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Adomian decomposition: a tool for solving a system of fractional differential equations. (English) Zbl 1061.34003
Summary: Adomian’s decomposition method is employed to obtain solutions of a system of fractional differential equations. The convergence of the method is discussed with some illustrative examples. In particular, for the initial value problem $$[D^{\alpha_1} y_1,\dots,D^{\alpha_n}y_n]^t =A(y_1, \dots,y_n)^t, \quad y_i(0)=c_i, \quad i=1,\dots,n,$$ where $A=[a_{ij}]$ is a real square matrix, the solution turns out to be $$\overline y(x)={\cal E}_{(\alpha_1,\dots, \alpha_n),1}(x^{\alpha_1}A_1,\dots, x^{\alpha_n}A_n) \overline y(0),$$ where ${\cal E}_{(\alpha_1,\dots,\alpha_n),1}$ denotes the multivariate Mittag-Leffler function defined for matrix arguments and $A_i$ is the matrix having $i$th row as $[a_i1\dots a_{in}]$, and all other entries are zero. Fractional oscillation and Bagley-Torvik equations are solved as illustrative examples.

##### MSC:
 34A12 Initial value problems for ODE, existence, uniqueness, etc. of solutions 26A33 Fractional derivatives and integrals (real functions)
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##### References:
 [1] Abboui, K.; Cherruault, Y.: New ideas for proving convergence of decomposition methods. Comput. appl. Math. 29, 103-105 (1995) · Zbl 0832.47051 [2] Adomian, G.: Solving frontier problems of physics: the decomposition method. (1994) · Zbl 0802.65122 [3] Atanackovic, T. M.; Stankovic, B.: On a system of differential equations with fractional derivatives arising in rod theory. J. phys. A 37, 1241-1250 (2004) · Zbl 1059.35011 [4] Babolian, E.; Biazar, J.; Islam, R.: Solution of the system of ordinary differential equations by Adomian decomposition method. Appl. math. Comput. 147, 713-719 (2004) · Zbl 1034.65053 [5] Chechkin, A. V.; Gorenflo, R.; Sokolov, I. M.; Gonchar, V. Yu.: Distributed order time fractional diffusion equation. Frac. calc. Appl. anal. 6, 259-279 (2003) · Zbl 1089.60046 [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] 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 [8] 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 [9] Luchko, Yu.; Gorenflo, R.: An operational method for solving fractional differential equations with the Caputo derivatives. Acta math. Vietnam 24, 207-233 (1999) · Zbl 0931.44003 [10] Miller, K. S.; Ross, B.: An introduction to the fractional calculus and fractional differential equations. (1993) · Zbl 0789.26002 [11] Podlubny, I.: Fractional differential equations. (1999) · Zbl 0924.34008 [12] Samko, S. G.; Kilbas, A. A.; Marichev, O. I.: Fractional integrals and derivatives: theory and applications. (1993) · Zbl 0818.26003 [13] Shawagfeh, N. T.: Analytical approximate solutions for nonlinear fractional differential equations. Appl. math. Comput. 131, 517-529 (2002) · Zbl 1029.34003 [14] Torvik, P. J.; Bagley, R. L.: On the appearance of the fractional derivative in the behaviour of real materials. J. appl. Mech. 51, 294-298 (1984) · Zbl 1203.74022 [15] West, B. J.; Bologna, M.; Grigolini, P.: Physics of fractal operators. (2003)