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Wazwaz, Abdul-Majid

The numerical solution of fifth-order boundary value problems by the decomposition method. (English) Zbl 0986.65072

J. Comput. Appl. Math. 136, No. 1-2, 259-270 (2001).
Summary: We present a fast and accurate numerical scheme for the solution of fifth-order boundary value problems with two-point boundary conditions. The Adomian decomposition method and a modified form of this method are applied to construct the numerical solution. The new approach provides the solution in the form of a rapidly convergent series and not at grid points. Two numerical illustrations are given to show the pertinent features of technique.

Cited in 54 Documents

MSC:

65L10 Numerical solution of boundary value problems involving ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations

Keywords:

numerical examples; convergence; fifth-order boundary value problems; two-point boundary conditions; Adomian decomposition method
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\textit{A.-M. Wazwaz}, J. Comput. Appl. Math. 136, No. 1--2, 259--270 (2001; Zbl 0986.65072)
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References:

[1] Adomian, G., A review of the decomposition method in applied mathematics, J. Math. Anal. Appl., 135, 501-544 (1988) · Zbl 0671.34053
[2] Adomian, G., Solving Frontier Problems of Physics: The Decomposition Method (1994), Kluwer: Kluwer Boston · Zbl 0802.65122
[3] Adomian, G., The diffusion-Brusselator equation, Comput. Math. Appl., 29, 5, 1-3 (1995) · Zbl 0827.35056
[4] Adomian, G.; Rach, R., Analytic solution of nonlinear boundary-value problems in several dimensions by decomposition, J. Math. Anal. Appl., 174, 118-137 (1993) · Zbl 0796.35017
[5] Agarwal, R. P., Boundary Value Problems for High Ordinary Differential Equations (1986), World Scientific: World Scientific Singapore · Zbl 0598.65062
[6] Bellomo, N.; Monaco, R., A comparison between Adomian’s decomposition methods and perturbation techniques for nonlinear random differential equations, J. Math. Anal. Appl., 110, 495-502 (1985) · Zbl 0575.60064
[7] Caglar, H. N.; Caglar, S. H.; Twizell, E. H., The numerical solution of fifth-order boundary value problems with sixth-degree B-spline functions, Appl. Math. Lett., 12, 25-30 (1999) · Zbl 0941.65073
[8] Cherruault, Y., Convergence of Adomian’s method, Math. Comput. Modelling, 14, 83-86 (1990) · Zbl 0728.65056
[9] Cherruault, Y.; Adomian, G., Decomposition method: a new proof of convergence, Math. Comput. Modelling, 18, 12, 103-106 (1993) · Zbl 0805.65057
[10] Cherrauault, Y.; Saccomandi, G.; Some, B., New results for convergence of Adomian’s method applied to integral equations, Math. Comput. Modelling, 16, 2, 85-93 (1992) · Zbl 0756.65083
[11] Datta, B. K., A new approach to the wave equation-an application of the decomposition method, J. Math. Anal. Appl., 142, 6-12 (1989) · Zbl 0685.35059
[12] Datta, B. K., A technique for approximate solutions to Schrodinger-loke equations, Comput. Math. Appl., 20, 1, 61-65 (1990) · Zbl 0707.65086
[13] Davis, A. R.; Karageorghis, A.; Phillips, T. N., Spectral Galerkin methods for the primary two-point boundary-value problem in modeling viscoelastic flows, Internat. J. Numer. Methods Eng., 26, 647-662 (1988) · Zbl 0635.73091
[14] Davis, A. R.; Karageorghis, A.; Phillips, T. N., Spectral collocation methods for the primary two-point boundary-value problem in modelling viscoelastic flows, Internat. J. Numer. Methods Eng., 26, 805-813 (1988) · Zbl 0637.76008
[16] Maleknejad, K.; Hadizadeh, M., A new computational method for Volterra-Fredholm integral equations, Comput. Math. Appl., 37, 1-8 (1999) · Zbl 0940.65151
[17] Rach, R., On the Adomian decomposition method and comparisons with Picards method, J. Math. Anal. Appl., 128, 480-483 (1987) · Zbl 0645.60067
[18] Shawagfeh, N. T., Nonperturbative approximate solution for Lane-Emden equation, J. Math. Phys., 34, 9, 4364-4369 (1993) · Zbl 0780.34007
[19] Venkatarangan, S. N.; Rajalashmi, K., A modification of Adomian’s solution for nonlinear oscillatory systems, Comput. Math. Appl., 29, 6, 67-73 (1995) · Zbl 0818.34006
[20] Venkatarangan, S. N.; Rajalashmi, K., Modification of Adomian’s decomposition method to solve equations containing radicals, Comput. Math. Appl., 29, 6, 75-80 (1995) · Zbl 0818.34007
[21] Wazwaz, A. M., A First Course in Integral Equations (1997), World Scientific: World Scientific New Jersey
[22] Wazwaz, A. M., A comparison between Adomian decomposition method and Taylor series method in the series solutions, Appl. Math. Comput., 79, 37-44 (1998) · Zbl 0943.65084
[23] Wazwaz, A. M., A reliable modification of Adomian’s decomposition method, Appl. Math. Comput., 102, 77-86 (1999) · Zbl 0928.65083
[24] Wazwaz, A. M., Analytical approximations and Pade’ approximants for Volterra’s population model, Appl. Math. Comput., 100, 13-25 (1999) · Zbl 0953.92026
[25] Wazwaz, A. M., The modified decomposition method and Pade’ approximants for solving Thomas-Fermi equation, Appl. Math. Comput., 105, 11-19 (1999) · Zbl 0956.65064
[26] Wazwaz, A. M., A new algorithm for calculating Adomian polynomials for nonlinear operators, Appl. Math. Comput., 111, 53-69 (2000) · Zbl 1023.65108
[27] Wazwaz, A. M., The decomposition method applied to systems of partial differential equations and to the reaction-diffusion Brusselator model, Appl. Math. Comput., 110, 251-264 (2000) · Zbl 1023.65109
[28] Wazwaz, A. M., The decomposition method for solving the diffusion equation subject to the specification of mass, Internat. J. Appl. Math., 3, 1, 25-34 (2000) · Zbl 1052.35049
[30] Yee, E., Application of the decomposition method to the solution of the reaction-convection-diffusion equation, Appl. Math. Comput., 56, 1-27 (1993) · Zbl 0773.76055
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