A numerical approach based on Vieta–Fibonacci polynomials to solve fractional order advection–reaction diffusion problem

Abstract

In this article, we attempt to provide the numerical solution for a non-linear reaction--advection diffusion equation with fractional-order space-time derivatives in a finite domain. In the proposed scheme, time fractional derivative in Caputo sense is approximated by using the non-standard finite difference method and the fractional space derivative is specifically approximated by using Vieta–Fibonacci polynomials. These approximations generate a system of ordinary differential equations which is converted into an equivalent system of algebraic equations by using collocation method. Finally, the obtained system of algebraic equations is solved to find the dependent variables (unknowns) of the considered problem. The stability and convergence related to the time discreatization of this approach are also discussed. In this study, the effectiveness and precision of the proposed scheme are analyzed with the help of examples, and it is observed that the proposed scheme is sufficiently accurate and efficient technique. Also, the effects of fractional-order derivatives on concentration profiles are discussed. © The Author(s), under exclusive licence to The Forum D’Analystes 2024.