Extremum principle for the Hadamard derivatives and its application to nonlinear fractional partial differential equations

TL;DR

This paper establishes new extremum principles for Hadamard fractional derivatives and applies them to prove uniqueness and continuous dependence of solutions for certain nonlinear fractional PDEs, including elliptic and diffusion equations.

Contribution

It introduces novel estimates for Hadamard fractional derivatives at extrema and applies these to demonstrate solution uniqueness and stability in fractional PDEs.

Findings

Uniqueness of solutions for linear and nonlinear time-fractional diffusion equations

Continuous dependence of solutions on initial and boundary data

Extremum principle for elliptic equations with Hadamard derivatives

Abstract

In this paper we obtain new estimates of the Hadamard fractional derivatives of a function at its extreme points. The extremum principle is then applied to show that the initial-boundary-value problem for linear and nonlinear time-fractional diffusion equations possesses at most one classical solution and this solution depends continuously on the initial and boundary conditions. The extremum principle for an elliptic equation with a fractional Hadamard derivative is also proved.