Special Solutions to the Space Fractional Diffusion Problem

TL;DR

This paper derives fundamental solutions for space-fractional diffusion equations involving Caputo derivatives, analyzing their properties, solution formulas, integrability, uniqueness, and demonstrating infinite signal propagation speed.

Contribution

It introduces explicit fundamental solutions for space-fractional diffusion on the half-line and studies their properties, solution representations, and propagation characteristics.

Findings

Fundamental solution $\\mathscr{E}$ derived for space-fractional diffusion.

Formulas for Dirichlet and Neumann problems using convolution with $\\mathscr{E}$.

Proved infinite speed of signal propagation.

Abstract

We derive a fundamental solution $\mathscr{E}$ to a space-fractional diffusion problem on the half-line. The equation involves the Caputo derivative. We establish properties of $\mathscr{E}$ as well as formulas for solutions to the Dirichlet and Neumann problems in terms of convolution of $\mathscr{E}$ with data. We also study integrability of derivative of solutions given in this way. We present conditions sufficient for uniqueness. Finally, we show the infinite speed of signal propagation.