Application of He’s variational iteration method to Helmholtz equation. (English) Zbl 1086.65113

Summary: We implement a new analytical technique, J. H. He’s variational iteration method [Varational iteration method – a kind of nonlinear analytical technique: Some examples. Int. J. Nonlinear Mech. 34, 699–708 (1999)] for solving the linear Helmholtz partial differential equation. In this method, general Lagrange multipliers are introduced to construct correction functionals for the problems. The multipliers in the functionals can be identified optimally via the variational theory. The initial approximations can be freely chosen with possible unknown constants, which can be determined by imposing the boundary/initial conditions. The results compare well with those obtained by the Adomian’s decomposition method.


65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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[1] Inokuti, M.; Sekine, H.; Mura, T., General use of the Lagrange multiplier in non-linear mathematical physics, (Nemat-Nasser, S., Variational method in the mechanics of solids (1978), Pergamon Press: Pergamon Press Oxford), 156-162
[2] He, J. H., Variational iteration method for delay differential equations, Commun Nonlinear Sci Numer Simulat, 2, 4, 235-236 (1997)
[3] He, J. H., Approximate solution of nonlinear differential equations with convolution product non-linearities, Comput Methods Appl Mech Eng, 167, 69-73 (1998) · Zbl 0932.65143
[4] He, J. H., Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput Methods Appl Mech Eng, 167, 57-68 (1998) · Zbl 0942.76077
[5] He, J. H., Variational iteration method—a kind of non-linear analytical technique: some examples, Int J Nonlin Mech, 34, 699-708 (1999) · Zbl 1342.34005
[6] He, J. H., Variational iteration method for autonomous ordinary differential systems, Appl Math Comput, 114, 115-123 (2000) · Zbl 1027.34009
[7] He, J. H.; Wan, Y. Q.; Guo, Q., An iteration formulation for normalized diode characteristics, Int J Circuit Theory Appl, 32, 6, 629-632 (2004) · Zbl 1169.94352
[8] He, J. H., Semi-inverse method of establishing generalized principles for fluid mechanics with emphasis on turbomachinery aerodynamics, Int J Turbo Jet-Engines, 14, 1, 23-28 (1997)
[9] He, J. H., Variational theory for linear magneto-electro-elasticity, Int J Nonlin Sci Numer Simulat, 2, 4, 309-316 (2001) · Zbl 1083.74526
[10] He, J. H., Generalized variational principles in fluids (2003), Science and Culture Publishing House of China: Science and Culture Publishing House of China Hong Kong, p. 222-30 [in Chinese]
[11] He, J. H., Variational principle for nano thin film lubrication, Int J Nonlin Sci Numer Simulat, 4, 3, 313-314 (2003)
[12] He, J. H., Variational principle for some nonlinear partial differential equations with variable coefficients, Chaos, Solitons & Fractals, 19, 4, 847-851 (2004) · Zbl 1135.35303
[13] Liu, H. M., Variational approach to nonlinear electrochemical system, Int J Nonlin Sci Numer Simulat, 5, 1, 95-96 (2004)
[14] Liu, H. M., Generalized variational principles for ion acoustic plasma waves by He’s semi-inverse method, Chaos, Solitons & Fractals, 23, 2, 573-576 (2005) · Zbl 1135.76597
[15] Hao, T. H., Search for variational principles in electrodynamics by Lagrange method, Int J Nonlin Sci Numer Simulat, 6, 2, 209-210 (2005) · Zbl 1401.78004
[16] Draˇgaˇnescu, Gh. E.; Caˇpaˇlnaˇsan, V., Nonlinear relaxation phenomena in polycrystalline solids, Int J Nonlin Sci Numer Simulat, 4, 3, 219-226 (2003)
[17] Marinca, V., An approximate solution for one-dimensional weakly nonlinear oscillations, Int J Nonlin Sci Numer Simulat, 3, 2, 107-110 (2002) · Zbl 1079.34028
[18] Burden, R. L.; Faires, J. D., Numerical analysis (1993), PWS Publishing Company: PWS Publishing Company Boston · Zbl 0788.65001
[19] Gerald, C. F.; Wheatley, P. O., Applied numerical analysis (1994), Addison Wesley: Addison Wesley California · Zbl 0877.65003
[20] EL-Sayed, S. M.; Kaya, D., Comparing numerical methods for Helmholtz equation model problem, Appl Math Comput, 150, 763-773 (2004) · Zbl 1051.65113
[21] Abassy, T. A.; El-Tawil, M. A.; Saleh, H. K., The solution of KdV and mKdV equations using Adomian Fade approximation, Int J Nonlin Sci Numer Simulat, 5, 4, 327-340 (2004) · Zbl 1401.65122
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