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

Hdl Handle:
http://hdl.handle.net/10034/620594
Title:
Authors:
Liu, Yanzhi; Roberts, Jason; Yan, Yubin
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.
Affiliation:
Lvliang University; University of Chester;
Citation:
Yanzhi, L., Roberts, J., & Yan, Y. (2017). Detailed error analysis for a fractional Adams method with graded meshes. Numerical Algorithms. https://doi.org/10.1007/s11075-017-0419-5
Publisher:
Springer
Journal:
Numerical Algorithms
Publication Date:
21-Sep-2017
URI:
http://hdl.handle.net/10034/620594
DOI:
10.1007/s11075-017-0419-5
Type:
Article
Language:
en
Description:
The final publication is available at Springer via http://dx.doi.org/10.1007/s11075-017-0419-5
ISSN:
1572-9265
Appears in Collections:
Mathematics

DC FieldValue Language
dc.contributor.authorLiu, Yanzhien
dc.contributor.authorRoberts, Jasonen
dc.contributor.authorYan, Yubinen
dc.date.accessioned2017-08-10T10:47:06Z-
dc.date.available2017-08-10T10:47:06Z-
dc.date.issued2017-09-21-
dc.identifier.citationYanzhi, L., Roberts, J., & Yan, Y. (2017). Detailed error analysis for a fractional Adams method with graded meshes. Numerical Algorithms. https://doi.org/10.1007/s11075-017-0419-5en
dc.identifier.issn1572-9265-
dc.identifier.doi10.1007/s11075-017-0419-5-
dc.identifier.urihttp://hdl.handle.net/10034/620594-
dc.descriptionThe final publication is available at Springer via http://dx.doi.org/10.1007/s11075-017-0419-5-
dc.description.abstractWe 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.en
dc.language.isoenen
dc.publisherSpringeren
dc.subjectFractional differential equationsen
dc.subjectCaputo derivativeen