## Search

Now showing items 1-2 of 2

JavaScript is disabled for your browser. Some features of this site may not work without it.

All of ChesterRepCommunitiesTitleAuthorsPublication DateSubmit DateSubjectsPublisherJournalThis CommunityTitleAuthorsPublication DateSubmit DateSubjectsPublisherJournal

Subjects

Caputo derivative (2)

Adams method (1)Error estimates (1)Finite difference method (1)Fractional differential equation (1)Fractional differential equations (1)View MoreJournalBIT Numerical Mathematics (1)Numerical Algorithms (1)Authors
Yan, Yubin (2)

Ford, Neville J. (1)Liu, Yanzhi (1)Pal, Kamal (1)Roberts, Jason (1)TypesArticle (2)

Now showing items 1-2 of 2

- List view
- Grid view
- Sort Options:
- Relevance
- Title Asc
- Title Desc
- Issue Date Asc
- Issue Date Desc
- Results Per Page:
- 5
- 10
- 20
- 40
- 60
- 80
- 100

Higher order numerical methods for solving fractional differential equations

Yan, Yubin; Pal, Kamal; Ford, Neville J. (Springer, 2013-10-05)

In this paper we introduce higher order numerical methods for solving fractional differential equations. We use two approaches to this problem. The first approach is based on a direct discretisation of the fractional differential operator: we obtain a numerical method for solving a linear fractional differential equation with order 0 < α < 1. The order of convergence of the numerical method is O(h^(3−α)). Our second approach is based on discretisation of the integral form of the fractional differential equation and we obtain a fractional Adams-type method for a nonlinear fractional differential equation of any order α >0. The order of convergence of the numerical method is O(h^3) for α ≥ 1 and O(h^(1+2α)) for 0 < α ≤ 1 for sufficiently smooth solutions. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

Detailed error analysis for a fractional adams method with graded meshes

Liu, Yanzhi; Roberts, Jason; Yan, Yubin (Springer, 2017-09-21)

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.

The export option will allow you to export the current search results of the entered query to a file. Different formats are available for download. To export the items, click on the button corresponding with the preferred download format.

By default, clicking on the export buttons will result in a download of the allowed maximum amount of items.

To select a subset of the search results, click "Selective Export" button and make a selection of the items you want to export. The amount of items that can be exported at once is similarly restricted as the full export.

After making a selection, click one of the export format buttons. The amount of items that will be exported is indicated in the bubble next to export format.