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Wazwaz, Abdul-Majid

Adomian decomposition method for a reliable treatment of the Bratu-type equations. (English) Zbl 1073.65068

Appl. Math. Comput. 166, No. 3, 652-663 (2005).
Summary: We present a framework to determine exact solutions of Bratu-type equations. The algorithm rests mainly on the Adomian decomposition method. The proposed scheme is illustrated by studying two boundary value problems and an initial value problem of Bratu-type. The first type gives a solution that blows up at the middle of the domain, whereas the other equations give bounded solutions.

Cited in 105 Documents

MSC:

65L10 Numerical solution of boundary value problems involving ordinary differential equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems

Keywords:

numerical examples; Bratu-type equations; algorithm; Adomian decomposition method; boundary value problems; initial value problem
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References:

[1] Ascher, U. M.; Matheij, R.; Russell, R. D., Numerical solution of boundary value problems for ordinary differential equations (1995), SIAM: SIAM Philadelphia, PA · Zbl 0843.65054
[2] Boyd, J. P., Chebyshev polynomial expansions for simultaneous approximation of two branches of a function with application to the one-dimensional Bratu equation, Applied Mathematics and Computation, 142, 189-200 (2003) · Zbl 1025.65042
[3] Boyd, J. P., An analytical and numerical study of the two-dimensional Bratu equation, Journal of Scientific Computing, 1, 2, 183-206 (1986) · Zbl 0649.65057
[4] Buckmire, R., Investigations of nonstandard Mickens-type finite-difference schemes for singular boundary value problems in cylindrical or spherical coordinates, Numerical Methods for partial Differential equations, 19, 3, 380-398 (2003) · Zbl 1079.76048
[5] Jacobson, J.; Schmitt, K., The Liouville-Bratu-Gelfand problem for radial operators, Journal of Differential Equations, 184, 283-298 (2002) · Zbl 1015.34013
[6] Y. Aregbesola, Numerical solution of Bratu problem using the method of weighted residual, in press.; Y. Aregbesola, Numerical solution of Bratu problem using the method of weighted residual, in press. · Zbl 0897.65053
[7] Adomian, G., Solving Frontier Problems of Physics: The Decomposition Method (1994), Kluwer: Kluwer Boston · Zbl 0802.65122
[8] Adomian, G., A review of the decomposition method in applied mathematics, Journal of Mathematical Analysis and Application, 135, 501-544 (1988) · Zbl 0671.34053
[9] Wazwaz, A. M., Partial Differential Equations: Methods and Applications (2002), Balkema Publishers: Balkema Publishers The Netherlands · Zbl 0997.35083
[10] Wazwaz, A. M., A new method for solving singular initial value problems in the second order differential equations, Applied Mathematics and Computation, 128, 47-57 (2002) · Zbl 1030.34004
[11] Wazwaz, A. M., A First Course in Integral Equations (1997), World Scientific: World Scientific Singapore · Zbl 0924.45001
[12] Wazwaz, A. M., Analytical approximations and Padé approximants for Volterra’s population model, Applied Mathematics and Computation, 100, 13-25 (1999) · Zbl 0953.92026
[13] Wazwaz, A. M., A new algorithm for calculating Adomian polynomials for nonlinear operators, Applied Mathematics and Computation, 111, 53-69 (2000) · Zbl 1023.65108
[14] Wazwaz, A. M., The decomposition method for solving the diffusion equation subject to the classification of mass, IJAM, 3, 1, 25-34 (2000) · Zbl 1052.35049
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