Adomian decomposition: a tool for solving a system of fractional differential equations

Abstract

Adomian decomposition method has been employed to obtain solutions of a system of fractional differential equations. Convergence of the method has been 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 i j ] is a real square matrix, the solution turns out to be y ¯ ( x ) = E ( α 1 , … , α n ) , 1 ( x α 1 A 1 , … , x α n A n ) y ¯ ( 0 ) , where E ( α 1 , … , α n ) , 1 denotes multivariate Mittag-Leffler function defined for matrix arguments and A i is the matrix having ith row as [ a i 1 … a i n ] , and all other entries are zero. Fractional oscillation and Bagley–Torvik equations are solved as illustrative examples.