Abstract
This paper proposes the computational approach for fractional-in-space reaction-diffusion equation, which is obtained by replacing the space second-order derivative in classical reaction-diffusion equation with the Riesz fractional derivative of order \begin{document}$ α $\end{document} in \begin{document}$ (0, 2] $\end{document} . The proposed numerical scheme for space fractional reaction-diffusion equations is based on the finite difference and Fourier spectral approximation methods. The paper utilizes a range of higher-order time stepping solvers which exhibit third-order accuracy in the time domain and spectral accuracy in the spatial domain to solve some fractional-in-space reaction-diffusion equations. The numerical experiment shows that the third-order ETD3RK scheme outshines its third-order counterparts, taking into account the computational time and accuracy. Applicability of the proposed methods is further tested with a higher dimensional system. Numerical simulation results show that pattern formation process in the classical sense is the same as in fractional scenarios.
Highlights
The concept of fractional differential equations has been considered as a useful tool in the modeling of many physical phenomena
We examine the solution of a fractional reaction-diffusion system of the form (1) in which the space fractional derivative is replaced by the Riesz derivative xDθα, and the resulting systems will be advanced in time with the family of third order schemes based on implicit-explicit and the exponential time differencing methods
It should be mentioned that among the Implicit-explicit Runge-Kutta (IMEXRK) schemes developed for (33), for instance, Ascher et al [8] constructed a family of L-stable two,three-stages diagonally implicit Runge-Kutta (DIRK) and four-stage, third-order combination schemes whose formulations were based on implicit-explicit Runge-Kutta methods for solving convection-diffusion problem
Summary
The concept of fractional differential equations has been considered as a useful tool in the modeling of many physical phenomena. ETD method, finite difference, implicit-explicit, fractional nonlinear PDEs, numerical simulations, Riesz derivative. We examine the solution of a fractional reaction-diffusion system of the form (1) in which the space fractional derivative is replaced by the Riesz derivative xDθα, and the resulting systems will be advanced in time with the family of third order schemes based on implicit-explicit and the exponential time differencing methods. Having done the discritization of the Riesz derivative above, we discuss the finite difference method (FDM) for the space fractional reaction-diffusion equation (1). It should be mentioned that among the IMEXRK schemes developed for (33), for instance, Ascher et al [8] constructed a family of L-stable two-,three-stages diagonally implicit Runge-Kutta (DIRK) and four-stage, third-order combination schemes whose formulations were based on implicit-explicit Runge-Kutta methods for solving convection-diffusion problem. Due to the remarkable performance displayed by the ETD3RK method when applied to solve one- and two-dimensional problems over other methods considered in this paper, the remainder experiments will be carried out with only the ETD3RK
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