Pitfalls in fast numerical solvers for fractional differential equations

We consider the problem of implementing fast algorithms for the numerical solution of initial value problems of the form x((alpha)) (t) = f (t, x (t)), x (0) = x(0), where x((alpha)) is the derivative of x of order alpha in the sense of Caputo and 0 < alpha < 1. We review some of the existing methods and explain their respective strengths and weaknesses. We identify and discuss potential problems in the development of generally applicable schemes. (c) 2005 Elsevier B.V. All rights reserved.