introduction ……………………………………………….. 62 § 3.2 Existence and uniqueness solution of the integral equation……… 67
§ 3.3 Continuity and normality of integral operator …………………... 72
§ 3.4 The kernel of position …………………………………………… 75
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Chapter 4 Series Method
§ 4.1 Separation of variables method …………………………………. 86
§ 4.2 Discussion and special cases …………………………………… 94
Chapter 5 Applications for Potential Kernel
§ 5.1 Electrostatic potential…………………………………………… 99
§ 5.2 Torsion of an isotropic elastic plate …………………………… 103
§ 5.3 Mechanics and mixed problem ………………………………… 107
§ 5.4 Raditions and molecular condition …………………………… 107
§ 5.5 Discussion and results ………………………………………….. 109
Appendix ………………………………………………………………. 114
References................................................................................................ 115
Arabic summary ……………………………………………………. أ ــ ج