Numerical comparison of methods for solving parabolic equations. (English) Zbl 1061.65098

Summary: We apply a new decomposition scheme to solve the linear heat equation. The approach is based on the choice of a suitable differential operator which may be ordinary or partial, linear or nonlinear, deterministic or stochastic. It does not require discretization and consequently of massive computation. In this scheme the solution is performed in the form of a convergent power series with easily computable components. This paper is particularly concerned with the Adomian decomposition method and the results obtained are compared to those obtained by a conventional finite-difference method and the Sinc method. The numerical results demonstrate that the new method is relatively accurate and easily implemented.


65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35K05 Heat equation
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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[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 Academic Publishers Boston, MA · Zbl 0802.65122
[3] Burden, R.L.; Faires, J.D., Numerical analysis, (1993), PWS Publishing Company Boston · Zbl 0788.65001
[4] Bellomo, N.; Sarafyan, D., On Adomian’s decomposition method and some comparisons with Picard’s iterative scheme, J. math. anal. appl, 123, 389-400, (1987) · Zbl 0624.60079
[5] 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
[6] Gerald, C.F.; Wheatley, P.O., Applied numerical analysis, (1994), Addison-Wesley Publishing Company California · Zbl 0877.65003
[7] Kaya, D., On the solution of a korteweg – de Vries like equation by the decomposition method, Int. J. comput. math, 72, 531-539, (1999) · Zbl 0948.65104
[8] Kaya, D., Explicit solution of a generalized nonlinear Boussinesq equation, J. appl. math, 1, 29-37, (2001) · Zbl 0976.35066
[9] Kaya, D.; Aassila, M., An application for a generalized KdV equation by decomposition method, Phys. lett. A, 299, 201-206, (2002) · Zbl 0996.35061
[10] D. Kaya, An explicit and numerical solutions of some-fifth-order KdV equation by decomposition method, Appl. Math. Comp., in press · Zbl 1024.65096
[11] Lund, J.; Bowers, K.L., Sinc methods for quadrature and differential equations, (1992), SIAM Philadelphia · Zbl 0753.65081
[12] Stenger, F., Numerical methods based on sinc and analytic functions, (1993), Springer-Verlag New York · Zbl 0803.65141
[13] Stenger, F., A sinc – galerkin method of solution of boundary value problems, Math. comp, 33, 85-109, (1979) · Zbl 0402.65053
[14] Wazwaz, A.M., The decomposition for approximate solution of the groursat problem, Appl. math. comp, 69, 299-311, (1995) · Zbl 0826.65077
[15] Wazwaz, A.M., A comparison between Adomian’s decomposition methods and Taylor series method in the series solutions, Appl. math. comp, 97, 37-44, (1998) · Zbl 0943.65084
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