Research Article Solution of Nonlinear Partial...
6
Research Article Solution of Nonlinear Partial Differential Equations by New Laplace Variational Iteration Method Eman M. A. Hilal 1 and Tarig M. Elzaki 2 1 Mathematics Department, Faculty of Sciences for Girls, King Abdulaziz University, Jeddah, Saudi Arabia 2 Mathematics Department, Faculty of Sciences and Arts, King Abdulaziz University, P.O. Box 110, Alkamil 21931, Saudi Arabia Correspondence should be addressed to Tarig M. Elzaki; [email protected] Received 17 December 2013; Revised 14 February 2014; Accepted 24 February 2014; Published 26 March 2014 Academic Editor: Leszek Olszowy Copyright © 2014 E. M. A. Hilal and T. M. Elzaki. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. e aim of this study is to give a good strategy for solving some linear and nonlinear partial differential equations in engineering and physics fields, by combining Laplace transform and the modified variational iteration method. is method is based on the variational iteration method, Laplace transforms, and convolution integral, introducing an alternative Laplace correction functional and expressing the integral as a convolution. Some examples in physical engineering are provided to illustrate the simplicity and reliability of this method. e solutions of these examples are contingent only on the initial conditions. 1. Introduction Nonlinear equations are of great importance to our contem- porary world. Nonlinear phenomena have important appli- cations in applied mathematics, physics, and issues related to engineering. Despite the importance of obtaining the exact solution of nonlinear partial differential equations in physics and applied mathematics, there is still the daunting problem of finding new methods to discover new exact or approximate solutions. In the recent years, many authors have devoted their attention to study solutions of nonlinear partial differential equations using various methods. Among these attempts are the Adomian decomposition method, homotopy pertur- bation method, variational iteration method [1–5], Laplace variational iteration method [6–8], differential transform method, and projected differential transform method. Many analytical and numerical methods have been pro- posed to obtain solutions for nonlinear PDEs with fractional derivatives, such as local fractional variational iteration meth- od [9], local fractional Fourier method, Yang-Fourier trans- form, and Yang-Laplace transform. Two Laplace variational iteration methods are currently suggested by Wu in [10–13]. In this work, we will use the new method termed He’s Laplace variational iteration method, and it will be employed in a straightforward manner. Also, the main result of this paper is to introduce an alternative Laplace correction functional and express the integral as a convolution. is approach can tackle functions with discontinuities and impulse functions effectively. 2. New Laplace Variational Iteration Method To illustrate the idea of new Laplace variational iteration method, we consider the following general differential equa- tions in physics: [ (, )] + [ (, )] = ℎ (, ) , (1) where is a linear partial differential operator given by 2 / 2 , is nonlinear operator, and ℎ(, ) is a known analytical function. Hindawi Publishing Corporation Journal of Function Spaces Volume 2014, Article ID 790714, 5 pages http://dx.doi.org/10.1155/2014/790714
Transcript of Research Article Solution of Nonlinear Partial...
Research ArticleSolution of Nonlinear Partial Differential Equations by NewLaplace Variational Iteration Method
Eman M A Hilal1 and Tarig M Elzaki2
1 Mathematics Department Faculty of Sciences for Girls King Abdulaziz University Jeddah Saudi Arabia2Mathematics Department Faculty of Sciences and Arts King Abdulaziz University PO Box 110 Alkamil 21931 Saudi Arabia
Correspondence should be addressed to Tarig M Elzaki tarigalzakigmailcom
Received 17 December 2013 Revised 14 February 2014 Accepted 24 February 2014 Published 26 March 2014
Academic Editor Leszek Olszowy
Copyright copy 2014 E M A Hilal and T M Elzaki This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited
The aim of this study is to give a good strategy for solving some linear and nonlinear partial differential equations in engineeringand physics fields by combining Laplace transform and the modified variational iteration method This method is based on thevariational iterationmethod Laplace transforms and convolution integral introducing an alternative Laplace correction functionaland expressing the integral as a convolution Some examples in physical engineering are provided to illustrate the simplicity andreliability of this method The solutions of these examples are contingent only on the initial conditions
1 Introduction
Nonlinear equations are of great importance to our contem-porary world Nonlinear phenomena have important appli-cations in applied mathematics physics and issues related toengineering Despite the importance of obtaining the exactsolution of nonlinear partial differential equations in physicsand applied mathematics there is still the daunting problemof finding newmethods to discover new exact or approximatesolutions
In the recent years many authors have devoted theirattention to study solutions of nonlinear partial differentialequations using various methods Among these attemptsare the Adomian decomposition method homotopy pertur-bation method variational iteration method [1ndash5] Laplacevariational iteration method [6ndash8] differential transformmethod and projected differential transform method
Many analytical and numerical methods have been pro-posed to obtain solutions for nonlinear PDEs with fractionalderivatives such as local fractional variational iterationmeth-od [9] local fractional Fourier method Yang-Fourier trans-form and Yang-Laplace transform Two Laplace variationaliteration methods are currently suggested by Wu in [10ndash13]
In this work we will use the new method termed HersquosLaplace variational iteration method and it will be employedin a straightforward manner
Also the main result of this paper is to introduce analternative Laplace correction functional and express theintegral as a convolution This approach can tackle functionswith discontinuities and impulse functions effectively
2 New Laplace Variational Iteration Method
To illustrate the idea of new Laplace variational iterationmethod we consider the following general differential equa-tions in physics
119871 [119906 (119909 119905)] + 119873 [119906 (119909 119905)] = ℎ (119909 119905) (1)
where 119871 is a linear partial differential operator given by12059721205971199052 119873 is nonlinear operator and ℎ(119909 119905) is a known
analytical function
Hindawi Publishing CorporationJournal of Function SpacesVolume 2014 Article ID 790714 5 pageshttpdxdoiorg1011552014790714
2 Journal of Function Spaces
According to the variational iteration method we canconstruct a correction function for (1) as follows119906119899+1
(119909 119905)
= 119906119899(119909 119905)
+ int
119905
0
120582 (119909 120589) [119871119906119899(119909 120589) + 119873
119899(119909 120589) minus ℎ (119909 120589)] 119889120589
119899 ge 0
(2)where 120582 is a general Lagrange multiplier which can beidentified optimally via the variational theory the subscripts119899 denote the 119899th approximation and119873
119899(119909 120589) is considered
