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Solution of singular and nonsingular initial and boundary value problems by modified variational iteration method. (English) Zbl 1155.65083

Summary: We apply the modified variational iteration method for solving the singular and nonsingular initial and boundary value problems. The proposed modification is made by introducing Adomian’s polynomials in the correct functional. The suggested algorithm is quite efficient and is practically well suited for use in such problems. The proposed iterative scheme finds the solution without any discretization, linearization, perturbation, or restrictive assumptions. Several examples are given to verify the efficiency and reliability of the suggested algorithm.

MSC:

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
35G30 Boundary value problems for nonlinear higher-order PDEs
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[1] S. Abbasbandy, “A new application of He’s variational iteration method for quadratic Riccati differential equation by using Adomian’s polynomials,” Journal of Computational and Applied Mathematics, vol. 207, no. 1, pp. 59-63, 2007. · Zbl 1120.65083 · doi:10.1016/j.cam.2006.07.012
[2] S. Abbasbandy, “Numerical solution of non-linear Klein-Gordon equations by variational iteration method,” International Journal for Numerical Methods in Engineering, vol. 70, no. 7, pp. 876-881, 2007. · Zbl 1194.65120 · doi:10.1002/nme.1924
[3] S. Abbasbandy, “Numerical method for non-linear wave and diffusion equations by the variational iteration method,” International Journal for Numerical Methods in Engineering, vol. 73, no. 12, pp. 1836-1843, 2008. · Zbl 1159.76372 · doi:10.1002/nme.2150
[4] M. A. Abdou and A. A. Soliman, “New applications of variational iteration method,” Physica D, vol. 211, no. 1-2, pp. 1-8, 2005. · Zbl 1084.35539 · doi:10.1016/j.physd.2005.08.002
[5] J. Biazar and H. Ghazvini, “He’s variational iteration method for fourth-order parabolic equations,” Computers & Mathematics with Applications, vol. 54, no. 7-8, pp. 1047-1054, 2007. · Zbl 1267.65147 · doi:10.1016/j.camwa.2006.12.049
[6] J.-H. He, “Homotopy perturbation technique,” Computer Methods in Applied Mechanics and Engineering, vol. 178, no. 3-4, pp. 257-262, 1999. · Zbl 0956.70017 · doi:10.1016/S0045-7825(99)00018-3
[7] J.-H. He, “Homotopy perturbation method for solving boundary value problems,” Physics Letters A, vol. 350, no. 1-2, pp. 87-88, 2006. · Zbl 1195.65207 · doi:10.1016/j.physleta.2005.10.005
[8] J.-H. He, “Comparison of homotopy perturbation method and homotopy analysis method,” Applied Mathematics and Computation, vol. 156, no. 2, pp. 527-539, 2004. · Zbl 1062.65074 · doi:10.1016/j.amc.2003.08.008
[9] J.-H. He, “Homotopy perturbation method for bifurcation of nonlinear problems,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 6, no. 2, pp. 207-208, 2005. · Zbl 1401.65085
[10] J.-H. He, “The homotopy perturbation method for nonlinear oscillators with discontinuities,” Applied Mathematics and Computation, vol. 151, no. 1, pp. 287-292, 2004. · Zbl 1039.65052 · doi:10.1016/S0096-3003(03)00341-2
[11] J.-H. He, “A coupling method of a homotopy technique and a perturbation technique for non-linear problems,” International Journal of Non-Linear Mechanics, vol. 35, no. 1, pp. 37-43, 2000. · Zbl 1068.74618 · doi:10.1016/S0020-7462(98)00085-7
[12] J.-H. He, “Some asymptotic methods for strongly nonlinear equations,” International Journal of Modern Physics B, vol. 20, no. 10, pp. 1141-1199, 2006. · Zbl 1102.34039 · doi:10.1142/S0217979206033796
[13] J.-H. He and X.-H. Wu, “Construction of solitary solution and compacton-like solution by variational iteration method,” Chaos, Solitons & Fractals, vol. 29, no. 1, pp. 108-113, 2006. · Zbl 1147.35338 · doi:10.1016/j.chaos.2005.10.100
[14] J.-H. He, “Variational iteration method-some recent results and new interpretations,” Journal of Computational and Applied Mathematics, vol. 207, no. 1, pp. 3-17, 2007. · Zbl 1119.65049 · doi:10.1016/j.cam.2006.07.009
[15] J.-H. He and X.-H. Wu, “Variational iteration method: new development and applications,” Computers & Mathematics with Applications, vol. 54, no. 7-8, pp. 881-894, 2007. · Zbl 1141.65372 · doi:10.1016/j.camwa.2006.12.083
[16] J.-H. He, “The variational iteration method for eighth-order initial-boundary value problems,” Physica Scripta, vol. 76, no. 6, pp. 680-682, 2007. · Zbl 1134.34307 · doi:10.1088/0031-8949/76/6/016
[17] J.-H. He, “Variational iteration method-a kind of non-linear analytical technique: some examples,” International Journal of Non-Linear Mechanics, vol. 34, no. 4, pp. 699-708, 1999. · Zbl 1342.34005 · doi:10.1016/S0020-7462(98)00048-1
[18] J.-H. He, “Variational iteration method for autonomous ordinary differential systems,” Applied Mathematics and Computation, vol. 114, no. 2-3, pp. 115-123, 2000. · Zbl 1027.34009 · doi:10.1016/S0096-3003(99)00104-6
[19] M. Inokuti, H. Sekine, and T. Mura, “General use of the Lagrange multiplier in nonlinear mathematical physics,” in Variational Method in the Mechanics of Solids, S. Nemat-Naseer, Ed., pp. 156-162, Pergamon Press, New York, NY, USA, 1978.
