## 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 $\overline y(x)={\mathcal E}_{(\alpha_1,\dots, \alpha_n),1}(x^{\alpha_1}A_1,\dots, x^{\alpha_n}A_n) \overline y(0),$ 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
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### References:

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