as a restricted variation that is 120575119873119899(119909 120589) = 0
Also we can find the Lagrange multipliers easily by usingintegration by parts of (1) but in this paper the Lagrangemultipliers are found to be of the form 120582 = 120582(119909 119905 minus 120589) and insuch a case the integration is basically the single convolutionwith respect to 119905 and hence Laplace transform is appropriateto use
Take Laplace transform of (2) then the correctionfunctional will be constructed in the formℓ [119906119899+1
(119909 119905)]
= ℓ [119906119899(119909 119905)]
+ ℓ [int
119905
0
120582 (119909 120589) [119871119906119899(119909 120589) + 119873
119899(119909 120589) minus ℎ (119909 120589)] 119889120589]
119899 ge 0
(3)thereforeℓ [119906119899+1
(119909 119905)]
= ℓ [119906119899(119909 119905)]
+ ℓ [120582 (119909 119905) lowast [119871119906119899(119909 119905) + 119873
119899(119909 119905) minus ℎ (119909 119905)]]
= ℓ [119906119899(119909 119905)]
+ ℓ [120582 (119909 119905)] ℓ [119871119906119899(119909 119905) + 119873
119899(119909 119905) minus ℎ (119909 119905)]
(4)where lowast is a single convolution with respect to 119905
To find the optimal value of 120582(119909 119905 minus 120589) we first take thevariation with respect to 119906
119899(119909 119905) Thus
120575
120575119906119899
ℓ [119906119899+1
(119909 119905)]
=120575
120575119906119899
ℓ [119906119899(119909 119905)]
+120575
120575119906119899
ℓ [120582 (119909 119905)] ℓ [119871119906119899(119909 119905) + 119873
119899(119909 119905) minus ℎ (119909 119905)]
(5)then (5) becomesℓ [120575119906119899+1
(119909 119905)] = ℓ [120575119906119899(119909 119905)] + 120575ℓ [120582 (119909 119905)] ℓ [119871119906
119899(119909 119905)]
(6)
In this paper we assume that 119871 is a linear partial differentialoperator given by 12059721205971199052 then (6) can be written in the form
ℓ [120575119906119899+1
(119909 119905)] = ℓ [120575119906119899(119909 119905)] + ℓ [120582 (119909 119905)] [119904
2ℓ120575119906119899(119909 119905)]
(7)
The extreme condition of 119906119899+1
(119909 119905) requires that 120575119906119899+1
(119909 119905) =
0 This means that the right hand side of (7) should be set tozero then we have the following condition
ℓ [120582 (119909 119905)] =minus1
1199042997904rArr 120582 (119909 119905) = minus119905 (8)
then we have the following iteration formula
ℓ [119906119899+1
(119909 119905)]
= ℓ [119906119899(119909 119905)]
minus ℓ [int
119905
0
(119905 minus 120589) [119871119906119899(119909 120589) + 119873
119899(119909 120589) minus ℎ (119909 120589)] 119889120589]
119899 ge 0
(9)
3 Applications
In this section we apply the Laplace variational iterationmethod for solving some linear and nonlinear partial differ-ential equations in physics
Example 1 Consider the initial linear partial differentialequation
119906119905119905(119909 119905) minus 119906
119909119909(119909 119905) + 119906 (119909 119905) = 0
119906 (119909 0) = 0120597119906 (119909 0)
120597119905= 119909
(10)
The Laplace variational iteration correction functional will beconstructed in the following manner
ℓ [119906119899+1
(119909 119905)]
= ℓ [119906119899(119909 119905)]
+ ℓ [int
119905
0
120582 (119909 119905 minus 120589)
times [(119906119899)119905119905(119909 120589) minus (119906
119899)119909119909
(119909 120589) + 119906119899(119909 120589)] 119889120589]
(11)
or
ℓ [119906119899+1
(119909 119905)]
= ℓ [119906119899(119909 119905)]
+ ℓ [120582 (119909 119905) lowast [(119906119899)119905119905(119909 119905) minus (119906
119899)119909119909
(119909 119905) + 119906119899(119909 119905)]]
= ℓ [119906119899(119909 119905)]
+ ℓ [120582 (119909 119905)] ℓ [(119906119899)119905119905(119909 119905) minus (119906
119899)119909119909
(119909 119905) + 119906119899(119909 119905)]
Journal of Function Spaces 3
= ℓ [119906119899(119909 119905)] + ℓ [120582 (119909 119905)]
times [1199042ℓ119906119899(119909 119905) minus 119904119906
119899(119909 0) minus
120597119906119899
120597119905(119909 0)
minus ℓ(119906119899)119909119909
(119909 119905) + ℓ119906119899(119909 119905) ]
(12)Take the variation with respect to 119906
119899(119909 119905) of (12) to obtain
120575
120575119906119899
ℓ [119906119899+1
(119909 119905)]
=120575
120575119906119899
ℓ [119906119899(119909 119905)]
+120575
120575119906119899
ℓ [120582 (119909 119905)]
times [1199042ℓ119906119899(119909 119905) minus 119904119906
119899(119909 0) minus
120597119906119899
120597119905(119909 0)
minus ℓ(119906119899)119909119909
(119909 119905) + ℓ119906119899(119909 119905) ]
(13)
then we haveℓ [120575119906119899+1
(119909 119905)] = ℓ [120575119906119899(119909 119905)]
+ ℓ [120582 (119909 119905)] [1199042ℓ119906119899(119909 119905) + ℓ119906
119899(119909 119905)]
= ℓ [120575119906119899(119909 119905)]
times 1 + ℓ [120582 (119909 119905)] (1199042+ 1)
(14)The extreme condition of 119906
119899+1(119909 119905) requires that
120575119906119899+1
(119909 119905) = 0 Hence we have
1 + (1199042+ 1) ℓ120582 (119909 119905) = 0
120582 (119909 119905) = ℓminus1
[minus1
1199042 + 1] = minus sin 119905
(15)
Substituting (15) into (11) we obtainℓ [119906119899+1
(119909 119905)]
= ℓ [119906119899(119909 119905)]
minus ℓ [int
119905
0
sin (119905 minus 120589)
times [(119906119899)119905119905(119909 120589) minus (119906
119899)119909119909
(119909 120589) + 119906119899(119909 120589)] 119889120589]
= ℓ [119906119899(119909 119905)]
minus ℓ [sin 119905] ℓ [(119906119899)119905119905 (119909 119905) minus (119906
119899)119909119909
(119909 119905) + 119906119899(119909 119905)]
(16)Let 1199060(119909 119905) = 119906(119909 0) + 119905(120597119906120597119905)(119909 0) = 119909119905 then from (16)
we have
ℓ [1199061(119909 119905)] = ℓ [119909119905] minus ℓ [sin 119905] ℓ [119909119905] =
119909
1199042minus
119909
1199042 (1199042 + 1)
(17)
The inverse Laplace transforms yields
1199061(119909 119905) = 119909 sin 119905 (18)
Substituting (18) into (11) we obtain
ℓ [1199062(119909 119905)] = ℓ [119909 sin 119905]
minus ℓ [sin 119905] ℓ [0] then 1199062(119909 119905) = 119909 sin 119905
(19)
then the exact solution of (10) is
119906 (119909 119905) = 119909 sin 119905 (20)
We see that the exact solution is coming very fast by usingonly few terms of the iterative scheme
Example 2 Consider the nonlinear partial differential equa-tion
119906119905119905(119909 119905) minus 119906
119909119909(119909 119905) + 119906
2(119909 119905) = 119909
21199052
119906 (119909 0) = 0120597119906 (119909 0)
120597119905= 119909
(21)
The Laplace variational iteration correction functional will beconstructed as follows
ℓ [119906119899+1
(119909 119905)]
= ℓ [119906119899(119909 119905)]
+ ℓ [int
119905
0
120582 (119909 119905 minus 120589) [(119906119899)119905119905(119909 120589) minus (119906
119899)119909119909
(119909 120589)
+ 1199062
119899(119909 120589) minus 119909
21199052] 119889120589]
(22)
or
ℓ [119906119899+1
(119909 119905)]
= ℓ [119906119899(119909 119905)]
+ ℓ [120582 (119909 119905)
lowast [(119906119899)119905119905(119909 119905) minus (119906
119899)119909119909
(119909 119905) + 1199062
119899(119909 119905) minus 119909
21199052]]
= ℓ [119906119899(119909 119905)]
+ ℓ [120582 (119909 119905)]
times ℓ [(119906119899)119905119905(119909 119905) minus (119906
119899)119909119909
(119909 119905) + 1199062
119899(119909 119905) minus 119909
21199052]
= ℓ [119906119899(119909 119905)] + ℓ [120582 (119909 119905)]
