Solving a system of nonlinear fractional differential equations using Adomian decomposition. (English) Zbl 1099.65137

Summary: The Adomian decomposition metbod is employed to obtain solutions of a system of nonlinear fractional differential equations: \[ D^{\alpha_i} y_i(x)=N_i(x,y_1, \dots,y_n),\quad y_i^{(k)}(0)=c^i_k,\quad 0\leq k\leq [\alpha_i],\;1\leq i\leq n, \] where \(D^{\alpha_i}\) denotes the Coputo fractional derivative. Some examples are solved as illustrations, using symbolic computation.


65R20 Numerical methods for integral equations
45J05 Integro-ordinary differential equations
26A33 Fractional derivatives and integrals
68W30 Symbolic computation and algebraic computation
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[1] Abboui, K.; Cherruault, Y., New ideas for proving convergence of decomposition methods, Comput. Appl. Math., 29, 7, 103-105 (1995) · Zbl 0832.47051
[2] Adomian, G., Solving Frontier Problems of Physics: The Decomposition Method (1994), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0802.65122
[3] Biazar, J.; Babolian, E.; Islam, R., Solution of the system of Volterra integral equations of the first kind by Adomian decomposition method, Appl. Math. Comput., 139, 249-258 (2003) · Zbl 1027.65180
[4] Biazar, J.; Babolian, E.; Islam, R., Solution of the system of ordinary differential equations by Adomian decomposition method, Appl. Math. Comput., 147, 3, 713-719 (2004) · Zbl 1034.65053
[5] Choi, H. W.; Shin, J. G., Symbolic implementation of the algorithm for calculating Adomian polynomials, Appl. Math. Comput., 146, 257-271 (2003) · Zbl 1033.65036
[6] Daftardar-Gejji, V.; Babakhani, A., Analysis of a system of fractional differential equations, J. Math. Anal. Appl., 293, 511-522 (2004) · Zbl 1058.34002
[7] Daftardar-Gejji, V.; Jafari, H., Adomian decomposition: a tool for solving a system of fractional differential equations, J. Math. Anal. Appl., 301, 2, 508-518 (2005) · Zbl 1061.34003
[8] Diethelm, K., An algorithm for the numerical solution of differential equations of fractional order, Electron. Trans. Numer. Anal., 5, 1-6 (1997) · Zbl 0890.65071
[9] Edwards, J. T.; Ford, N. J.; Simpson, A. C., The numerical solution of linear multi-term fractional differential equations: systems of equations, J. Comput. Appl. Math., 148, 401-418 (2002) · Zbl 1019.65048
[10] Luchko, Y.; Gorenflo, R., An operational method for solving fractional differential equations with the Caputo derivatives, Acta Math Vietnam., 24, 2, 207-233 (1999) · Zbl 0931.44003
[11] Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), Wiley: Wiley New York · Zbl 0789.26002
[12] Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press San Diego · Zbl 0918.34010
[13] Samko, G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives: Theory and Applications (1993), Gordon and Breach: Gordon and Breach Yverdon · Zbl 0818.26003
[14] Shawagfeh, N. T., Analytical approximate solutions for nonlinear fractional differential equations, Appl. Math. Comput., 131, 517-529 (2002) · Zbl 1029.34003
[15] Wazwaz, A. M., A reliable technique for solving the wave equation in infinite one-dimensional medium, Appl. Math. Comput., 92, 1-7 (1998) · Zbl 0942.65107
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