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 $\alpha$ may be an arbitrary positive number and $ \lceil \alpha \rceil$ denotes the smallest integer no less than $\alpha$ 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. \cite[Theorem 3.2]{dieforfre} introduced a fractional Adams method with the uniform meshes $t_{n}= T (n/N), n=0, 1, 2, \dots, N$ 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 $\alpha \leq 1$. 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 $y \in C^{m} [0, T]$ for some $m \in \mathbb{N}$ and $ 0 < \alpha 1$, we show that the optimal convergence order of this method can be recovered uniformly in $t_{n}$ even if $\, ^{C}_{0}D^{\alpha}_{t} y$ behaves as $t^{\sigma}, 0< \sigma <1$. Numerical examples are given to show that the numerical results are consistent with the theoretical results.