times [1199042ℓ119906119899(119909 119905) minus 119904119906
119899(119909 0) minus
120597119906119899
120597119905(119909 0) minus ℓ(119906
119899)119909119909
(119909 119905)
+ ℓ1199062
119899(119909 119905) minus ℓ (119909
21199052) ]
(23)
4 Journal of Function Spaces
Take the variation with respect to 119906119899(119909 119905) of (23) and make
the correction functional stationary to obtain
ℓ [120575119906119899+1
(119909 119905)] = ℓ [120575119906119899(119909 119905)] + ℓ [120582 (119909 119905)] [119904
2ℓ120575119906119899(119909 119905)]
= ℓ [120575119906119899(119909 119905)] 1 + 119904
2ℓ [120582 (119909 119905)]
(24)
This implies that
1 + 1199042ℓ120582 (119909 119905) = 0 120582 (119909 119905) = ℓ
minus1[minus1
1199042] = minus119905 (25)
Substituting (25) into (22) we obtain
ℓ [119906119899+1
(119909 119905)] = ℓ [119906119899(119909 119905)]
minus ℓ [int
119905
0
(119905 minus 120589) [(119906119899)119905119905(119909 120589) minus (119906
119899)119909119909
(119909 120589)
+ 1199062
119899(119909 120589) minus 119909
21205892] 119889120589]
(26)or
ℓ [119906119899+1
(119909 119905)]
= ℓ [119906119899(119909 119905)] + ℓ [minus119905]
times ℓ [(119906119899)119905119905(119909 119905) minus (119906
119899)119909119909
(119909 119905) + 1199062
119899(119909 119905) minus 119909
21199052]
(27)
Let 1199060(119909 119905) = 119906(119909 0) + 119905(120597119906120597119905)(119909 0) = 119909119905 then from (27)
we have
ℓ [1199061(119909 119905)] = ℓ [119909119905] + ℓ [minus119905] ℓ [0 minus 0 + 119909
21199052minus 11990921199052]
1199061(119909 119905) = 119909119905
(28)
then the exact solution of (21) is
119906 (119909 119905) = 119909119905 (29)
Again the exact solution is coming very fast by using only fewterms of the iterative scheme
Example 3 Consider the physics nonlinear boundary valueproblem
119906119905minus 6119906119906
119909+ 119906119909119909119909
= 0 119906 (119909 0) =6
1199092 119909 = 0 (30)
The Laplace variational iteration correction functional is asfollowsℓ [119906119899+1
(119909 119905)]
= ℓ [119906119899(119909 119905)]
+ ℓ [int
119905
0
120582 (119909 119905 minus 120589) [(119906119899)119905(119909 120589) minus 6119906
119899(119909 120589) (119906
119899)119909(119909 120589)
+ (119906119899)119909119909119909
(119909 120589)] 119889120589]
(31)
orℓ [119906119899+1
(119909 119905)]
= ℓ [119906119899(119909 119905)]
+ ℓ [120582 (119909 119905) lowast [(119906119899)119905(119909 119905) minus 6 (119906
119899) (119909 119905) (119906
119899)119909(119909 119905)
+(119906119899)119909119909119909
(119909 119905)] ]
= ℓ [119906119899(119909 119905)] + ℓ [120582 (119909 119905)]
timesℓ[(119906119899)119905(119909 119905) minus 6 (119906
119899) (119909 119905) (119906
119899)119909(119909 119905) + (119906
119899)119909119909119909
(119909 119905)]
= ℓ [119906119899(119909 119905)] + ℓ [120582 (119909 119905)]
times [119904ℓ119906119899(119909 119905) minus 119906
119899(119909 0)
minus ℓ [6 (119906119899) (119909 119905) (119906
119899)119909(119909 119905) minus (119906
119899)119909119909119909
(119909 119905)]]
(32)
Take the variation with respect to 119906119899(119909 119905) of the last equation
and make the correction functional stationary to obtain
ℓ [120575119906119899+1
(119909 119905)] = ℓ [120575119906119899(119909 119905)] + ℓ [120582 (119909 119905)] [119904ℓ120575119906
119899(119909 119905)]
= ℓ [120575119906119899(119909 119905)] 1 + 119904ℓ [120582 (119909 119905)]
(33)
this implies that
1 + 119904ℓ120582 (119909 119905) = 0 120582 (119909 119905) = ℓminus1
[minus1
119904] = minus1 (34)
Substituting (34) into (31) we obtain
ℓ [119906119899+1
(119909 119905)] = ℓ [119906119899(119909 119905)]
+ ℓ [int
119905
0
(minus1) [(119906119899)119905(119909 120589)
minus 6 (119906119899) (119909 120589) (119906
119899)119909(119909 120589)
+(119906119899)119909119909119909
(119909 120589)] 119889120589]
(35)or
ℓ [119906119899+1
(119909 119905)]
= ℓ [119906119899] + ℓ [minus1] ℓ [(119906119899)119905
minus (119906119899) (119906119899)119909+ (119906119899)119909119909119909
]
(36)
Let 1199060(119909 119905) = 119906(119909 0) = 6119909
2 then from (36) we have
ℓ [1199061(119909 119905)] = ℓ [
6
1199092] + ℓ [minus1] ℓ [
288
1199095] =
6
1199092minus
288
1199095119905
1199062(119909 119905) =
6
1199092minus
288
1199095119905 minus
6048
11990981199052
(37)
and then the exact solution of (30) is
119906 (119909 119905) =
6119909 (1199093minus 24119905)
(1199093 minus 12119905)2
(38)
Journal of Function Spaces 5
4 Conclusion
Themethod of combining Laplace transforms and variationaliteration method is proposed for the solution of linearand nonlinear partial differential equations This method isapplied in a direct waywithout employing linearization and issuccessfully implemented by using the initial conditions andconvolution integral
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work was funded by the Deanship of Scientific Research(DSR) King AbdulazizUniversity Jeddah under Grant no(363-008-D1434) The authors therefore acknowledge withthanks DSR technical and financial support
References
[1] J Biazar andH Ghazvini ldquoHersquos variational iterationmethod forsolving linear and non-linear systems of ordinary differentialequationsrdquo Applied Mathematics and Computation vol 191 no1 pp 287ndash297 2007
[2] J He ldquoVariational iteration method for delay differential equa-tionsrdquo Communications in Nonlinear Science and NumericalSimulation vol 2 no 4 pp 235ndash236 1997
[3] J-H He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999
[4] J-H He ldquoVariational iteration method for autonomous ordi-nary differential systemsrdquo Applied Mathematics and Computa-tion vol 114 no 2-3 pp 115ndash123 2000
[5] J-H He and X-H Wu ldquoVariational iteration method newdevelopment and applicationsrdquo Computers ampMathematics withApplications vol 54 no 7-8 pp 881ndash894 2007
[6] SAKhuri andA Sayfy ldquoALaplace variational iteration strategyfor the solution of differential equationsrdquo Applied MathematicsLetters vol 25 no 12 pp 2298ndash2305 2012
[7] E Hesameddini and H Latifizadeh ldquoReconstruction of varia-tional iteration algorithms using the laplace transformrdquo Interna-tional Journal of Nonlinear Sciences and Numerical Simulationvol 10 no 11-12 pp 1377ndash1382 2009
[8] G-C Wu and D Baleanu ldquoVariational iteration method forfractional calculusmdasha universal approach by Laplace trans-formrdquo Advances in Difference Equations vol 2013 article 182013
[9] X J Yang and D Baleanu ldquoFractal heat conduction problemsolved by local fractional variation iteration methodrdquo ThermalScience vol 17 no 2 pp 625ndash628 2013
[10] G C Wu ldquoVariational iteration method for solving the time-fractional diffusion equations in porous mediumrdquo ChinesePhysics B vol 21 no 12 Article ID 120504 2012
[11] G-C Wu and D Baleanu ldquoVariational iteration method forthe Burgersrsquo flow with fractional derivativesmdashnew Lagrangemultipliersrdquo Applied Mathematical Modelling vol 37 no 9 pp6183ndash6190 2013
[12] G C Wu ldquoChallenge in the variational iteration method-a newapproach to identification of the Lagrange mutipliersrdquo Journalof King Saud UniversitymdashScience vol 25 pp 175ndash178 2013
[13] G C Wu ldquoLaplace transform overcoming principle drawbacksin application of the variational iteration method to fractionalheat equationsrdquo Thermal Science vol 16 no 4 pp 1257ndash12612012
