Adomian decomposition: a tool for solving a system of fractional differential equations. (English) Zbl 1061.34003

Daftardar-Gejji, Varsha; Jafari, Hossein

J. Math. Anal. Appl. 301, No. 2, 508-518 (2005).

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 \[ \overline y(x)={\mathcal E}_{(\alpha_1,\dots, \alpha_n),1}(x^{\alpha_1}A_1,\dots, x^{\alpha_n}A_n) \overline y(0), \] where \({\mathcal E}_{(\alpha_1,\dots,\alpha_n),1}\) 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, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
26A33	Fractional derivatives and integrals

Keywords:

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

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