On Numerical Solution Of The Time Fractional Advection-Diffusion Equation Involving Atangana-Baleanu-Caputo Derivative

Abstract

Abstract A powerful algorithm is proposed to get the solutions of the time fractional Advection-Diffusion equation(TFADE): A B C D 0 + , t β u ( x , t ) = ζ u x x ( x , t ) − κ u x ( x , t ) + $^{ABC}\mathcal{D}_{0^+,t}^{\beta}u(x,t) =\zeta u_{xx}(x,t)- \kappa u_x(x,t)+$ F(x, t), 0 < β ≤ 1. The time-fractional derivative A B C D 0 + , t β u ( x , t ) $^{ABC}\mathcal{D}_{0^+,t}^{\beta}u(x,t)$ is described in the Atangana-Baleanu Caputo concept. The basis of our approach is transforming the original equation into a new equation by imposing a transformation involving a fictitious coordinate. Then, a geometric scheme namely the group preserving scheme (GPS) is implemented to solve the new equation by taking an initial guess. Moreover, in order to present the power of the presented approach some examples are solved, successfully.