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Journal of Function Spaces
According to the variational iteration method we canconstruct a correction function for (1) as follows119906119899+1
(119909 119905)
= 119906119899(119909 119905)
+ int
119905
0
120582 (119909 120589) [119871119906119899(119909 120589) + 119873
119899(119909 120589) minus ℎ (119909 120589)] 119889120589
119899 ge 0
(2)where 120582 is a general Lagrange multiplier which can beidentified optimally via the variational theory the subscripts119899 denote the 119899th approximation and119873
119899(119909 120589) is considered
as a restricted variation that is 120575119873119899(119909 120589) = 0
Also we can find the Lagrange multipliers easily by usingintegration by parts of (1) but in this paper the Lagrangemultipliers are found to be of the form 120582 = 120582(119909 119905 minus 120589) and insuch a case the integration is basically the single convolutionwith respect to 119905 and hence Laplace transform is appropriateto use
Take Laplace transform of (2) then the correctionfunctional will be constructed in the formℓ [119906119899+1
(119909 119905)]
= ℓ [119906119899(119909 119905)]
+ ℓ [int
119905
0
120582 (119909 120589) [119871119906119899(119909 120589) + 119873
119899(119909 120589) minus ℎ (119909 120589)] 119889120589]
119899 ge 0
(3)thereforeℓ [119906119899+1
(119909 119905)]
= ℓ [119906119899(119909 119905)]
+ ℓ [120582 (119909 119905) lowast [119871119906119899(119909 119905) + 119873
119899(119909 119905) minus ℎ (119909 119905)]]
= ℓ [119906119899(119909 119905)]
+ ℓ [120582 (119909 119905)] ℓ [119871119906119899(119909 119905) + 119873
119899(119909 119905) minus ℎ (119909 119905)]
(4)where lowast is a single convolution with respect to 119905
To find the optimal value of 120582(119909 119905 minus 120589) we first take thevariation with respect to 119906
119899(119909 119905) Thus
120575
120575119906119899
ℓ [119906119899+1
(119909 119905)]
=120575
120575119906119899
ℓ [119906119899(119909 119905)]
+120575
120575119906119899
ℓ [120582 (119909 119905)] ℓ [119871119906119899(119909 119905) + 119873
119899(119909 119905) minus ℎ (119909 119905)]
(5)then (5) becomesℓ [120575119906119899+1
(119909 119905)] = ℓ [120575119906119899(119909 119905)] + 120575ℓ [120582 (119909 119905)] ℓ [119871119906
119899(119909 119905)]
(6)
In this paper we assume that 119871 is a linear partial differentialoperator given by 12059721205971199052 then (6) can be written in the form
ℓ [120575119906119899+1
(119909 119905)] = ℓ [120575119906119899(119909 119905)] + ℓ [120582 (119909 119905)] [119904
2ℓ120575119906119899(119909 119905)]
(7)
The extreme condition of 119906119899+1
(119909 119905) requires that 120575119906119899+1
(119909 119905) =
0 This means that the right hand side of (7) should be set tozero then we have the following condition
ℓ [120582 (119909 119905)] =minus1
1199042997904rArr 120582 (119909 119905) = minus119905 (8)
then we have the following iteration formula
ℓ [119906119899+1
(119909 119905)]
= ℓ [119906119899(119909 119905)]
minus ℓ [int
119905
0
(119905 minus 120589) [119871119906119899(119909 120589) + 119873
119899(119909 120589) minus ℎ (119909 120589)] 119889120589]
119899 ge 0
(9)
3 Applications
In this section we apply the Laplace variational iterationmethod for solving some linear and nonlinear partial differ-ential equations in physics
Example 1 Consider the initial linear partial differentialequation
119906119905119905(119909 119905) minus 119906
119909119909(119909 119905) + 119906 (119909 119905) = 0
119906 (119909 0) = 0120597119906 (119909 0)
120597119905= 119909
(10)
The Laplace variational iteration correction functional will beconstructed in the following manner
ℓ [119906119899+1
(119909 119905)]
= ℓ [119906119899(119909 119905)]
+ ℓ [int
119905
0
120582 (119909 119905 minus 120589)
times [(119906119899)119905119905(119909 120589) minus (119906
119899)119909119909
(119909 120589) + 119906119899(119909 120589)] 119889120589]
(11)
or
ℓ [119906119899+1
(119909 119905)]
= ℓ [119906119899(119909 119905)]
+ ℓ [120582 (119909 119905) lowast [(119906119899)119905119905(119909 119905) minus (119906
119899)119909119909
(119909 119905) + 119906119899(119909 119905)]]
= ℓ [119906119899(119909 119905)]
+ ℓ [120582 (119909 119905)] ℓ [(119906119899)119905119905(119909 119905) minus (119906
119899)119909119909
(119909 119905) + 119906119899(119909 119905)]
Journal of Function Spaces 3
= ℓ [119906119899(119909 119905)] + ℓ [120582 (119909 119905)]
times [1199042ℓ119906119899(119909 119905) minus 119904119906
119899(119909 0) minus
120597119906119899
120597119905(119909 0)
minus ℓ(119906119899)119909119909
(119909 119905) + ℓ119906119899(119909 119905) ]
(12)Take the variation with respect to 119906
119899(119909 119905) of (12) to obtain
120575
120575119906119899
ℓ [119906119899+1
(119909 119905)]
=120575
120575119906119899
ℓ [119906119899(119909 119905)]
+120575
120575119906119899
ℓ [120582 (119909 119905)]
times [1199042ℓ119906119899(119909 119905) minus 119904119906
119899(119909 0) minus
120597119906119899
120597119905(119909 0)
minus ℓ(119906119899)119909119909
(119909 119905) + ℓ119906119899(119909 119905) ]
(13)
then we haveℓ [120575119906119899+1
(119909 119905)] = ℓ [120575119906119899(119909 119905)]
+ ℓ [120582 (119909 119905)] [1199042ℓ119906119899(119909 119905) + ℓ119906
119899(119909 119905)]
= ℓ [120575119906119899(119909 119905)]
times 1 + ℓ [120582 (119909 119905)] (1199042+ 1)
(14)The extreme condition of 119906
119899+1(119909 119905) requires that
120575119906119899+1
(119909 119905) = 0 Hence we have
1 + (1199042+ 1) ℓ120582 (119909 119905) = 0
120582 (119909 119905) = ℓminus1
[minus1
1199042 + 1] = minus sin 119905
(15)
Substituting (15) into (11) we obtainℓ [119906119899+1
(119909 119905)]
= ℓ [119906119899(119909 119905)]
minus ℓ [int
119905
0
sin (119905 minus 120589)
times [(119906119899)119905119905(119909 120589) minus (119906
119899)119909119909
(119909 120589) + 119906119899(119909 120589)] 119889120589]
= ℓ [119906119899(119909 119905)]
minus ℓ [sin 119905] ℓ [(119906119899)119905119905 (119909 119905) minus (119906
119899)119909119909
(119909 119905) + 119906119899(119909 119905)]
(16)Let 1199060(119909 119905) = 119906(119909 0) + 119905(120597119906120597119905)(119909 0) = 119909119905 then from (16)
we have
ℓ [1199061(119909 119905)] = ℓ [119909119905] minus ℓ [sin 119905] ℓ [119909119905] =
119909
1199042minus
119909
1199042 (1199042 + 1)
(17)
The inverse Laplace transforms yields
1199061(119909 119905) = 119909 sin 119905 (18)
Substituting (18) into (11) we obtain
ℓ [1199062(119909 119905)] = ℓ [119909 sin 119905]
minus ℓ [sin 119905] ℓ [0] then 1199062(119909 119905) = 119909 sin 119905
(19)
then the exact solution of (10) is
119906 (119909 119905) = 119909 sin 119905 (20)
We see that the exact solution is coming very fast by usingonly few terms of the iterative scheme
Example 2 Consider the nonlinear partial differential equa-tion
119906119905119905(119909 119905) minus 119906
119909119909(119909 119905) + 119906
2(119909 119905) = 119909
21199052
119906 (119909 0) = 0120597119906 (119909 0)
120597119905= 119909
(21)
The Laplace variational iteration correction functional will beconstructed as follows
ℓ [119906119899+1
(119909 119905)]
= ℓ [119906119899(119909 119905)]
+ ℓ [int
119905
0
120582 (119909 119905 minus 120589) [(119906119899)119905119905(119909 120589) minus (119906
119899)119909119909
