## 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 *t*
_{
n
} = *T*(*n*/*N*),*n* = 0,1,2,…,*N* and proved that this method has the optimal convergence order uniformly in *t*
_{
n
}, 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* ∈ *C*
^{m}[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 *C*
^{2}[0,*T*]. By using the graded meshes *t*
_{
n
} = *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 *t*
_{
n
} even if \(\,^{C}_{0}D^{\alpha }_{t} y\) behaves as *t*
^{σ},0 < *σ* < 1. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

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## Acknowledgments

The work of the first author was carried out during her stay at the University of Chester, which is supported financially by Shanxi province government, P. R. China. She thanks the Department of Mathematics, University of Chester, for its warm hospitality and providing a very good working condition for her during her stay in Chester.

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Liu, Y., Roberts, J. & Yan, Y. Detailed error analysis for a fractional Adams method with graded meshes.
*Numer Algor* **78**, 1195–1216 (2018). https://doi.org/10.1007/s11075-017-0419-5

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DOI: https://doi.org/10.1007/s11075-017-0419-5