Abstract
In this paper, a high-order numerical scheme is proposed for solving the two-dimensional fractional reaction–subdiffusion equation. The method is based on adopting a third-order weighted and shifted Grünwald difference (WSGD) operator to approximate the time Caputo fractional derivative and applying the orthogonal spline collocation (OSC) method to approximate the spatial derivative. Stability and convergence analysis of the proposed method are rigorously proved. Several numerical examples in one variable and in two space variables are presented to validate our theoretical analysis.
Highlights
1 Introduction Fractional equations can be used to describe some physical phenomenon more accurately than the classical integer-order differential equation, one of which fractional reaction– diffusion equations have been researched in recent years in many areas of science and engineering
Huang [17] proposed a numerical algorithm for a two-dimensional fractional reaction subdiffusion equation with τ 1+γ order in time and second order in space
In [32], a temporal second-order fully discrete two-grid finite element (FE) scheme is presented to solve nonlinear fractional Cable equation, in which the spatial direction is approximated by two-grid FE method and the integer and fractional derivatives in time are discretized by second-order two-step backward difference method and second-order weighted and shifted Grünwald difference (WSGD) operator
Summary
Fractional equations can be used to describe some physical phenomenon more accurately than the classical integer-order differential equation, one of which fractional reaction– diffusion equations have been researched in recent years in many areas of science and engineering. Following the idea of the WSGD operator, Wang and Vong [29] used compact finite difference schemes for the modified time sub-diffusion equation with Riemann–Liouville fractional derivative and the temporal Caputo fractional diffusion-wave equation. In [32], a temporal second-order fully discrete two-grid finite element (FE) scheme is presented to solve nonlinear fractional Cable equation, in which the spatial direction is approximated by two-grid FE method and the integer and fractional derivatives in time are discretized by second-order two-step backward difference method and second-order WSGD operator. In [33], Yang discussed a new numerical approximation for the two-dimensional distributed-order time fractional reaction–diffusion equation, which combines the idea of a WSGD operator with the second order in time direction and the orthogonal spline collocation method in the space direction.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
