On the convergence of He’s variational iteration method. (English) Zbl 1120.65112

Summary: We consider J.-H. He’s variational iteration method [Int. J. Mod. Phys. B 20, No. 10, 1141–1199 (2006; Zbl 1102.34039)] for solving second-order initial value problems. We discuss the use of this approach for solving several important partial differential equations. This method is based on the use of Lagrange multipliers for identification of the optimal value of a parameter in a functional. This procedure is a powerful tool for solving the large amount of problems. Using the variational iteration method, it is possible to find the exact solution or an approximate solution of the problem. This technique provides a sequence of functions which converges to the exact solution of the problem. Our emphasis will be on the convergence of the variational iteration method. In the current paper this scheme will be investigated in details and efficiency of the approach will be shown by applying the procedure on several interesting and important models.


65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35G25 Initial value problems for nonlinear higher-order PDEs


Zbl 1102.34039
Full Text: DOI


[1] Abdou, M.A.; Soliman, A.A., Variational iteration method for solving burger’s and coupled Burger’s equations, J. comput. appl. math., 181, 245-251, (2005) · Zbl 1072.65127
[2] Adomian, G., Solving frontier problems of physics: the decomposition method, (1994), Kluwer Academic Publishers Boston, MA · Zbl 0802.65122
[3] Adomian, G.; Rach, R., Modified decomposition solution of linear and nonlinear boundary-value problems, Nonlinear anal., 23, 615-619, (1994) · Zbl 0810.34015
[4] Debnath, L., Nonlinear partial differential equations for scientists and engineers, (1997), Birkhauser Boston · Zbl 0892.35001
[5] Dehghan, M.; Tatari, M., The use of He’s variational iteration method for solving the fokker – planck equation, Phys. scripta, 74, 310-316, (2006) · Zbl 1108.82033
[6] Elsgolts, L., Differential equations and the calculus of variations, translated from the Russian by G. yankovsky, (1977), Mir Moscow
[7] Evans, L.C., Partial differential equations, (1998), American Mathematical Society Providence, RI
[8] Guellal, S.; Grimalt, P.; Cherruault, Y., Numerical study of Lorenz’s equation by the Adomian method, Comput. math. appl., 33, 25-29, (1997) · Zbl 0869.65044
[9] He, J.H., Variational iteration method for delay differential equations, Comm. non-linear. sci. numer. simulation, 2, 235-236, (1997)
[10] He, J.H., Approximate solution of nonlinear differential equations with convolution product non-linearities, Comput. methods. appl. mech. engng., 167, 69-73, (1998) · Zbl 0932.65143
[11] He, J.H., Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput. methods. appl. mech. engng., 167, 57-68, (1998) · Zbl 0942.76077
[12] He, J.H., Variational iteration method—a kind of non-linear analytical technique: some examples, Internat. J. non-linear mech., 34, 699-708, (1999) · Zbl 1342.34005
[13] He, J.H., Variational iteration method for autonomous ordinary differential systems, Appl. math. comput., 114, 115-123, (2000) · Zbl 1027.34009
[14] He, J.H., Some asymptotic methods for strongly nonlinear equations, Internat. J. modern phys. B, 20, 10, 1141-1199, (2006) · Zbl 1102.34039
[15] J.H. He, Non-perturbative methods for strongly nonlinear problems, dissertation de-Verlag im Internet GmbH, Berlin, 2006.
[16] He, J.H.; Wan, Y.Q.; Guo, Q., An iteration formulation for normalized diode characteristics, Internat. J. circuit. theory. appl., 32, 629-632, (2004) · Zbl 1169.94352
[17] Inokuti, M.; Sekine, H.; Mura, T., General use of the Lagrange multiplier in non-linear mathematical physics, (), 156-162
[18] 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
[19] M. Tatari, M. Dehghan, He’s variational iteration method for computing a control parameter in a semi-linear inverse parabolic equation, Chaos, Solitons and Fractals, in press, corrected proof, available online 15 March 2006.
[20] Wazwaz, A.M., The modified Adomian decomposition method for solving linear and nonlinear boundary value problems of tenth-order and twelfth-order, Internat. J. nonlinear sci. numer. simulation, 1, 17-24, (2000) · Zbl 0966.65058
[21] Wazwaz, A.M., Construction of solitary wave solutions and rational solutions for the KdV equation by Adomian decomposition method, Chaos, solitons and fractals, 12, 2283-2293, (2001) · Zbl 0992.35092
[22] Wazwaz, A.M., An analytical study on the third-order dispersive partial differential equation, Appl. math. comput., 142, 511-520, (2003) · Zbl 1109.35312
[23] Tatari, M.; Dehghan, M., Numerical solution of Laplace equation in a disk using the Adomian decomposition method, Phys. scripta, 72, 5, 345-348, (2005) · Zbl 1128.65311
[24] Dehghan, M.; Tatari, M., The use of Adomian decomposition method for solving problems in calculus of variations, Math. problem eng., 1-12, (2006) · Zbl 1200.65050
[25] Dehghan, M., Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices, Math. comput. simul., 71, 1, 16-30, (2006) · Zbl 1089.65085
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.