Detailed error analysis for a fractional Adams method with graded meshes

We consider a fractional Adams method for solving the nonlinear fractional differential equation , equipped with the initial conditions . Here, alpha may be an arbitrary positive number and aOE alpha aOEe denotes the smallest integer no less than alpha and the differential operator is the Caputo derivative. Under the assumption , Diethelm et al. (Numer. Algor. 36, 31-52, 2004) introduced a fractional Adams method with the uniform meshes t (n) = T(n/N),n = 0,1,2,aEuro broken vertical bar,N and proved that this method has the optimal convergence order uniformly in t (n) , that is O(N (-2)) if alpha > 1 and O(N (-1-alpha) ) if alpha a 1. They also showed that if , the optimal convergence order of this method cannot be obtained with the uniform meshes. However, it is well-known that for y a C (m) [0,T] for some and 0 < alpha < m, the Caputo fractional derivative takes the form "" (Diethelm et al. Numer. Algor. 36, 31-52, 2004), which implies that behaves as t (aOE alpha aOEe-alpha) which is not in C (2)[0,T]. By using the graded meshes t (n) = T(n/N) (r) ,n = 0,1,2,aEuro broken vertical bar,N with some suitable r > 1, we show that the optimal convergence order of this method can be recovered uniformly in t (n) even if behaves as t (sigma) ,0 < sigma < 1. Numerical examples are given to show that the numerical results are consistent with the theoretical results.