On Theoretical and Numerical Aspect of Fractional Differential Equations with Purely Integral Conditions

TL;DR

This paper investigates a fractional Caputo advection-diffusion equation with integral boundary conditions, establishing theoretical existence and uniqueness results and developing a finite difference numerical method with supporting examples.

Contribution

It introduces a novel approach combining energy inequalities for theoretical analysis and finite difference methods for numerical solutions of fractional PDEs with integral boundary conditions.

Findings

Proved existence and uniqueness of solutions.

Developed a finite difference numerical scheme.

Provided numerical examples demonstrating effectiveness.

Abstract

In this paper, we are interested in the study of a problem with fractional derivatives having boundary conditions of integral types. The problem represents a Caputo type advection-diffusion equation where the fractional order derivative with respect to time with $1<\alpha <2$. The method of the energy inequalities is used to prove the existence and the uniqueness of solutions of the problem. The finite difference method is also introduced to study the problem numerically in order to find an approximate solution of the considered problem. Some numerical examples are presented to show satisfactory results.