This paper applies the variational iteration method to solve two systems of nonlinear partial differential equations, numerically and/or analytically. Two examples are given to illustrate the accuracy and effectiveness of the method. Comparison with Adomian decomposition method reveals that the method is easier to be implemented.
HeJ.H., A New Approach to Nonlinear Partial Differential Equations, Commun. Nonlinear Sci. Numer. Simul., 1997, 2(4), 230–235.
2.
HeJ.H., Approximate Analytical Solution for Seepage Flow with Fractional Derivatives in Porous Media, Comput. Methods Appl. Mech. Eng., 1998, 167, 57–68.
3.
HeJ.H., Approximate Solution of Nonlinear Differential Equations with Convolution Product Nonlinearities, Comput. Methods Appl. Mech. Eng., 1998, 167, 69–73.
4.
HeJ.H., Variational Iteration Method-a Kind of Non-linear Analytical Technique: Some Examples, Int. J. Nonlinear Mech., 1999, 34, 699–708.
5.
HeJ.H. and WuX.H., Construction of solitary solution and compacton-like solution by Variational iteration method, Chaos, Solitons and Fractals, 2006, 29, 108–113.
6.
DarvishiM.T.KhaniF. and SolimanA.A., The Numerical Simulation for Stiff Systems of Ordinary Differential Equations, Comput. Math. Appl., 2007, 54(7–8), 1055–1063.
7.
DarvishiM.T. and KhaniF., Numerical and Explicit Solutions of the Fifth-Order Korteweg-de Vries Equations, Chaos, Solitons and Fractals, 2009, 39, 2484–2490.
8.
DebtnathL., Nonlinear Partial Differential Equations for Scientists and Engineers, Birkhauser, Boston, 1997.
9.
WazwazA. M., Partial Differential Equations Methods and Applications, Rotterdam, Balkema, 2002.
10.
HirotaR., Exact Solution of the Kortewegde Vries Equation for Multiple Collisions of Solitons, Phys. Rev. Lett., 1971, 27, 1192–1194.
11.
YanC.T., A Simple Transformation for Nonlinear Waves, Phys. Lett. A, 1996, 224, 77–82.
12.
WangM.L., Exact Solution for a Compound KdV-Burgers Equation, Phys. Lett. A, 1996, 213, 279–287.
13.
YanZ.Y. and ZhangH.Q., Symbolic Computation and New Families of Exact Soliton-Like Solutions to the Integrable Broer-Kaup (BK) Equations in (2+1)-Dimensional Spaces, J. Phys. A, 2001, 34, 1785–1792.
14.
YanZ.Y. and ZhangH.Q., New Explicit Solitary Wave Solutions and Periodic Wave Solutions for Whitham-Broer-Kaup Equation in Shallow Water, Phys. Lett. A, 2001, 285, 355–362.
15.
El-ShahedM., Application of He's Homotopy Perturbation Method to Volterra's Integro-Differential Equation, Int. J. of Nonlinear Sciences and Numerical Simulation, 2005, 6(2), 163–168.
16.
HeJ.H., Application of Homotopy Perturbation Method to Nonlinear Wave Equations, Appl. Math. Comput., 2005, 26(3), 695–700.
17.
HeJ.H., Limit Cycle and Bifurcation of Nonlinear Problems, Chaos, Solitons and Fractals, 2005, 26(3), 827–833.
18.
HeJ.H., Homotopy Perturbation Method for Bifurcation of Nonlinear Problems, Int. J. of Nonlinear Sciences and Numerical Simulation, 2005, 6(2), 207–208.
19.
HeJ.H., Homotopy Perturbation Method for Solving Boundary Value Problems, Phys. Lett. A, 2006, 350(1–2), 87–88.
20.
SeddiquiA.M.MahmoodR. and GhoriQ.K., Thin Film Flow of a Third Grade Fluid on a Moving Belt by He's Homotopy Perturbation Method, Int. J. of Nonlinear Sciences and Numerical Simulation, 2006, 7(1), 7–14.
21.
SeddiquiA.M.AhmedM. and GhoriQ.K., Couette and Poiseuille Flows for Non-Newtonian Fluids, Int. J. of Nonlinear Sciences and Numerical Simulation, 2006, 7(1), 15–26.
22.
