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An aftertreatment technique for improving the accuracy of Adomian’s decomposition method. (English) Zbl 1005.34006

Summary: Adomian’s decomposition method (ADM) is a nonnumerical method that can be adapted for solving nonlinear ordinary differential equations. Here, the principle of the decomposition method is described, and its advantages as well as drawbacks are discussed. Then an aftertreatment technique (AT) is proposed that yields an analytic approximate solution with fast convergence rate and high accuracy through the application of Padé approximation to the series solution derived from ADM. Some concrete examples are studied to show with numerical results how the AT works efficiently.

MSC:

34A45 Theoretical approximation of solutions to ordinary differential equations
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
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[1] Adomian, G., Nonlinear stochastic systems theory and applications to physics, (1989), Kluwer Academic · Zbl 0659.93003
[2] Adomian, G., A review of the decomposition method and some recent results for nonlinear equations, Computers math. applic., 21, 5, 101-127, (1991) · Zbl 0732.35003
[3] Adomian, G.; Rach, R.C.; Meyers, R.E., An efficient methodology for the physical sciences, Kybernetes, 20, 7, 24-34, (1991) · Zbl 0744.65039
[4] Adomian, G., Solving frontier problems of physics: the decomposition method, (1994), Kluwer Academic · Zbl 0802.65122
[5] Adomian, G., Solution of physical problems by decomposition, Computers math. applic., 27, 9/10, 145-154, (1994) · Zbl 0803.35020
[6] Guellal, S.; Cherruault, Y., Application of the decomposition method to identify the distributed parameters of an elliptical equation, Mathl. comput. modelling, 21, 4, 51-55, (1995) · Zbl 0822.65120
[7] Jiao, Y.C.; Xiao, G.X.; Hao, Y., Solution of one-dimensional Poisson’s equation by decomposition method, Discussion paper series, (April 1996), Institute of Policy and Planning Sciences, University of Tsukuba, No. 678
[8] Mavoungou, T.; Cherruault, Y., Numerical study of Fisher’s equation by Adomian’s method, Mathl. comput. modelling, 19, 1, 89-95, (1994) · Zbl 0799.65099
[9] Jiao, Y.C.; Hao, Y.; Yamamoto, Y., An extension of the decomposition method for solving nonlinear equations and its convergence, Discussion paper series, (April 1996), Institute of Policy and Planning Sciences, University of Tsukuba, No. 677
[10] Venkatarangan, S.N.; Rajalakshmi, K., A modification of Adomian’s solution for nonlinear oscillatory systems, Computers math. applic., 29, 6, 67-73, (1995) · Zbl 0818.34006
[11] Baker, G.A., Essentials of Padé approximants, (1975), Academic Press
[12] Baker, G.A.; Graves-Morris, P., Padé approximants, part I: basic theory, (1981), Addison-Wesley · Zbl 0603.30044
[13] Cherruault, Y.; Adomian, G.; Abbaoui, K.; Rach, R., Further remarks on convergence of decomposition method, International journal of bio-medical computing, 38, 89-93, (1995)
[14] Abbaoui, K.; Cherruault, Y., Convergence of Adomian’s method applied to differential equations, Computers math. applic., 28, 5, 103-109, (1994) · Zbl 0809.65073
[15] Churchhill, R.V., ()
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