[20] R. E. Kidder, “Unsteady flow of gas through a semi-infinite porous medium,” Journal of Applied Mechanics, vol. 24, pp. 329-332, 1957. · Zbl 0078.40903
[21] S. Momani and S. Abuasad, “Application of He’s variational iteration method to Helmholtz equation,” Chaos, Solitons & Fractals, vol. 27, no. 5, pp. 1119-1123, 2006. · Zbl 1086.65113 · doi:10.1016/j.chaos.2005.04.113
[22] S. Momani and V. S. Ertürk, “Solutions of non-linear oscillators by the modified differential transform method,” Computers & Mathematics with Applications, vol. 55, no. 4, pp. 833-842, 2008. · Zbl 1142.65058 · doi:10.1016/j.camwa.2007.05.009
[23] S. T. Mohyud-Din, “A reliable algorithm for Blasius equation,” in Proceedings of the International Conference of Mathematical Sciences (ICMS ’07), pp. 616-626, Selangor, Malaysia, November 2007.
[24] M. A. Noor and S. T. Mohyud-Din, “Homotopy perturbation method for nonlinear higher-order boundary value problems,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 9, no. 4, pp. 395-408, 2008. · Zbl 1142.65386
[25] S. T. Mohyud-Din and M. A. Noor, “Homotopy perturbation method for solving fourth-order boundary value problems,” Mathematical Problems in Engineering, vol. 2007, Article ID 98602, 15 pages, 2007. · Zbl 1144.65311 · doi:10.1155/2007/98602
[26] M. A. Noor and S. T. Mohyud-Din, “An efficient algorithm for solving fifth-order boundary value problems,” Mathematical and Computer Modelling, vol. 45, no. 7-8, pp. 954-964, 2007. · Zbl 1133.65052 · doi:10.1016/j.mcm.2006.09.004
[27] M. A. Noor and S. T. Mohyud-Din, “Homotopy perturbation method for solving sixth-order boundary value problems,” Computers & Mathematics with Applications, vol. 55, no. 12, pp. 2953-2972, 2008. · Zbl 1142.65386 · doi:10.1016/j.camwa.2007.11.026
[28] M. A. Noor and S. T. Mohyud-Din, “Homotopy method for solving eighth order boundary value problems,” Journal of Mathematical Analysis and Approximation Theory, vol. 1, no. 2, pp. 161-169, 2006. · Zbl 1204.65086
[29] M. A. Noor and S. T. Mohyud-Din, “Approximate solutions of Flieral-Petviashivili equation and its variants,” International Journal of Mathematics and Computer Science, vol. 2, no. 4, pp. 345-360, 2007. · Zbl 1136.65073
[30] M. A. Noor and S. T. Mohyud-Din, “Variational iteration technique for solving higher order boundary value problems,” Applied Mathematics and Computation, vol. 189, no. 2, pp. 1929-1942, 2007. · Zbl 1122.65374 · doi:10.1016/j.amc.2006.12.071
[31] M. A. Noor and S. T. Mohyud-Din, “An efficient method for fourth-order boundary value problems,” Computers & Mathematics with Applications, vol. 54, no. 7-8, pp. 1101-1111, 2007. · Zbl 1141.65375 · doi:10.1016/j.camwa.2006.12.057
[32] M. A. Noor and S. T. Mohyud-Din, “Variational iteration method for solving higher-order nonlinear boundary value problems using He’s polynomials,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 9, no. 2, 2008. · Zbl 1151.65334
[33] M. A. Noor and S. T. Mohyud-Din, “Variational iteration method for fifth-order boundary value problems using He’s polynomials,” Mathematical Problems in Engineering, vol. 2008, Article ID 954794, 12 pages, 2008. · Zbl 1151.65334 · doi:10.1155/2008/954794
[34] M. A. Noor and S. T. Mohyud-Din, “Variational iteration method for unsteady flow of gas through a porous medium using He’s polynomials and Pade approximants,” Computer and Mathmetics with Applications. In press. · Zbl 1189.65169 · doi:10.1016/j.camwa.2009.03.016
[35] M. A. Noor and S. T. Mohyud-Din, “Solution of twelfth-order boundary value problems by variational iteration technique,” Journal of Applied Mathematics and Computing. In press. · Zbl 1152.65084 · doi:10.1007/s12190-008-0081-0
