High-order numerical methods for solving time fractional partial differential equations
type: Item
instance of: scholarly article
title: High-order numerical methods for solving time fractional partial differential equations
zbMATH Open document ID: 1387.65106
author: Zongqi Liang
author: Y. Yan
author: Zhiqiang Li
published in: Journal of Scientific Computing
publication date: 2018-03-13T00:00:00Z
full work available at URL: http://hdl.handle.net/10034/620273
review text: The authors develop a fully discrete numerical scheme to solve the time fractional partial differential equation \[ \begin{alignedat}{3} _0^C D_{t}^{\alpha } u (x, t) - \Delta u (x, t) &= f(x, t), &\qquad &x \in \Omega, \quad t \in (0, T], \\ u(x, 0) &= u_{0}, &&x \in \Omega, \\ u(x, t) &= q(x, t), &&x \in \partial \Omega, \quad t \in (0, T],\end{alignedat} \] ``where \(\Omega \subset \mathbb{R}^{d}\), \(d=1, 2, 3\) is a convex polygonal/polyhedral domain and \(\Delta\) denotes the Laplacian. Here \(f\) and \(q\) are two given functions and \(_0^C D_{t}^{\alpha } x(t)\), \(0< \alpha <1\) denotes the Caputo fractional order derivative.'' The time discretization is based the Diethelm's method and the space discretization is based on the standard finite element method. The error estimates are obtained with the convergence of second order in space and (2-3) order in time. Some numerical examples are considered to show that the numerical results are consistent with the theoretical results. Several lemmas and theorems are stated and proved.
Mathematics Subject Classification ID: 65M60
Mathematics Subject Classification ID: 65M15
Mathematics Subject Classification ID: 65M70
Mathematics Subject Classification ID: 35R11
zbMATH DE Number: 6849491
zbMATH Keywords: time fractional partial differential equations
zbMATH Keywords: finite element
zbMATH Keywords: error estimates
Wikidata QID: Q59614672
MaRDI profile type: Publication
OpenAlex ID: W2551065309
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DOI: 10.1007/S10915-016-0319-1
description: scientific article; zbMATH DE number 6849491