(119909 120589)
+ 1199062
119899(119909 120589) minus 119909
21199052] 119889120589]
(22)
or
ℓ [119906119899+1
(119909 119905)]
= ℓ [119906119899(119909 119905)]
+ ℓ [120582 (119909 119905)
lowast [(119906119899)119905119905(119909 119905) minus (119906
119899)119909119909
(119909 119905) + 1199062
119899(119909 119905) minus 119909
21199052]]
= ℓ [119906119899(119909 119905)]
+ ℓ [120582 (119909 119905)]
times ℓ [(119906119899)119905119905(119909 119905) minus (119906
119899)119909119909
(119909 119905) + 1199062
119899(119909 119905) minus 119909
21199052]
= ℓ [119906119899(119909 119905)] + ℓ [120582 (119909 119905)]
times [1199042ℓ119906119899(119909 119905) minus 119904119906
119899(119909 0) minus
120597119906119899
120597119905(119909 0) minus ℓ(119906
119899)119909119909
(119909 119905)
+ ℓ1199062
119899(119909 119905) minus ℓ (119909
21199052) ]
(23)
4 Journal of Function Spaces
Take the variation with respect to 119906119899(119909 119905) of (23) and make
the correction functional stationary to obtain
ℓ [120575119906119899+1
(119909 119905)] = ℓ [120575119906119899(119909 119905)] + ℓ [120582 (119909 119905)] [119904
2ℓ120575119906119899(119909 119905)]
= ℓ [120575119906119899(119909 119905)] 1 + 119904
2ℓ [120582 (119909 119905)]
(24)
This implies that
1 + 1199042ℓ120582 (119909 119905) = 0 120582 (119909 119905) = ℓ
minus1[minus1
1199042] = minus119905 (25)
Substituting (25) into (22) we obtain
ℓ [119906119899+1
(119909 119905)] = ℓ [119906119899(119909 119905)]
minus ℓ [int
119905
0
(119905 minus 120589) [(119906119899)119905119905(119909 120589) minus (119906
119899)119909119909
(119909 120589)
+ 1199062
119899(119909 120589) minus 119909
21205892] 119889120589]
(26)or
ℓ [119906119899+1
(119909 119905)]
= ℓ [119906119899(119909 119905)] + ℓ [minus119905]
times ℓ [(119906119899)119905119905(119909 119905) minus (119906
119899)119909119909
(119909 119905) + 1199062
119899(119909 119905) minus 119909
21199052]
(27)
Let 1199060(119909 119905) = 119906(119909 0) + 119905(120597119906120597119905)(119909 0) = 119909119905 then from (27)
we have
ℓ [1199061(119909 119905)] = ℓ [119909119905] + ℓ [minus119905] ℓ [0 minus 0 + 119909
21199052minus 11990921199052]
1199061(119909 119905) = 119909119905
(28)
then the exact solution of (21) is
119906 (119909 119905) = 119909119905 (29)
Again the exact solution is coming very fast by using only fewterms of the iterative scheme
Example 3 Consider the physics nonlinear boundary valueproblem
119906119905minus 6119906119906
119909+ 119906119909119909119909
= 0 119906 (119909 0) =6
1199092 119909 = 0 (30)
The Laplace variational iteration correction functional is asfollowsℓ [119906119899+1
(119909 119905)]
= ℓ [119906119899(119909 119905)]
+ ℓ [int
119905
0
120582 (119909 119905 minus 120589) [(119906119899)119905(119909 120589) minus 6119906
119899(119909 120589) (119906
119899)119909(119909 120589)
+ (119906119899)119909119909119909
(119909 120589)] 119889120589]
(31)
orℓ [119906119899+1
(119909 119905)]
= ℓ [119906119899(119909 119905)]
+ ℓ [120582 (119909 119905) lowast [(119906119899)119905(119909 119905) minus 6 (119906
119899) (119909 119905) (119906
119899)119909(119909 119905)
+(119906119899)119909119909119909
(119909 119905)] ]
= ℓ [119906119899(119909 119905)] + ℓ [120582 (119909 119905)]
timesℓ[(119906119899)119905(119909 119905) minus 6 (119906
119899) (119909 119905) (119906
119899)119909(119909 119905) + (119906
119899)119909119909119909
(119909 119905)]
= ℓ [119906119899(119909 119905)] + ℓ [120582 (119909 119905)]
times [119904ℓ119906119899(119909 119905) minus 119906
119899(119909 0)
minus ℓ [6 (119906119899) (119909 119905) (119906
119899)119909(119909 119905) minus (119906
119899)119909119909119909
(119909 119905)]]
(32)
Take the variation with respect to 119906119899(119909 119905) of the last equation
and make the correction functional stationary to obtain
ℓ [120575119906119899+1
(119909 119905)] = ℓ [120575119906119899(119909 119905)] + ℓ [120582 (119909 119905)] [119904ℓ120575119906
119899(119909 119905)]
= ℓ [120575119906119899(119909 119905)] 1 + 119904ℓ [120582 (119909 119905)]
(33)
this implies that
1 + 119904ℓ120582 (119909 119905) = 0 120582 (119909 119905) = ℓminus1
[minus1
119904] = minus1 (34)
Substituting (34) into (31) we obtain
ℓ [119906119899+1
(119909 119905)] = ℓ [119906119899(119909 119905)]
+ ℓ [int
119905
0
(minus1) [(119906119899)119905(119909 120589)
minus 6 (119906119899) (119909 120589) (119906
119899)119909(119909 120589)
+(119906119899)119909119909119909
(119909 120589)] 119889120589]
(35)or
ℓ [119906119899+1
(119909 119905)]
= ℓ [119906119899] + ℓ [minus1] ℓ [(119906119899)119905
minus (119906119899) (119906119899)119909+ (119906119899)119909119909119909
]
(36)
Let 1199060(119909 119905) = 119906(119909 0) = 6119909
2 then from (36) we have
ℓ [1199061(119909 119905)] = ℓ [
6
1199092] + ℓ [minus1] ℓ [
288
1199095] =
6
1199092minus
288
1199095119905
1199062(119909 119905) =
6
1199092minus
288
1199095119905 minus
6048
11990981199052
(37)
and then the exact solution of (30) is
119906 (119909 119905) =
6119909 (1199093minus 24119905)
(1199093 minus 12119905)2
(38)
Journal of Function Spaces 5
4 Conclusion
Themethod of combining Laplace transforms and variationaliteration method is proposed for the solution of linearand nonlinear partial differential equations This method isapplied in a direct waywithout employing linearization and issuccessfully implemented by using the initial conditions andconvolution integral
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work was funded by the Deanship of Scientific Research(DSR) King AbdulazizUniversity Jeddah under Grant no(363-008-D1434) The authors therefore acknowledge withthanks DSR technical and financial support
References
[1] J Biazar andH Ghazvini ldquoHersquos variational iterationmethod forsolving linear and non-linear systems of ordinary differentialequationsrdquo Applied Mathematics and Computation vol 191 no1 pp 287ndash297 2007
[2] J He ldquoVariational iteration method for delay differential equa-tionsrdquo Communications in Nonlinear Science and NumericalSimulation vol 2 no 4 pp 235ndash236 1997
[3] J-H He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999
[4] J-H He ldquoVariational iteration method for autonomous ordi-nary differential systemsrdquo Applied Mathematics and Computa-tion vol 114 no 2-3 pp 115ndash123 2000
[5] J-H He and X-H Wu ldquoVariational iteration method newdevelopment and applicationsrdquo Computers ampMathematics withApplications vol 54 no 7-8 pp 881ndash894 2007
[6] SAKhuri andA Sayfy ldquoALaplace variational iteration strategyfor the solution of differential equationsrdquo Applied MathematicsLetters vol 25 no 12 pp 2298ndash2305 2012
[7] E Hesameddini and H Latifizadeh ldquoReconstruction of varia-tional iteration algorithms using the laplace transformrdquo Interna-tional Journal of Nonlinear Sciences and Numerical Simulationvol 10 no 11-12 pp 1377ndash1382 2009
[8] G-C Wu and D Baleanu ldquoVariational iteration method forfractional calculusmdasha universal approach by Laplace trans-formrdquo Advances in Difference Equations vol 2013 article 182013
[9] X J Yang and D Baleanu ldquoFractal heat conduction problemsolved by local fractional variation iteration methodrdquo ThermalScience vol 17 no 2 pp 625ndash628 2013
[10] G C Wu ldquoVariational iteration method for solving the time-fractional diffusion equations in porous mediumrdquo ChinesePhysics B vol 21 no 12 Article ID 120504 2012
