Cauchy Problem for Fractional Diffusion Equations

TL;DR

This paper studies a fractional diffusion evolution equation with variable coefficients, constructing and analyzing its fundamental solution to model diffusion processes on complex inhomogeneous fractals.

Contribution

It introduces a fundamental solution for a fractional diffusion equation with variable coefficients, advancing understanding of diffusion on fractal-like structures.

Findings

Constructed the fundamental solution for the fractional diffusion equation.

Analyzed properties of the fundamental solution.

Applied to diffusion processes on inhomogeneous fractals.

Abstract

We consider an evolution equation with the regularized fractional derivative of an order $\alpha \in (0,1)$ with respect to the time variable, and a uniformly elliptic operator with variable coefficients acting in the spatial variables. Such equations describe diffusion on inhomogeneous fractals. A fundamental solution of the Cauchy problem is constructed and investigated.