Study of the initial boundary value problem for a two-dimensional convection-diffusion equation with a fractional time derivative in the sense of Caputo-Fabrizio

Abstract

In this paper, we study an initial boundary value problem for a differential equation with a fractional order derivative in time in the Caputo-Fabrizio sense. This equation is of great practical importance in modeling the processes of fluid motion in porous media and anomalous dispersion. The uniqueness and continuous dependence of the solution of the problem on the input data in differential form is proved. A computationally efficient implicit difference scheme with weights is
proposed. A priori estimates are obtained for the solution of the problem under the assumption that the solution exists in the class of sufficiently smooth functions. The uniqueness of the solution and the stability of the difference scheme with respect to the initial data and the right-hand side of the equation follows from the obtained estimates. The convergence of the difference problem solution to the differential problem solution with the second order in time and space variables
is proved. The results of computational experiments confirming the reliability of the theoretical analysis are presented.

Key words: Fractional differential equation, fractional derivative in the sense of Caputo-Fabrizio, finite difference method, energy inequality method, stability, convergence, a priori estimate.