On Fractional Diffusion-Advection-Reaction Equation In $\mathbb{R}$

TL;DR

This paper analyzes the mathematical properties of a fractional differential equation modeling anomalous diffusion, advection, and reaction on the real line, focusing on existence, uniqueness, and smoothness of solutions using advanced functional analysis techniques.

Contribution

It provides a rigorous analysis of solutions to a class of fractional differential equations with Riemann-Liouville operators, including existence, uniqueness, and smoothness results.

Findings

Established existence and uniqueness of solutions.

Characterized Sobolev spaces with fractional derivatives.

Proved smoothness properties of solutions.

Abstract

We present an analysis of existence, uniqueness, and smoothness of the solution to a class of fractional ordinary differential equations posed on the whole real line that models a steady state behavior of a certain anomalous diffusion, advection, and reaction. The anomalous diffusion is modeled by the fractional Riemann-Liouville differential operators. The strong solution of the equation is sought in a Sobolev space defined by means of Fourier Transform. The key component of the analysis hinges on a characterization of this Sobolev space with the Riemann-Liouville derivatives that are understood in a weak sense. The existence, uniqueness, and smoothness of the solution is demonstrated with the assistance of several tools from functional and harmonic analyses.