A novel numerical method for solving a two-dimensional variable-order time fractional advection-diffusion problem

Abstract

The convection-diffusion equation describes the transport of physical quantities such as particles, energy, and pollutants by incorporating both diffusion and convection phenomena. In this study, a numerical method is proposed to approximate solutions to the two-dimensional variable-order time fractional advection-diffusion equation (VO-TFADE). The method combines Taylor's functions and two-dimensional Laguerre polynomials to approximate the temporal and spatial components. Caputo's fractional derivative definition is applied, with the fractional derivatives being approximated using an operational matrix of differentiation. The problem is then transformed into a system of algebraic equations. Additionally, the error bound for the proposed method is provided. Numerical examples validate the accuracy and efficiency of the approach.