Analysis of a system of fractional differential equations

Abstract

We prove existence and uniqueness theorems for the initial value problem for the system of fractional differential equations Dα[x̄(t) - x̄(0)] = Ax̄(t), x̄(0) = x̄0, where Dα denotes standard Riemann-Liouville fractional derivative, 0 < α < 1, x̄(t) = [x1(t),..., xn(t)]t and A is a square matrix. The unique solution to this initial value problem turns out to be Eα(tα A)x̄ 0, where Eα denotes the Mittag-Leffler function generalized for matrix arguments. Further we analyze the system Dα[x̄(t) -x̄(0)] = f̄(t, x̄), x̄(0) = x̄0, 0 < α < 1, and investigate dependence of the solutions on the initial conditions. © 2004 Elsevier Inc. All rights reserved.