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 ,^{C}_{0}D^{alpha }_{t} y(t) = f(t, y(t)), , alpha >0, equipped with the initial conditions y^{(k)} (0) = y_{0}^{(k)}, k=0, 1, dots , lceil alpha rceil -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 ,^{C}_{0}D^{alpha }_{t} y in C^{2}[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 ,^{C}_{0}D^{alpha }_{t} y(t) notin C^{2}[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 in mathbb {N} and 0 < α < m, the Caputo fractional derivative ,^{C}_{0}D^{alpha }_{t} y(t) takes the form “,^{C}_{0}D^{alpha }_{t} y(t) = c t^{lceil alpha rceil -alpha } + text {smoother terms}” (Diethelm et al. Numer. Algor. 36, 31–52, 2004), which implies that ,^{C}_{0}D^{alpha }_{t} y behaves as t⌈α⌉−α which is not in C2[0,T]. By using the graded meshes tn = T(n/N)r,n = 0,1,2,…,N with some suitable r > 1, we show that the optimal convergence order of this method can be recovered uniformly in tn even if ,^{C}_{0}D^{alpha }_{t} y behaves as tσ,0 < σ < 1. Numerical examples are given to show that the numerical results are consistent with the theoretical results.