Compactly supported solution of the time-fractional porous medium equation on the half-line

TL;DR

This paper proves the existence and uniqueness of a compactly supported solution to the time-fractional porous medium equation on the half-line, using a transformation to a Volterra integral equation and a shooting method.

Contribution

It introduces a novel approach transforming the fractional PDE into a nonlinear Volterra integral equation and applies a shooting method to establish solution uniqueness.

Findings

Unique compactly supported solution exists

Transformation to Volterra integral equation is effective

Shooting method identifies initial conditions for no-flux boundary

Abstract

In this work we prove that the time-fractional porous medium equation on the half-line with Dirichlet boundary condition has a unique compactly supported solution. The approach we make is based on a transformation of the fractional integro-differential equation into a nonlinear Volterra integral equation. Then, the shooting method is applied in order to facilitate the analysis of the free-boundary problem. We further show that there exists an exactly one choice of initial conditions for which the solution has a zero which guarantees the no-flux condition. Then, our previous considerations imply the unique solution of the original problem.