NUMERICAL TREATMENT OF THE SPACE–TIME FRACTAL–FRACTIONAL MODEL OF NONLINEAR ADVECTION–DIFFUSION–REACTION EQUATION THROUGH THE BERNSTEIN POLYNOMIALS

Abstract

In this paper, the nonlinear space–time fractal–fractional advection–diffusion–reaction equation is introduced and a highly accurate methodology is presented for its numerical solution. In the time direction, the fractal–fractional derivative in the Atangana–Riemann–Liouville concept is utilized whereas the fractional derivatives in the Caputo and Atangana–Baleanu–Caputo senses are mutually used in the space variable to define this new class of problems. The presented method utilizes the Bernstein polynomials (BPs) and their operational matrices of fractional and fractal–fractional derivatives (which are generated in this study). To this end, the unknown solution is expanded by the BP and is replaced in the equation. Then, the generated operational matrices and the collocation method are employed to generate a system of algebraic equations. Eventually, by solving this system a numerical solution is obtained for the problem. The validity of the designed method is investigated through three numerical examples.