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Improvement of He’s variational iteration method for solving systems of differential equations. (English) Zbl 1189.65178

Summary: In recent years a lot of attention from researchers has been attracted to the various aspects of the well known He’s variational iteration method. This method is a very powerful method for solving a large amount of problems. It provides a sequence which converges to the solution of the problem without discretization of the variables. In this work an idea is proposed that accelerates the convergence of the sequences which result from the variational iteration method for solving systems of differential equations. Illustrative examples are presented to show the validity of the new method.

MSC:

65L99 Numerical methods for ordinary differential equations
34A45 Theoretical approximation of solutions to ordinary differential equations
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[1] He, J.H., Some asymptotic methods for strongly nonlinear equations, International journal of modern physics B, 20, 10, 1141-1199, (2006) · Zbl 1102.34039
[2] He, J.H., Variational iteration method for autonomous ordinary differential systems, Applied mathematics and computation, 114, 115-123, (2000) · Zbl 1027.34009
[3] Momani, S.; Abuasad, S., Application of he’s variational iteration method to Helmholtz equation, Chaos, solitons and fractals, 27, 1119-1123, (2006) · Zbl 1086.65113
[4] Abdou, M.A.; Soliman, A.A., Variational iteration method for solving burgers’ and coupled burgers’ equations, Journal of computational and applied mathematics, 181, 245-251, (2005) · Zbl 1072.65127
[5] Momani, S.; Odibat, Z., Analytical approach to linear fractional partial differential equations arising in fluid mechanics, Physics letters A, 355, 271-279, (2006) · Zbl 1378.76084
[6] Inc, M., Numerical simulation of KdV and mkdv equations with initial conditions by the variational iteration method, Chaos, solitons and fractals, 34, 1075-1081, (2007) · Zbl 1142.35572
[7] He, J.H; Wu, X.H., Construction of solitary solution and compacton-like solution by variational iteration method, Chaos, solitons and fractals, 29, 108-113, (2006) · Zbl 1147.35338
[8] Soliman, A.A., A numerical simulation and explicit solutions of kdv – burgers’ and lax’s seventh-order KdV equations, Chaos, solitons and fractals, 29, 294-302, (2006) · Zbl 1099.35521
[9] He, J.H., Variational iteration method—a kind of non-linear analytical technique: some examples, International journal of nonlinear mechanics, 34, 699-708, (1999) · Zbl 1342.34005
[10] Tatari, M.; Dehghan, M., He’s variational iteration method for computing a control parameter in a semi-linear inverse parabolic equation, Chaos, solitons and fractals, 33, 671-677, (2007) · Zbl 1131.65084
[11] Dehghan, M.; Tatari, M., Identifying an unknown function in a parabolic equation with overspecified data via he’s variational iteration method, Chaos, solitons and fractals, 36, 157-166, (2008) · Zbl 1152.35390
[12] Dehghan, M.; Shakeri, S., Application of he’s variational iteration method for solving the Cauchy reaction – diffusion problem, Journal of computational and applied mathematics, 214, 435-446, (2008) · Zbl 1135.65381
[13] Shakeri, F.; Dehghan, M., Numerical solution of a biological population model using he’s variational iteration method, Computers and mathematics with applications, 54, 1197-1209, (2007) · Zbl 1137.92033
[14] Shakeri, F.; Dehghan, M., Solution of a model describing biological species living together using the variational iteration method, Mathematical and computer modelling, 48, 685-699, (2008) · Zbl 1156.92332
[15] Tatari, M.; Dehghan, M., On the convergence of he’s variational iteration method, Journal of computational and applied mathematics, 207, 121-128, (2007) · Zbl 1120.65112
[16] He, J.H.; Wu, X.H., Variational iteration method: new development and applications, Computers and mathematics with applications, 54, 881-894, (2007) · Zbl 1141.65372
