Detailed error analysis for a fractional Adams method with graded meshes

Abstract

We consider a fractional Adams method for solving the nonlinear fractional differential equation 0CDtαy(t)=f(t,y(t)),α>0, equipped with the initial conditions y(k)(0)=y0(k),k=0,1,…,⌈α⌉−1. Here, α may be an arbitrary positive number and ⌈α⌉ denotes the smallest integer no less than α and the differential operator is the Caputo derivative. Under the assumption 0CDtαy∈C2[0,T], Diethelm et al. (Numer. Algor. 36, 31–52, 2004) introduced a fractional Adams method with the uniform meshes tn = T(n/N),n = 0,1,2,…,N and proved that this method has the optimal convergence order uniformly in tn, that is O(N−2) if α > 1 and O(N−1−α) if α ≤ 1. They also showed that if 0CDtαy(t)∉C2[0,T], the optimal convergence order of this method cannot be obtained with the uniform meshes. However, it is well-known that for y ∈ Cm[0,T] for some m∈ ℕ and 0 < α 1, we show that the optimal convergence order of this method can be recovered uniformly in tn even if 0CDtαy behaves as tσ,0 < σ < 1. Numerical examples are given to show that the numerical results are consistent with the theoretical results.