DarvishiM.T. and KhaniF., Application of He's Homotopy Perturbation Method to Stiff Systems of Ordinary Differential Equations, Zeitschrift fuer Naturforschung A, 2008, 63a(l–2), 19–23.
23.
DarvishiM.T.KhaniF.Hamedi-NezhadS. and RyuSang-Wan, New Modification of the HPM for Numerical Solutions of the Sine-Gordon and Coupled Sine-Gordon Equations, Int. J. Comput. Math., (in press).
24.
InokuttiM.SekineH. and MuraT., General Use of the Lagrange Multiplier in Nonlinear Mathematics, in Nemat-Nassers editor, Variational Method in the Mechanics of Solids, Oxford: Pergamon Press, 1978, 156–162.
DarvishiM.T., Preconditioning and Domain Decomposition Schemes to Solve PDEs, International J. of Pure and Applied Mathematics, 2004, 1(4), 419–439.
27.
DarvishiM.T. and JavidiM., A Numerical Solution of Burgers' Equation by Pseudospectral Method and Darvishi's Preconditioning, Appl. Math. Comput., 2006, 173(1), 421–429.
28.
DarvishiM.T.KheybariS. and KhaniF., A Numerical Solution of the Korteweg-de Vries Equation by Pseudospectral Method using Darvishi's Preconditionings, Appl. Math. Comput., 2006182(1), 98–105.
29.
DarvishiM.T.KhaniF. and KheybariS., Spectral Collocation Solution of a Generalized Hirota-Satsuma KdV Equation, Int. J. Comput. Math., 2007, 84(4), 541–551.
30.
DarvishiM.T.KhaniF. and KheybariS., Spectral Collocation Method and Darvishi's Preconditionings to Solve the Generalized Burgers-Huxley Equation, Commun. Nonlinear Sci. Numer. Simul., 2008, 13(10), 2091–2103.
31.
HeJ.H., Bookkeeping Parameter in Perturbation Methods, Int. J. Nonlinear Sci. Numer. Simul., 2001, 2, 257.
32.
DarvishiM.T.KaramiA. and ShinB.-C., Application of He's Parameter-Expansion Method for Oscillators with Smooth Odd Non-linearities, Phys. Lett. A, 2008, 372(33), 5381–5384.
33.
ShinB.-C.DarvishiM.T. and KaramiA., Application of He's parameter-expansion method to a nonlinear self-excited oscillator system, Int. J. Nonlinear Sci. Numer. Simul., 2009, 10(1), 137–143.
HeJ.H.WanY.Q. and GouQ., An Iteration Formulation for Normalized Diode Characteristics, Int. J. Circuit Theory Appl., 2004, 32(6), 629–632.
36.
HeJ.H., Some Assymptotic Methods for Strongly Nonlinear Equations, Int. J. of Modern Physics B, 2006, 29(10), 1141–1199.
37.
Al-KhalledK. and AllanF., Construction of Solutions for the Shallow Water Equations by the Decomposition Method, Math. Comput. Simul., 2004, 66(6), 479–486.
38.
KayaD., An Application for the Higher Order Modified KdV Equation by Decomposition Method, Commun. Nonlinear Sci. Numer. Simul., 2005, 10(6), 693–702.
39.
GarvenF.. The maple book, Chapman and Hall/CRC, 2001.
40.
LuZ. and ZhangH., New Application of Further Extend Tanh Method, Phys. Lett. A, 2004, 324, 293–298.
41.
SolimanA.A., On the Solution of Two-dimensional Coupled Burgers Equations by Variational Iteration Method, Chaos, Solitons and Fractals, 2009, 40, 1146–1155.
42.
SalernaM., On the Phase Manifold Geometry of the Two-Dimensional Burgers Equations, Phys. Lett. A, 1987, 121, 15–18.
43.
KayaD. and YokusA., A Decomposition Method for Finding Solitary and Periodic Solutions for a Coupled Higher-Dimensional Burgers Equation, Appl. Math. Comput., 2005, 164, 857–864.
44.
BillinghamJ., Dynamics of a Strongly Nonlocal Reaction Diffusion Population Model, Nonlinearity, 2004, 17, 313–346.
45.
El-WakilS.A. and AbdouM.A., New Applications of Adomian Decomposition Method, Chaos, Solitons and Fractals, 2007, 33(2), 513–522.
