Adomian decomposition method for non-smooth initial value problems. (English) Zbl 1235.65105

Wu, Guo-Cheng

Math. Comput. Modelling 54, No. 9-10, 2104-2108 (2011).

Summary: The Adomian decomposition method is extended to the calculations of the non-differentiable functions. The iteration procedure is based on Jumarie’s Taylor series. A specific fractional differential equation is used to elucidate the solution procedure and the results are compared with the exact solution of the corresponding ordinary differential equations, revealing high accuracy and efficiency.

MSC:

65L99	Numerical methods for ordinary differential equations
34A08	Fractional ordinary differential equations
34A25	Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
45J05	Integro-ordinary differential equations

Keywords:

modified Riemann-Liouville derivative; Adomian decomposition method; non-smooth functions

References:

[1]	Adomian, G., Solving Frontier Problems of Physics: The Decomposition Method (1994), Kluwer Academic Publishers: Kluwer Academic Publishers Boston · Zbl 0802.65122
[2]	He, J. H., Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput. Methods Appl. Mech. Engrg., 167, 57-68 (1998) · Zbl 0942.76077
[3]	Ray, S. S.; Bera, R. K., Analytical solution of a fractional diffusion equation by Adomian decomposition method, Appl. Math. Comput., 174, 329-336 (2006) · Zbl 1089.65108
[4]	Jafari, H.; Daftardar-Gejji, V., Solving a system of nonlinear fractional differential equations using Adomian decomposition, J. Comput. Appl. Math., 196, 644-651 (2006) · Zbl 1099.65137
[5]	Wang, Q., Numerical solutions for fractional KdV-Burgers equation by Adomian decomposition method, Appl. Math. Comput., 182, 1048-1055 (2006) · Zbl 1107.65124
[6]	Li, C. P.; Wang, Y. H., Numerical algorithm based on Adomian decomposition for fractional differential equations, Comput. Math. Appl., 57, 1672-1681 (2009) · Zbl 1186.65110
[7]	Momani, S.; Shawagfeh, N., Decomposition method for solving fractional Riccati differential equations, Appl. Math. Comput., 182, 1083-1092 (2006) · Zbl 1107.65121
[8]	Wang, Q., Homotopy perturbation method for fractional KdV-Burgers equation, Chaos Solitons Fractals, 35, 843-850 (2008) · Zbl 1132.65118
[9]	Jumarie, G., New stochastic fractional models for Malthusian growth, the Poissonian birth process and optimal management of populations, Math. Comput. Modelling, 44, 231-254 (2006) · Zbl 1130.92043
[10]	Jumarie, G., Laplaces transform of fractional order via the Mittag-Leffler function and modified Riemann-Liouville derivative, Appl. Math. Lett., 22, 1659-1664 (2009) · Zbl 1181.44001
[11]	Odzijewicz, T.; Torres, D. F.M., Fractional calculus of variations for double integrals, Balkan J. Geom. Appl., 16, 2, 102-113 (2011) · Zbl 1221.49041
[13]	Jumarie, G., Modified Riemann-Liouville derivative and fractional Taylor series of non-differentiable functions Further results, Comput. Math. Appl., 51, 1367-1376 (2006) · Zbl 1137.65001
[14]	Adomian, G.; Rach, R., Inversion of nonlinear stochastic operators, J. Math. Anal. Appl., 91, 39-46 (1983) · Zbl 0504.60066
[15]	Rach, R., A convenient computational form for the Adomian polynomials, J. Math. Anal. Appl., 102, 415-419 (1984) · Zbl 0552.60061
[16]	Duan, J. S., Recurrence triangle for Adomian polynomials, Appl. Math. Comput., 216, 1235-1241 (2010) · Zbl 1190.65031
[17]	Duan, J. S., An efficient algorithm for the multivariable Adomian polynomials, Appl. Math. Comput., 217, 2456-2467 (2010) · Zbl 1204.65022
[18]	Shawagfeh, N. T., Analytical approximate solutions for nonlinear fractional differential equations, Appl. Math. Comput., 131, 517-529 (2002) · Zbl 1029.34003
[19]	Carpinter, A.; Sapora, A., Diffusion problems in fractal media defined on Cantor Sets, ZAMM Z. Angew. Math. Mech., 90, 3, 203-210 (2010) · Zbl 1187.80011
[20]	Kolwankar, K. M.; Gangal, A. D., Fractional differentiability of nowhere differentiable functions and dimensions, Chaos, 6, 505-513 (1996) · Zbl 1055.26504