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Zbl 1061.34003
Daftardar-Gejji, Varsha; Jafari, Hossein
Adomian decomposition: a tool for solving a system of fractional differential equations.
(English)
[J] J. Math. Anal. Appl. 301, No. 2, 508-518 (2005). ISSN 0022-247X

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 2000:
*34A12 Initial value problems for ODE
26A33 Fractional derivatives and integrals (real functions)

Keywords: Caputo fractional derivative; System of fractional differential equations; Adomian decomposition; Bagley-Torvik equation; Fractional oscillation equation; Mittag-Leffler function

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