Some second-order \(\theta\) schemes combined with finite element method for nonlinear fractional cable equation
type: Item
instance of: scholarly article
title: Some second-order \(\theta\) schemes combined with finite element method for nonlinear fractional cable equation
zbMATH Open document ID: 1433.65218
author: Yang Liu
author: Hong Li
author: Fawang Liu
author: Yajun Wang
author: Yanwei Du
published in: Numerical Algorithms
publication date: 2019-02-07T00:00:00Z
full work available at URL: P9 NUMA Y18 LDLLW.pdf
review text: The authors investigate numerical methods for solving fractional partial differential equations. Specifically, they develop a second-order numerical method for the fractional cable equation \[ \frac{\partial u}{\partial t}+\mathcal K D^\alpha_tu-D^\beta_t\Delta u+g(u)=f, \] where \(\mathcal K\) is a constant and \(D^\alpha_t\) denotes the Riemann-Liouville fractional derivative with respect to time. The authors approximate the fractional time derivatives using the weighted and shifted Grünwald difference (WSGD) operators developed in [\textit{W. Tian} et al., Math. Comput. 84, No. 294, 1703--1727 (2015; Zbl 1318.65058)]. For the space discretization, they use standard finite element methods. For the integer time derivative \(\frac{\partial u}{\partial t}\), the authors use the \(\theta\) method, a generalization of the backwards difference method and the Crank-Nicolson method. Combining these three ingredients, the authors develop their new numerical method and prove stability and error estimates. The authors illustrate their method with numerical examples in one and two dimensions, verifying the rates of convergence.
reviewed by: Yakov Berchenko-Kogan
Mathematics Subject Classification ID: 65M60
Mathematics Subject Classification ID: 65N15
Mathematics Subject Classification ID: 65N30
Mathematics Subject Classification ID: 35R11
Mathematics Subject Classification ID: 65N12
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 92C37
Mathematics Subject Classification ID: 92C20
zbMATH DE Number: 7015337
zbMATH Keywords: second-order \(\theta\) scheme
zbMATH Keywords: nonlinear fractional cable equation
zbMATH Keywords: finite element algorithm
zbMATH Keywords: stability
zbMATH Keywords: error estimates
zbMATH Keywords: fractional partial differential equation
zbMATH Keywords: Riemann-Liouville derivative
MaRDI profile type: Publication
OpenAlex ID: W2789967849
cites work: Superconvergence analysis of nonconforming finite element method for two-dimensional time fractional diffusion equations
cites work: A new fractional finite volume method for solving the fractional diffusion equation
cites work: A Petrov-Galerkin finite element method for variable-coefficient fractional diffusion equations
cites work: A finite difference method with non-uniform timesteps for fractional diffusion equations
cites work: Finite difference approximations for fractional advection-dispersion flow equations
cites work: Developing Finite Element Methods for Maxwell's Equations in a Cole–Cole Dispersive Medium
cites work: Galerkin finite element approximation of symmetric space-fractional partial differential equations
cites work: Moving finite element methods for time fractional partial differential equations
cites work: Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion
cites work: Finite difference/finite element method for a nonlinear time-fractional fourth-order reaction-diffusion problem
cites work: A two-grid mixed finite element method for a nonlinear fourth-order reaction-diffusion problem with time-fractional derivative
cites work: Finite element method for space-time fractional diffusion equation
cites work: The Galerkin finite element method for a multi-term time-fractional diffusion equation
cites work: A class of second order difference approximations for solving space fractional diffusion equations
cites work: Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation
cites work: Finite difference/finite element method for two-dimensional space and time fractional Bloch-Torrey equations
cites work: The Use of Finite Difference/Element Approaches for Solving the Time-Fractional Subdiffusion Equation
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cites work: Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation
cites work: Galerkin finite element method and error analysis for the fractional cable equation
cites work: A new fully discrete finite difference/element approximation for fractional cable equation
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cites work: Local discontinuous Galerkin method for a nonlinear time-fractional fourth-order partial differential equation
cites work: Some temporal second order difference schemes for fractional wave equations
cites work: A new difference scheme for the time fractional diffusion equation
cites work: Stability and convergence of finite difference schemes for a class of time-fractional sub-diffusion equations based on certain superconvergence
cites work: High-order local discontinuous Galerkin method combined with WSGD-approximation for a fractional subdiffusion equation
cites work: A high-order compact finite difference scheme for the fractional sub-diffusion equation
cites work: Galerkin finite element method for nonlinear fractional Schrödinger equations
DOI: 10.1007/S11075-018-0496-0
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description: scientific article; zbMATH DE number 7015337