[11] G-C Wu and D Baleanu ldquoVariational iteration method forthe Burgersrsquo flow with fractional derivativesmdashnew Lagrangemultipliersrdquo Applied Mathematical Modelling vol 37 no 9 pp6183ndash6190 2013
[12] G C Wu ldquoChallenge in the variational iteration method-a newapproach to identification of the Lagrange mutipliersrdquo Journalof King Saud UniversitymdashScience vol 25 pp 175ndash178 2013
[13] G C Wu ldquoLaplace transform overcoming principle drawbacksin application of the variational iteration method to fractionalheat equationsrdquo Thermal Science vol 16 no 4 pp 1257ndash12612012
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Function Spaces 3
= ℓ [119906119899(119909 119905)] + ℓ [120582 (119909 119905)]
times [1199042ℓ119906119899(119909 119905) minus 119904119906
119899(119909 0) minus
120597119906119899
120597119905(119909 0)
minus ℓ(119906119899)119909119909
(119909 119905) + ℓ119906119899(119909 119905) ]
(12)Take the variation with respect to 119906
119899(119909 119905) of (12) to obtain
120575
120575119906119899
ℓ [119906119899+1
(119909 119905)]
=120575
120575119906119899
ℓ [119906119899(119909 119905)]
+120575
120575119906119899
ℓ [120582 (119909 119905)]
times [1199042ℓ119906119899(119909 119905) minus 119904119906
119899(119909 0) minus
120597119906119899
120597119905(119909 0)
minus ℓ(119906119899)119909119909
(119909 119905) + ℓ119906119899(119909 119905) ]
(13)
then we haveℓ [120575119906119899+1
(119909 119905)] = ℓ [120575119906119899(119909 119905)]
+ ℓ [120582 (119909 119905)] [1199042ℓ119906119899(119909 119905) + ℓ119906
119899(119909 119905)]
= ℓ [120575119906119899(119909 119905)]
times 1 + ℓ [120582 (119909 119905)] (1199042+ 1)
(14)The extreme condition of 119906
119899+1(119909 119905) requires that
120575119906119899+1
(119909 119905) = 0 Hence we have
1 + (1199042+ 1) ℓ120582 (119909 119905) = 0
120582 (119909 119905) = ℓminus1
[minus1
1199042 + 1] = minus sin 119905
(15)
Substituting (15) into (11) we obtainℓ [119906119899+1
(119909 119905)]
= ℓ [119906119899(119909 119905)]
minus ℓ [int
119905
0
sin (119905 minus 120589)
times [(119906119899)119905119905(119909 120589) minus (119906
119899)119909119909
(119909 120589) + 119906119899(119909 120589)] 119889120589]
= ℓ [119906119899(119909 119905)]
minus ℓ [sin 119905] ℓ [(119906119899)119905119905 (119909 119905) minus (119906
119899)119909119909
(119909 119905) + 119906119899(119909 119905)]
(16)Let 1199060(119909 119905) = 119906(119909 0) + 119905(120597119906120597119905)(119909 0) = 119909119905 then from (16)
we have
ℓ [1199061(119909 119905)] = ℓ [119909119905] minus ℓ [sin 119905] ℓ [119909119905] =
119909
1199042minus
119909
1199042 (1199042 + 1)
(17)
The inverse Laplace transforms yields
1199061(119909 119905) = 119909 sin 119905 (18)
Substituting (18) into (11) we obtain
ℓ [1199062(119909 119905)] = ℓ [119909 sin 119905]
minus ℓ [sin 119905] ℓ [0] then 1199062(119909 119905) = 119909 sin 119905
(19)
then the exact solution of (10) is
119906 (119909 119905) = 119909 sin 119905 (20)
We see that the exact solution is coming very fast by usingonly few terms of the iterative scheme
Example 2 Consider the nonlinear partial differential equa-tion
119906119905119905(119909 119905) minus 119906
119909119909(119909 119905) + 119906
2(119909 119905) = 119909
21199052
119906 (119909 0) = 0120597119906 (119909 0)
120597119905= 119909
(21)
The Laplace variational iteration correction functional will beconstructed as follows
ℓ [119906119899+1
(119909 119905)]
= ℓ [119906119899(119909 119905)]
+ ℓ [int
119905
0
120582 (119909 119905 minus 120589) [(119906119899)119905119905(119909 120589) minus (119906
119899)119909119909
(119909 120589)
+ 1199062
119899(119909 120589) minus 119909
21199052] 119889120589]
(22)
or
ℓ [119906119899+1
(119909 119905)]
= ℓ [119906119899(119909 119905)]
+ ℓ [120582 (119909 119905)
lowast [(119906119899)119905119905(119909 119905) minus (119906
119899)119909119909
(119909 119905) + 1199062
119899(119909 119905) minus 119909
21199052]]
= ℓ [119906119899(119909 119905)]
+ ℓ [120582 (119909 119905)]
times ℓ [(119906119899)119905119905(119909 119905) minus (119906
119899)119909119909
(119909 119905) + 1199062
119899(119909 119905) minus 119909
21199052]
= ℓ [119906119899(119909 119905)] + ℓ [120582 (119909 119905)]
times [1199042ℓ119906119899(119909 119905) minus 119904119906
119899(119909 0) minus
120597119906119899
120597119905(119909 0) minus ℓ(119906
119899)119909119909
(119909 119905)
+ ℓ1199062
119899(119909 119905) minus ℓ (119909
21199052) ]
(23)
4 Journal of Function Spaces
Take the variation with respect to 119906119899(119909 119905) of (23) and make
the correction functional stationary to obtain
ℓ [120575119906119899+1
(119909 119905)] = ℓ [120575119906119899(119909 119905)] + ℓ [120582 (119909 119905)] [119904
2ℓ120575119906119899(119909 119905)]
= ℓ [120575119906119899(119909 119905)] 1 + 119904
2ℓ [120582 (119909 119905)]
(24)
This implies that
1 + 1199042ℓ120582 (119909 119905) = 0 120582 (119909 119905) = ℓ
minus1[minus1
1199042] = minus119905 (25)
Substituting (25) into (22) we obtain
ℓ [119906119899+1
(119909 119905)] = ℓ [119906119899(119909 119905)]
minus ℓ [int
119905
0
(119905 minus 120589) [(119906119899)119905119905(119909 120589) minus (119906
119899)119909119909
(119909 120589)
+ 1199062
119899(119909 120589) minus 119909
21205892] 119889120589]
(26)or
ℓ [119906119899+1
(119909 119905)]
= ℓ [119906119899(119909 119905)] + ℓ [minus119905]
times ℓ [(119906119899)119905119905(119909 119905) minus (119906
119899)119909119909
(119909 119905) + 1199062
119899(119909 119905) minus 119909
21199052]
(27)
Let 1199060(119909 119905) = 119906(119909 0) + 119905(120597119906120597119905)(119909 0) = 119909119905 then from (27)
we have
ℓ [1199061(119909 119905)] = ℓ [119909119905] + ℓ [minus119905] ℓ [0 minus 0 + 119909
21199052minus 11990921199052]
1199061(119909 119905) = 119909119905
(28)
then the exact solution of (21) is
119906 (119909 119905) = 119909119905 (29)
Again the exact solution is coming very fast by using only fewterms of the iterative scheme
Example 3 Consider the physics nonlinear boundary valueproblem
119906119905minus 6119906119906
119909+ 119906119909119909119909
= 0 119906 (119909 0) =6
1199092 119909 = 0 (30)
The Laplace variational iteration correction functional is asfollowsℓ [119906119899+1
(119909 119905)]
= ℓ [119906119899(119909 119905)]
+ ℓ [int
119905
0
120582 (119909 119905 minus 120589) [(119906119899)119905(119909 120589) minus 6119906
119899(119909 120589) (119906
119899)119909(119909 120589)
+ (119906119899)119909119909119909
(119909 120589)] 119889120589]
(31)
orℓ [119906119899+1
(119909 119905)]
= ℓ [119906119899(119909 119905)]
+ ℓ [120582 (119909 119905) lowast [(119906119899)119905(119909 119905) minus 6 (119906
119899) (119909 119905) (119906
119899)119909(119909 119905)
+(119906119899)119909119909119909
(119909 119905)] ]
= ℓ [119906119899(119909 119905)] + ℓ [120582 (119909 119905)]
timesℓ[(119906119899)119905(119909 119905) minus 6 (119906
119899) (119909 119905) (119906
119899)119909(119909 119905) + (119906
119899)119909119909119909
(119909 119905)]
= ℓ [119906119899(119909 119905)] + ℓ [120582 (119909 119905)]
times [119904ℓ119906119899(119909 119905) minus 119906
119899(119909 0)
minus ℓ [6 (119906119899) (119909 119905) (119906
119899)119909(119909 119905) minus (119906
119899)119909119909119909
(119909 119905)]]
