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(α)(t)=f(t,x(t)), x(0)=x0x(0)=x0, where x(α)x(α) is the derivative of x   of order αα in the sense of Caputo and 0<α<10<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.