ChesterRep Collection:
http://hdl.handle.net/10034/6981
Wed, 20 Sep 2017 10:42:24 GMT2017-09-20T10:42:24ZAn approach to construct higher order time discretisation schemes for time fractional partial differential equations with nonsmooth data
http://hdl.handle.net/10034/620610
Title: An approach to construct higher order time discretisation schemes for time fractional partial differential equations with nonsmooth data
Authors: Ford, Neville J.; Yan, Yubin
Abstract: In this paper, we shall review an approach by which we can seek higher order time discretisation schemes for solving time fractional partial differential equations with nonsmooth data. The low regularity of the solutions of time fractional partial differential equations implies standard time discretisation schemes only yield first order accuracy. To obtain higher order time discretisation schemes when the solutions of time fractional partial differential equations have low regularities, one may correct the starting steps of the standard time discretisation schemes to capture the singularities of the solutions. We will consider these corrections of some higher order time discretisation schemes obtained by using Lubich's fractional multistep methods, L1 scheme and its modification, discontinuous Galerkin methods, etc. Numerical examples are given to show that the theoretical results are consistent with the numerical results.
Description: Invited review article for Anniversary Edition of Journal.Fri, 01 Sep 2017 00:00:00 GMThttp://hdl.handle.net/10034/6206102017-09-01T00:00:00ZA note on finite difference methods for nonlinear fractional differential equations with non-uniform meshes
http://hdl.handle.net/10034/620595
Title: A note on finite difference methods for nonlinear fractional differential equations with non-uniform meshes
Authors: Yanzhi, Liu; Roberts, Jason; Yan, Yubin
Abstract: We consider finite difference methods for solving nonlinear fractional differential equations in the Caputo fractional derivative sense with non-uniform meshes. Under the assumption that the Caputo derivative of the solution of the fractional differential equation is suitably smooth, Li et al. \lq \lq Finite difference methods with non-uniform meshes for nonlinear fractional differential equations\rq\rq, Journal of Computational Physics, 316(2016), 614-631, obtained the error estimates of finite difference methods with non-uniform meshes. However the Caputo derivative of the solution of the fractional differential equation in general has a weak singularity near the initial time. In this paper, we obtain the error estimates of finite difference methods with non-uniform meshes when the Caputo fractional derivative of the solution of the fractional differential equation has lower smoothness. The convergence result shows clearly how the regularity of the Caputo fractional derivative of the solution affect the order of convergence of the finite difference methods. Numerical results are presented that confirm the sharpness of the error analysis.Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10034/6205952017-01-01T00:00:00ZDetailed error analysis for a fractional adams method with graded meshes
http://hdl.handle.net/10034/620594
Title: Detailed error analysis for a fractional adams method with graded meshes
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.Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10034/6205942017-01-01T00:00:00ZA Note on the Well-Posedness of Terminal Value Problems for Fractional Differential Equations.
http://hdl.handle.net/10034/620550
Title: A Note on the Well-Posedness of Terminal Value Problems for Fractional Differential Equations.
Authors: Diethelm, Kai; Ford, Neville J.
Abstract: This note is intended to clarify some im-
portant points about the well-posedness of terminal value
problems for fractional di erential equations. It follows the
recent publication of a paper by Cong and Tuan in this jour-
nal in which a counter-example calls into question the earlier
results in a paper by this note's authors. Here, we show in
the light of these new insights that a wide class of terminal
value problems of fractional differential equations is well-
posed and we identify those cases where the well-posedness
question must be regarded as open.Sat, 17 Jun 2017 00:00:00 GMThttp://hdl.handle.net/10034/6205502017-06-17T00:00:00Z