On the Dirichlet BVP of fractional diffusion advection reaction equation in bounded interval: Structure of solution, integral equation and approximation

Abstract

We revisit a Dirichlet boundary value problem that involves a fractional diffusion, advection, and reaction differential equation in a one-dimensional bounded domain. By utilizing a variational formulation, we establish the existence of a weak solution to this problem. A procedure that involves raising regularity is employed to transform the variational formulation solution into the classical one. Furthermore, we demonstrate that such a solution has a closed-form expression, which is decomposable into two components: a completely known non-smooth part and a smooth part expressed as an integral of an unknown function. Availability of this decomposition allows for the development of a numerical approximation of the solution indirectly via an approximation of the aforementioned unknown function. This function satisfies a type of Volterra–Fredholm integral equation. We propose a two-step finite element method that transforms the integral equation into solving two decoupled Volterra integral equations of the second kind. Once this is in place, an approximate solution to the boundary value problem is gathered from the previously mentioned closed-form. The performance of the proposed method is substantiated by a convergence analysis and a set of numerical experiments.