Existence and Regularity of solutions to 1-D Fractional Order Diffusion Equations

TL;DR

This paper analyzes the existence and regularity of solutions to one-dimensional steady-state fractional diffusion equations, exploring different models and boundary conditions to determine well-posedness and solution smoothness.

Contribution

It provides explicit regularity results for solutions of Riemann-Liouville and Riemann-Liouville-Caputo fractional diffusion models under various boundary conditions.

Findings

Regularity of solutions depends on the RHS function.

Identifies boundary conditions for well-posedness.

Establishes conditions for solution existence and smoothness.

Abstract

In this article we investigate the existence and regularity of 1-d steady state fractional order diffusion equations. Two models are investigated:the Riemann-Liouville fractional diffusion equation, and the Riemann-Liouville-Caputo fractional diffusion equation. For these models we explicitly show how the regularity of the solution depends upon the right hand side (rhs) function. We also establish for which Dirichlet and Neumann boundary conditions the models are well posed.