(32)
Take the variation with respect to 119906119899(119909 119905) of the last equation
and make the correction functional stationary to obtain
ℓ [120575119906119899+1
(119909 119905)] = ℓ [120575119906119899(119909 119905)] + ℓ [120582 (119909 119905)] [119904ℓ120575119906
119899(119909 119905)]
= ℓ [120575119906119899(119909 119905)] 1 + 119904ℓ [120582 (119909 119905)]
(33)
this implies that
1 + 119904ℓ120582 (119909 119905) = 0 120582 (119909 119905) = ℓminus1
[minus1
119904] = minus1 (34)
Substituting (34) into (31) we obtain
ℓ [119906119899+1
(119909 119905)] = ℓ [119906119899(119909 119905)]
+ ℓ [int
119905
0
(minus1) [(119906119899)119905(119909 120589)
minus 6 (119906119899) (119909 120589) (119906
119899)119909(119909 120589)
+(119906119899)119909119909119909
(119909 120589)] 119889120589]
(35)or
ℓ [119906119899+1
(119909 119905)]
= ℓ [119906119899] + ℓ [minus1] ℓ [(119906119899)119905
minus (119906119899) (119906119899)119909+ (119906119899)119909119909119909
]
(36)
Let 1199060(119909 119905) = 119906(119909 0) = 6119909
2 then from (36) we have
ℓ [1199061(119909 119905)] = ℓ [
6
1199092] + ℓ [minus1] ℓ [
288
1199095] =
6
1199092minus
288
1199095119905
1199062(119909 119905) =
6
1199092minus
288
1199095119905 minus
6048
11990981199052
(37)
and then the exact solution of (30) is
119906 (119909 119905) =
6119909 (1199093minus 24119905)
(1199093 minus 12119905)2
(38)
Journal of Function Spaces 5
4 Conclusion
Themethod of combining Laplace transforms and variationaliteration method is proposed for the solution of linearand nonlinear partial differential equations This method isapplied in a direct waywithout employing linearization and issuccessfully implemented by using the initial conditions andconvolution integral
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work was funded by the Deanship of Scientific Research(DSR) King AbdulazizUniversity Jeddah under Grant no(363-008-D1434) The authors therefore acknowledge withthanks DSR technical and financial support
References
[1] J Biazar andH Ghazvini ldquoHersquos variational iterationmethod forsolving linear and non-linear systems of ordinary differentialequationsrdquo Applied Mathematics and Computation vol 191 no1 pp 287ndash297 2007
[2] J He ldquoVariational iteration method for delay differential equa-tionsrdquo Communications in Nonlinear Science and NumericalSimulation vol 2 no 4 pp 235ndash236 1997
[3] J-H He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999
[4] J-H He ldquoVariational iteration method for autonomous ordi-nary differential systemsrdquo Applied Mathematics and Computa-tion vol 114 no 2-3 pp 115ndash123 2000
[5] J-H He and X-H Wu ldquoVariational iteration method newdevelopment and applicationsrdquo Computers ampMathematics withApplications vol 54 no 7-8 pp 881ndash894 2007
[6] SAKhuri andA Sayfy ldquoALaplace variational iteration strategyfor the solution of differential equationsrdquo Applied MathematicsLetters vol 25 no 12 pp 2298ndash2305 2012
[7] E Hesameddini and H Latifizadeh ldquoReconstruction of varia-tional iteration algorithms using the laplace transformrdquo Interna-tional Journal of Nonlinear Sciences and Numerical Simulationvol 10 no 11-12 pp 1377ndash1382 2009
[8] G-C Wu and D Baleanu ldquoVariational iteration method forfractional calculusmdasha universal approach by Laplace trans-formrdquo Advances in Difference Equations vol 2013 article 182013
[9] X J Yang and D Baleanu ldquoFractal heat conduction problemsolved by local fractional variation iteration methodrdquo ThermalScience vol 17 no 2 pp 625ndash628 2013
[10] G C Wu ldquoVariational iteration method for solving the time-fractional diffusion equations in porous mediumrdquo ChinesePhysics B vol 21 no 12 Article ID 120504 2012
[11] G-C Wu and D Baleanu ldquoVariational iteration method forthe Burgersrsquo flow with fractional derivativesmdashnew Lagrangemultipliersrdquo Applied Mathematical Modelling vol 37 no 9 pp6183ndash6190 2013
[12] G C Wu ldquoChallenge in the variational iteration method-a newapproach to identification of the Lagrange mutipliersrdquo Journalof King Saud UniversitymdashScience vol 25 pp 175ndash178 2013
[13] G C Wu ldquoLaplace transform overcoming principle drawbacksin application of the variational iteration method to fractionalheat equationsrdquo Thermal Science vol 16 no 4 pp 1257ndash12612012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Journal of Function Spaces
Take the variation with respect to 119906119899(119909 119905) of (23) and make
the correction functional stationary to obtain
ℓ [120575119906119899+1
(119909 119905)] = ℓ [120575119906119899(119909 119905)] + ℓ [120582 (119909 119905)] [119904
2ℓ120575119906119899(119909 119905)]
= ℓ [120575119906119899(119909 119905)] 1 + 119904
2ℓ [120582 (119909 119905)]
(24)
This implies that
1 + 1199042ℓ120582 (119909 119905) = 0 120582 (119909 119905) = ℓ
minus1[minus1
1199042] = minus119905 (25)
Substituting (25) into (22) we obtain
ℓ [119906119899+1
(119909 119905)] = ℓ [119906119899(119909 119905)]
minus ℓ [int
119905
0
(119905 minus 120589) [(119906119899)119905119905(119909 120589) minus (119906
119899)119909119909
(119909 120589)
+ 1199062
119899(119909 120589) minus 119909
21205892] 119889120589]
(26)or
ℓ [119906119899+1
(119909 119905)]
= ℓ [119906119899(119909 119905)] + ℓ [minus119905]
times ℓ [(119906119899)119905119905(119909 119905) minus (119906
119899)119909119909
(119909 119905) + 1199062
119899(119909 119905) minus 119909
21199052]
(27)
Let 1199060(119909 119905) = 119906(119909 0) + 119905(120597119906120597119905)(119909 0) = 119909119905 then from (27)
we have
ℓ [1199061(119909 119905)] = ℓ [119909119905] + ℓ [minus119905] ℓ [0 minus 0 + 119909
21199052minus 11990921199052]
1199061(119909 119905) = 119909119905
(28)
then the exact solution of (21) is
119906 (119909 119905) = 119909119905 (29)
Again the exact solution is coming very fast by using only fewterms of the iterative scheme
Example 3 Consider the physics nonlinear boundary valueproblem
119906119905minus 6119906119906
119909+ 119906119909119909119909
= 0 119906 (119909 0) =6
1199092 119909 = 0 (30)
The Laplace variational iteration correction functional is asfollowsℓ [119906119899+1
(119909 119905)]
= ℓ [119906119899(119909 119905)]
+ ℓ [int
119905
0
120582 (119909 119905 minus 120589) [(119906119899)119905(119909 120589) minus 6119906
119899(119909 120589) (119906
119899)119909(119909 120589)
+ (119906119899)119909119909119909
(119909 120589)] 119889120589]
(31)
orℓ [119906119899+1
(119909 119905)]
= ℓ [119906119899(119909 119905)]
+ ℓ [120582 (119909 119905) lowast [(119906119899)119905(119909 119905) minus 6 (119906
119899) (119909 119905) (119906
119899)119909(119909 119905)
+(119906119899)119909119909119909
(119909 119905)] ]
= ℓ [119906119899(119909 119905)] + ℓ [120582 (119909 119905)]
timesℓ[(119906119899)119905(119909 119905) minus 6 (119906
119899) (119909 119905) (119906
119899)119909(119909 119905) + (119906
119899)119909119909119909
(119909 119905)]
= ℓ [119906119899(119909 119905)] + ℓ [120582 (119909 119905)]
times [119904ℓ119906119899(119909 119905) minus 119906
119899(119909 0)
minus ℓ [6 (119906119899) (119909 119905) (119906
119899)119909(119909 119905) minus (119906
119899)119909119909119909
(119909 119905)]]
(32)
Take the variation with respect to 119906119899(119909 119905) of the last equation
and make the correction functional stationary to obtain
ℓ [120575119906119899+1
