A numerical study for three-dimensional fractional advection–diffusion equation of variable order with modified Atangana–Baleanu–Caputo derivative

Abstract

This paper is concerned about the numerical solution of a three-dimensional fractional advection–diffusion equation with Modified Atangana–Baleanu–Caputo derivative of variable order. The presented numerical scheme is based on the approximations of the generalized Lucas operational matrix and the collocation method. This paper derives the operational matrix of generalized Lucas polynomials for modified Atangana–Baleanu–Caputo fractional derivative of variable order. After utilizing these operational matrices, we collocate the residual, initial, and boundary conditions at certain points of the domain, which yields a system of algebraic equations. By solving this system of equations with the help of Newton’s method, we get an approximate solution to the proposed problem. The stability and convergence for the proposed technique are also presented. The validation of the chosen scheme is demonstrated through the error analyses of a few numerical problems with known solutions, which confirms the accuracy of the proposed numerical scheme for the presented model. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2025.