A Numerical Study for Nonlinear Time-Space Fractional Reaction-Diffusion Model of Fourth-Order

Abstract

In this article, we discuss the fractional temporal-spatial reaction-diffusion model with Neumann boundary conditions in one- and two-dimensional cases. The problem is solved by using a novel approach that depends on the approximation of a variable-order (VO) Caputo fractional derivative in the form of an operational matrix based on the shifted Vieta-Fibonacci (SVF) and collocation procedures. In this proposed scheme, first, the shifted Vieta-Fibonacci and operational matrix are used to approximate the dependent variable and Caputo derivatives of variable order, respectively, to construct the residual connected with the proposed problem. After that, the residual is collocated at some points of the domain, which produces a system of algebraic equations, and this system is solved by an appropriate numerical technique. The convergence and error analysis of the scheme are also analyzed. In this article, we also analyze the order of convergence for the solution of the considered problem. For validation purposes, the proposed scheme is applied to some particular cases of the proposed model, and the comparisons are made with the exact solution. It is found that the scheme is sufficiently accurate, and the accuracy enhances as the degree of approximating polynomials improves.