A numerical approach to solve 2D fractional RADE of variable-order with Vieta–Lucas polynomials

Abstract

Here, we present a novel numerical algorithm based on the operational matrix and collocation approach for solving the 2D non-linear fractional reaction–advection diffusion equations (RADE) of variable order that is related to the groundwater pollution problem. In this article, the operational matrices of variable-order derivatives are derived with the aid of Vieta–Lucas polynomials for 2D problem, which is used in the proposed algorithm. To find a solution to the considered problem, the terms are approximated by a series of triple-shifted Vieta–Lucas polynomials to construct the residual of the RADE. Then the collocation technique is applied to transform the problem into algebraic equations, which are solved by Newton’s method. Moreover, the convergency and upper bound of the derived error formula for the approximate solution are discussed. Finally, some examples are presented to show the efficiency of the considered scheme, and the results are shown by using graphical presentation and tabular representations.