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 ( α ) ( t ) = f ( t , x ( t ) ) , x ( 0 ) = x 0 , where x ( α ) is the derivative of x of order α in the sense of Caputo and 0 < α < 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.