[36] M. A. Noor and S. T. Mohyud-Din, “Variational iteration decomposition method for solving eighth-order boundary value problems,” Differential Equations and Nonlinear Mechanics, vol. 2007, Article ID 19529, 16 pages, 2007. · Zbl 1143.49023 · doi:10.1155/2007/19529
[37] M. A. Noor and S. T. Mohyud-Din, “Modified variational iteration method for solving fourth-order boundary value problems,” Journal of Applied Mathematics and Computing. In press. · Zbl 1176.65083 · doi:10.1007/s12190-008-0090-z
[38] M. A. Noor and S. T. Mohyud-Din, “Variational decomposition method for singular initial value problems,” International Journal of Pure and Applied Mathematics. In press. · Zbl 1161.35458
[39] M. A. Noor and S. T. Mohyud-Din, “Variational iteration method for solving initial value problems of Bratu type,” Applications and Applied Mathematics, vol. 3, pp. 89-99, 2008. · Zbl 1177.65113
[40] M. A. Noor and S. T. Mohyud-Din, “Variational iteration decomposition method for solving sixth-order boundary value problems,” International Journal of Nonlinear Sciences and Numerical Simulation. In press. · Zbl 1221.65176
[41] J. I. Ramos, “On the variational iteration method and other iterative techniques for nonlinear differential equations,” Applied Mathematics and Computation, vol. 199, no. 1, pp. 39-69, 2008. · Zbl 1142.65082 · doi:10.1016/j.amc.2007.09.024
[42] A.-M. Wazwaz, “Analytic treatment for variable coefficient fourth-order parabolic partial differential equations,” Applied Mathematics and Computation, vol. 123, no. 2, pp. 219-227, 2001. · Zbl 1027.35006 · doi:10.1016/S0096-3003(00)00070-9
[43] A.-M. Wazwaz, “A study on a boundary-layer equation arising in an incompressible fluid,” Applied Mathematics and Computation, vol. 87, no. 2-3, pp. 199-204, 1997. · Zbl 0904.76067 · doi:10.1016/S0096-3003(96)00281-0
[44] A.-M. Wazwaz, “The modified decomposition method applied to unsteady flow of gas through a porous medium,” Applied Mathematics and Computation, vol. 118, no. 2-3, pp. 123-132, 2001. · Zbl 1024.76056 · doi:10.1016/S0096-3003(99)00209-X
[45] L. Xu, “He’s homotopy perturbation method for a boundary layer equation in unbounded domain,” Computers & Mathematics with Applications, vol. 54, no. 7-8, pp. 1067-1070, 2007. · Zbl 1267.76089 · doi:10.1016/j.camwa.2006.12.052
[46] L. Xu, “Variational iteration method for solving integral equations,” Computers & Mathematics with Applications, vol. 54, no. 7-8, pp. 1071-1078, 2007. · Zbl 1141.65400 · doi:10.1016/j.camwa.2006.12.053
[47] J. P. Boyd, “Padé approximant algorithm for solving nonlinear ordinary differential equation boundary value problems on an unbounded domain,” Computers in Physics, vol. 11, no. 3, pp. 299-303, 1997. · doi:10.1063/1.168606
[48] D. D. Ganji, G. A. Afrouzi, and R. A. Talarposhti, “Application of He’s variational iteration method for solving the reaction-diffusion equation with ecological parameters,” Computers & Mathematics with Applications, vol. 54, no. 7-8, pp. 1010-1017, 2007. · Zbl 1267.65151 · doi:10.1016/j.camwa.2006.12.069
[49] A. Ghorbani and J. Saberi-Nadjafi, “He’s homotopy perturbation method for calculating adomian polynomials,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, no. 2, pp. 229-232, 2007. · Zbl 1401.65056
[50] A. Ghorbani, “Beyond Adomian polynomials: He polynomials,” Chaos, Solitons & Fractals. In press. · Zbl 1197.65061 · doi:10.1016/j.chaos.2007.06.034
[51] M. A. Hajji and K. Al-Khaled, “Analytic studies and numerical simulations of the generalized Boussinesq equation,” Applied Mathematics and Computation, vol. 191, no. 2, pp. 320-333, 2007. · Zbl 1193.35188 · doi:10.1016/j.amc.2007.02.090
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