Multi-order fractional differential equations and their numerical solution

We consider the numerical solution of (possibly nonlinear) fractional differential equations of the form y((alpha))(t) = f(t,y(t),y((beta1))(t),y((beta2))(t), ...,y((betan))(t)) with alpha > beta(n) > beta(n-1) > ... > beta(1) and alpha - beta(n) less than or equal to 1, beta(j) - beta(j-1) less than or equal to 1, beta < beta(1) less than or equal to 1, combined with suitable initial conditions. The derivatives are understood in the Caputo sense. We begin by discussing the analytical questions of existence and uniqueness of solutions, and we investigate how the solutions depend on the given data. Moreover we propose convergent and stable numerical methods for such initial value problems. (C) 2003 Elsevier Inc. All rights reserved.