(119909 119905)] = ℓ [120575119906119899(119909 119905)] + ℓ [120582 (119909 119905)] [119904ℓ120575119906
119899(119909 119905)]
= ℓ [120575119906119899(119909 119905)] 1 + 119904ℓ [120582 (119909 119905)]
(33)
this implies that
1 + 119904ℓ120582 (119909 119905) = 0 120582 (119909 119905) = ℓminus1
[minus1
119904] = minus1 (34)
Substituting (34) into (31) we obtain
ℓ [119906119899+1
(119909 119905)] = ℓ [119906119899(119909 119905)]
+ ℓ [int
119905
0
(minus1) [(119906119899)119905(119909 120589)
minus 6 (119906119899) (119909 120589) (119906
119899)119909(119909 120589)
+(119906119899)119909119909119909
(119909 120589)] 119889120589]
(35)or
ℓ [119906119899+1
(119909 119905)]
= ℓ [119906119899] + ℓ [minus1] ℓ [(119906119899)119905
minus (119906119899) (119906119899)119909+ (119906119899)119909119909119909
]
(36)
Let 1199060(119909 119905) = 119906(119909 0) = 6119909
2 then from (36) we have
ℓ [1199061(119909 119905)] = ℓ [
6
1199092] + ℓ [minus1] ℓ [
288
1199095] =
6
1199092minus
288
1199095119905
1199062(119909 119905) =
6
1199092minus
288
1199095119905 minus
6048
11990981199052
(37)
and then the exact solution of (30) is
119906 (119909 119905) =
6119909 (1199093minus 24119905)
(1199093 minus 12119905)2
(38)
Journal of Function Spaces 5
4 Conclusion
Themethod of combining Laplace transforms and variationaliteration method is proposed for the solution of linearand nonlinear partial differential equations This method isapplied in a direct waywithout employing linearization and issuccessfully implemented by using the initial conditions andconvolution integral
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work was funded by the Deanship of Scientific Research(DSR) King AbdulazizUniversity Jeddah under Grant no(363-008-D1434) The authors therefore acknowledge withthanks DSR technical and financial support
References
[1] J Biazar andH Ghazvini ldquoHersquos variational iterationmethod forsolving linear and non-linear systems of ordinary differentialequationsrdquo Applied Mathematics and Computation vol 191 no1 pp 287ndash297 2007
[2] J He ldquoVariational iteration method for delay differential equa-tionsrdquo Communications in Nonlinear Science and NumericalSimulation vol 2 no 4 pp 235ndash236 1997
[3] J-H He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999
[4] J-H He ldquoVariational iteration method for autonomous ordi-nary differential systemsrdquo Applied Mathematics and Computa-tion vol 114 no 2-3 pp 115ndash123 2000
[5] J-H He and X-H Wu ldquoVariational iteration method newdevelopment and applicationsrdquo Computers ampMathematics withApplications vol 54 no 7-8 pp 881ndash894 2007
[6] SAKhuri andA Sayfy ldquoALaplace variational iteration strategyfor the solution of differential equationsrdquo Applied MathematicsLetters vol 25 no 12 pp 2298ndash2305 2012
[7] E Hesameddini and H Latifizadeh ldquoReconstruction of varia-tional iteration algorithms using the laplace transformrdquo Interna-tional Journal of Nonlinear Sciences and Numerical Simulationvol 10 no 11-12 pp 1377ndash1382 2009
[8] G-C Wu and D Baleanu ldquoVariational iteration method forfractional calculusmdasha universal approach by Laplace trans-formrdquo Advances in Difference Equations vol 2013 article 182013
[9] X J Yang and D Baleanu ldquoFractal heat conduction problemsolved by local fractional variation iteration methodrdquo ThermalScience vol 17 no 2 pp 625ndash628 2013
[10] G C Wu ldquoVariational iteration method for solving the time-fractional diffusion equations in porous mediumrdquo ChinesePhysics B vol 21 no 12 Article ID 120504 2012
[11] G-C Wu and D Baleanu ldquoVariational iteration method forthe Burgersrsquo flow with fractional derivativesmdashnew Lagrangemultipliersrdquo Applied Mathematical Modelling vol 37 no 9 pp6183ndash6190 2013
[12] G C Wu ldquoChallenge in the variational iteration method-a newapproach to identification of the Lagrange mutipliersrdquo Journalof King Saud UniversitymdashScience vol 25 pp 175ndash178 2013
[13] G C Wu ldquoLaplace transform overcoming principle drawbacksin application of the variational iteration method to fractionalheat equationsrdquo Thermal Science vol 16 no 4 pp 1257ndash12612012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Function Spaces 5
4 Conclusion
Themethod of combining Laplace transforms and variationaliteration method is proposed for the solution of linearand nonlinear partial differential equations This method isapplied in a direct waywithout employing linearization and issuccessfully implemented by using the initial conditions andconvolution integral
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
This work was funded by the Deanship of Scientific Research(DSR) King AbdulazizUniversity Jeddah under Grant no(363-008-D1434) The authors therefore acknowledge withthanks DSR technical and financial support
References
[1] J Biazar andH Ghazvini ldquoHersquos variational iterationmethod forsolving linear and non-linear systems of ordinary differentialequationsrdquo Applied Mathematics and Computation vol 191 no1 pp 287ndash297 2007
[2] J He ldquoVariational iteration method for delay differential equa-tionsrdquo Communications in Nonlinear Science and NumericalSimulation vol 2 no 4 pp 235ndash236 1997
[3] J-H He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999
[4] J-H He ldquoVariational iteration method for autonomous ordi-nary differential systemsrdquo Applied Mathematics and Computa-tion vol 114 no 2-3 pp 115ndash123 2000
[5] J-H He and X-H Wu ldquoVariational iteration method newdevelopment and applicationsrdquo Computers ampMathematics withApplications vol 54 no 7-8 pp 881ndash894 2007
[6] SAKhuri andA Sayfy ldquoALaplace variational iteration strategyfor the solution of differential equationsrdquo Applied MathematicsLetters vol 25 no 12 pp 2298ndash2305 2012
[7] E Hesameddini and H Latifizadeh ldquoReconstruction of varia-tional iteration algorithms using the laplace transformrdquo Interna-tional Journal of Nonlinear Sciences and Numerical Simulationvol 10 no 11-12 pp 1377ndash1382 2009
[8] G-C Wu and D Baleanu ldquoVariational iteration method forfractional calculusmdasha universal approach by Laplace trans-formrdquo Advances in Difference Equations vol 2013 article 182013
[9] X J Yang and D Baleanu ldquoFractal heat conduction problemsolved by local fractional variation iteration methodrdquo ThermalScience vol 17 no 2 pp 625ndash628 2013
[10] G C Wu ldquoVariational iteration method for solving the time-fractional diffusion equations in porous mediumrdquo ChinesePhysics B vol 21 no 12 Article ID 120504 2012
[11] G-C Wu and D Baleanu ldquoVariational iteration method forthe Burgersrsquo flow with fractional derivativesmdashnew Lagrangemultipliersrdquo Applied Mathematical Modelling vol 37 no 9 pp6183ndash6190 2013
[12] G C Wu ldquoChallenge in the variational iteration method-a newapproach to identification of the Lagrange mutipliersrdquo Journalof King Saud UniversitymdashScience vol 25 pp 175ndash178 2013
[13] G C Wu ldquoLaplace transform overcoming principle drawbacksin application of the variational iteration method to fractionalheat equationsrdquo Thermal Science vol 16 no 4 pp 1257ndash12612012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Stochastic AnalysisInternational Journal of
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Complex AnalysisJournal of
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OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
