Numerical analysis and pattern formation process for space-fractional superdiffusive systems

Kolade M Owolabi

doi:10.3934/dcdss.2019036

Abstract

In this paper, we consider the numerical solution of fractional-in-space reaction-diffusion equation, which is obtained from the classical reaction-diffusion equation by replacing the second-order spatial derivative with a fractional derivative of order \begin{document}$ α∈(1, 2] $\end{document} . We adopt a class of second-order approximations, based on the weighted and shifted Grunwald difference operators in Riemann-Liouville sense to numerically simulate two multicomponent systems with fractional-order in higher dimensions. The efficiency and accuracy of the numerical schemes are justified by reporting the norm infinity and norm relative errors as well as their convergence. The complexity of the dynamics in the equation is theoretically discussed by conducting its local and global stability analysis and Numerical experiments are performed to back-up the theoretical claims.

Highlights

The subject of fractional calculus is as old as classical calculus, which is based on a generalisation of integration and ordinary differentiation of arbitrary non-integer order
Many researchers have shown their interest to study the properties of fractional calculus and provide robust and accurate analytical and numerical techniques for solving fractional differential equations, for example, the differential transform method [51], the finite element and finite difference methods [15,33,34,35,43,66], the spectral collocation and tau methods [12,13,14,18,23,29,59], Chen [19] studied fractional diffusion equations by using the Kansa method, in which the MultiQuadrics and thin plate spline served as the radial basis function
In order to give a good working guidelines on the appropriate choice of parameters for the numerical simulation of full fractional reaction-diffusion system, it is mandatory to consider the local dynamics of the system

Summary

Introduction

The subject of fractional calculus is as old as classical calculus, which is based on a generalisation of integration and ordinary differentiation of arbitrary non-integer order. We consider the time-dependent fractional-in-space reaction-diffusion system using a class of second-order approximations, known as the weighted and shifted Grunwald difference schemes in the sense of Riemann-Liouville fractional operators, subject to the Crank-Nicholson technique for the time discretization. The L-R Riemann-Liouville fractional order derivatives of u(x) at point x can be approximated by the weighted and shifted Grunwald difference operators with second order accuracy [ x−a ]+r φa α gk(α)u(x − (k − r) ). For the discretization in space, we use the weighted and shifted Grunwald difference operators LDα,r,su(x, t) and RDα,r,su(x, t) for the respective approximation of left and right Riemann-Liouvillve fractional derivatives aDxαu(x, t) and xDbαu(x, t), with pair (r, s) = (1, 0) or (1, −1), it shows that δtuni γLDα,r,suni + γLDα,r,suni +1 + βRDα,r,suni + βRDα,r,suni +1. We use the weighted and shifted Grunwald difference operators

Dα x

IMEX scheme called

Conclusion