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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 α 1 y 1 ,,D α n y n ] t =A(y 1 ,,y n ) t ,y i (0)=c i ,i=1,,n,

where A=[a ij ] is a real square matrix, the solution turns out to be

y ¯(x)= (α 1 ,,α n ),1 (x α 1 A 1 ,,x α n A n )y ¯(0),

where (α 1 ,,α n ),1 denotes the multivariate Mittag-Leffler function defined for matrix arguments and A i is the matrix having ith row as [a i 1a in ], and all other entries are zero. Fractional oscillation and Bagley-Torvik equations are solved as illustrative examples.

34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions
26A33Fractional derivatives and integrals (real functions)