×

An efficient method for solving Bratu equations. (English) Zbl 1093.65108

Summary: We present a numerical technique for solving the Bratu equation. It is based on the Laplace Adomian decomposition method which produces an implicit equation in two variables. We use the predictor corrector technique to trace the solution curve generated from this equation. Numerical results and conclusions are presented.

MSC:

65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35A22 Transform methods (e.g., integral transforms) applied to PDEs
65H20 Global methods, including homotopy approaches to the numerical solution of nonlinear equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Adomain, G., Solving Frontier Problems on Physics: The Decomposition Method (1994), Kluwer Acadimic Publisher: Kluwer Acadimic Publisher Boston · Zbl 0802.65122
[2] Adomain, G., A review of the decomposition method in applied mathematics, J. Math. Anal. Appl., 135, 501-544 (1988) · Zbl 0671.34053
[3] Allgower, E. L.; Georg, K., Continuation and path following, Acta Numerica (1993), Cambridge University Press: Cambridge University Press Cambridge, pp. 1-64 · Zbl 0792.65034
[4] Allgower, E. L.; Chien, C. S.; Georg, K.; Wang, C.-F., Conjugate gradient methods for continuation problems, J. Comput. Appl. Math., 38, 1-16 (1991) · Zbl 0753.65046
[5] Abbott, J. P., An efficient algorithm for the determination of certain bifurcation points, J. Comput. Appl. Math., 4, 19-27 (1978) · Zbl 0384.65022
[6] Ascher, U. M.; Matheij, R. M.M.; Russell, R. D., Numerical Solution of Boundary Value Problems for Ordinary Differential Equations (1995), Society for Industrial and Applied Mathematics: Society for Industrial and Applied Mathematics Philadelphia, PA · Zbl 0843.65054
[7] Attili, B. S., Efficient finite differences for parameter-dependent singularities in two-point boundary value problems, Comput. Methods Appl. Mech. Engrg., 190, 5543-5549 (2001) · Zbl 0993.65085
[8] Bank, R. E.; Chan, T. F., PLTMGC: a multi-grid continuation program for parametrized nonlinear elliptic systems, SIAM J. Sci. Statist. Comput., 7, 540-559 (1986) · Zbl 0589.65074
[9] Bolstad, J. H.; Keller, H. B., A multigrid continuation method for elliptic problems with folds, SIAM J. Sci. Statist. Comput., 7, 1081-1104 (1986) · Zbl 0631.65052
[10] Chan, T. F., Newton-like pseudo-arclength methods for computing simple turning points, SIAM J. Sci. Statist. Comput., 5, 135-148 (1984) · Zbl 0536.65029
[11] Deeba, E.; Khuri, S. A.; Xie, S., An algorithm for solving boundary value problems, J. Comput. Phys., 159, 125-138 (2000) · Zbl 0959.65091
[12] Fedoseyev, A. I.; Friedman, M. J.; Kansa, E. J., Continuation for nonlinear elliptic partial differential equations discretized by the multiquardic method, Int. J. Bifurcat. Chaos, 10, 2, 481-492 (2000) · Zbl 1090.65550
[13] Gidas, B.; Ni, W.; Nirenberg, L., Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68, 209-243 (1979) · Zbl 0425.35020
[14] Keller, H. B.; Cohen, D. S., Some position problems suggested by nonlinear heat generation, J. Math. Mech., 16, 1361-1376 (1967) · Zbl 0152.10401
[15] Rosen, J. B., Approximate solution and error bounds for quasilinear elliptic boundary value problems, SIAM J. Numer. Anal., 8, 80-103 (1970) · Zbl 0206.47501
[16] Siyyam, H.; Syam, M., The modified trapezoidal rule for line integrals, J. Computat. Appl. Math., 84, 1-14 (1997) · Zbl 0885.65017
[17] Syam, M.; Siyyam, H., Numerical differentiation of implicitly defined curves, J. Computat. Appl. Math., 108, 131-144 (1999) · Zbl 0939.65029
[18] Wazwaz, A. M., A new modification of the Adomian decomposition method for linear and nonlinear operators, Appl. Math. Comput., 122, 393-405 (2001) · Zbl 1027.35008
[19] Wazwaz, A. M., A comparison between Adomain decomposition method and Taylor series method in the series solutions, Appl. Math. Comput., 97, 37-44 (1998) · Zbl 0943.65084
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.