[17] He, J.H., Variational iteration method — some recent results and new interpretations, Journal of computational and applied mathematics, 207, 3-17, (2007) · Zbl 1119.65049
[18] Ozer, H., Application of the variational iteration method to the boundary value problems with jump discontinuities arising in solid mechanics, international, Journal of nonlinear sciences and numerical simulation, 8, 513-518, (2007)
[19] Biazar, J.; Ghazvini, H., He’s variational iteration method for solving hyperbolic differential equations, International journal of nonlinear sciences and numerical simulation, 8, 311-314, (2007) · Zbl 1193.65144
[20] Odibat, Z.M.; Momani, S., Application of variational iteration method to nonlinear differential equations of fractional order, International journal of nonlinear sciences and numerical simulation, 7, 27-34, (2007) · Zbl 1401.65087
[21] Dehghan, M.; Tatari, M., The use of he’s variational iteration method for solving a fokker – planck equation, Physica scripta, 74, 310-316, (2006) · Zbl 1108.82033
[22] Dehghan, M.; Shakeri, F., Approximate solution of a differential equation arising in astrophysics using the variational iteration method, New astronomy, 13, 53-59, (2008)
[23] Tatari, M.; Dehghan, M., Solution of problems in calculus of variations via he’s variational iteration method, Physics letters A, 362, 401-406, (2007) · Zbl 1197.65112
[24] Shakeri, F.; Dehghan, M., Numerical solution of the klein – gordon equation via he’s variational iteration method, Nonlinear dynamics, 51, 89-97, (2008) · Zbl 1179.81064
[25] He, J.H., Variational iteration method for delay differential equations, Communications in nonlinear science and numerical simulation, 2, 235-2356, (1997)
[26] He, J.H., Approximate solution of nonlinear differential equations with convolution product non-linearities, Computer methods in applied mechanics and engineering, 167, 69-73, (1998) · Zbl 0932.65143
[27] He, J.H., Approximate analytical solution for seepage flow with fractional derivatives in porous media, Computer methods in applied mechanics and engineering, 167, 57-68, (1998) · Zbl 0942.76077
[28] M. Dehghan, F. Shakeri, Solution of parabolic integro-differential equations arising in heat conduction in materials with memory via He’s variational iteration technique, Communications in Numerical Methods in Engineering (2008) (in press) · Zbl 1192.65158
[29] Saadatmandi, A.; Dehghan, M., Variational iteration method for solving a generalized pantograph equation, Computers and mathematics with applications, 58, 11-12, 2190-2196, (2009) · Zbl 1189.65172
[30] Dehghan, M.; Shakeri, F., The use of the decomposition procedure of Adomian for solving a delay differential equation arising in electrodynamics, Physica scripta, 78, 1-11, (2008), Article No. 065004 · Zbl 1159.78319
[31] M. Dehghan, A. Saadatmandi, Variational iteration method for solving the wave equation subject to an integral conservation condition, Chaos, Solitons and Fractals (2008) (in press) · Zbl 1198.65202
[32] Yousefi, S.A.; Lotfi, A.; Dehghan, M., He’s variational iteration method for solving the nonlinear mixed volterra – fredholm integral equations, Computers and mathematics with applications, 58, 11-12, 2172-2176, (2009) · Zbl 1189.65317
[33] S.A. Yousefi, M. Dehghan, The use of He’s variational iteration method for solving variational problems, International Journal of Computer Mathematics (2008) (in press) · Zbl 1191.65078
[34] Dehghan, M., Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices, Mathematics and computers in simulation, 71, 16-30, (2006) · Zbl 1089.65085
[35] M. Dehghan, F. Shakeri, The numerical solution of the second Painleve equation, Numerical Method for Partial Differential Equations, in press · Zbl 1172.65037
[36] Shakourifar, M.; Dehghan, M., On the numerical solution of nonlinear systems of Volterra integro-differential equations with delay arguments, Computing, 82, 241-260, (2008) · Zbl 1154.65098
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