Fast difference scheme for the reaction-diffusion-advection equation with exact artificial boundary conditions

In this paper, we derive the exact artificial boundary conditions for one-dimensional reaction-diffusion-advection equation on an unbounded domain. By employing the Laplace transform, we reduce the original unbound domain problem into a bounded domain problem. The exact artificial boundary conditions are given by Caputo-tempered fractional derivatives in the reduced initial-boundary value problem. We show that the reduced initial-boundary value problem is stable with the exact artificial boundary conditions. We design a finite difference scheme for the reduced finite domain problem. To save the computational cost, we developed a fast algorithm to solve Caputo-tempered derivatives arise in the boundary conditions. We prove that the present difference schemes are uniquely solvable and unconditionally stable in the energy norm. Finally, we demonstrate the effectiveness of the proposed methods by some numerical examples.