## Detailed error analysis for a fractional adams method with graded meshes

dc.contributor.author | Liu, Yanzhi | * |

dc.contributor.author | Roberts, Jason A. | * |

dc.contributor.author | Yan, Yubin | * |

dc.date.accessioned | 2017-08-10T10:47:06Z | |

dc.date.available | 2017-08-10T10:47:06Z | |

dc.date.issued | 2017-09-21 | |

dc.identifier.citation | Yanzhi, L., Roberts, J., & Yan, Y. (2018). Detailed error analysis for a fractional Adams method with graded meshes. Numerical Algorithms, 78(4), 1195-1216. https://doi.org/10.1007/s11075-017-0419-5 | en |

dc.identifier.issn | 1572-9265 | |

dc.identifier.doi | 10.1007/s11075-017-0419-5 | |

dc.identifier.uri | http://hdl.handle.net/10034/620594 | |

dc.description | The final publication is available at Springer via http://dx.doi.org/10.1007/s11075-017-0419-5 | |

dc.description.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. | |

dc.language.iso | en | en |

dc.publisher | Springer | en |

dc.relation.url | https://link.springer.com/article/10.1007/s11075-017-0419-5 | en |

dc.rights.uri | http://creativecommons.org/licenses/by/4.0/ | en |

dc.subject | Fractional differential equations | en |

dc.subject | Caputo derivative | en |

dc.subject | Adams method | en |

dc.title | Detailed error analysis for a fractional adams method with graded meshes | en |

dc.type | Article | en |

dc.contributor.department | Lvliang University; University of Chester; | en |

dc.identifier.journal | Numerical Algorithms | |

dc.date.accepted | 2017-07-21 | |

or.grant.openaccess | Yes | en |

rioxxterms.funder | Unfunded | en |

rioxxterms.identifier.project | Unfunded | en |

rioxxterms.version | AM | en |

rioxxterms.licenseref.startdate | 2018-09-21 | |

html.description.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 <m$, the Caputo fractional derivative $\, ^{C}_{0}D^{\alpha}_{t} y(t) $ takes the form \lq \lq $\, ^{C}_{0}D^{\alpha}_{t} y(t) = c t^{\lceil \alpha \rceil -\alpha} + \mbox{\emph{smoother terms}}$\rq\rq \cite[Theorem 2.2]{dieforfre}, which implies that $\, ^{C}_{0}D^{\alpha}_{t} y $ behaves as $t^{\lceil \alpha \rceil -\alpha}$ which is not in $ C^{2}[0, T]$. By using the graded meshes $t_{n}= T (n/N)^{r}, n=0, 1, 2, \dots, N$ 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 $t^{\sigma}, 0< \sigma <1$. Numerical examples are given to show that the numerical results are consistent with the theoretical results. | |

rioxxterms.publicationdate | 2017-09-21 |