Maximum principle and its application for the nonlinear time-fractional diffusion equations with Cauchy-Dirichlet conditions

TL;DR

This paper establishes a maximum principle for one-dimensional sub-diffusion equations with Atangana-Baleanu fractional derivatives, ensuring uniqueness and continuous dependence of solutions for nonlinear time-fractional diffusion equations.

Contribution

It introduces a maximum principle specific to Atangana-Baleanu fractional derivatives and applies it to prove uniqueness and stability of solutions for nonlinear time-fractional diffusion equations.

Findings

Maximum principle formulated and proved for Atangana-Baleanu fractional derivative

Uniqueness of classical solutions established for the initial-boundary-value problem

Solutions depend continuously on initial and boundary data

Abstract

In this paper, a maximum principle for the one-dimensional sub-diffusion equation with Atangana-Baleanu fractional derivative is formulated and proved. The proof of the maximum principle is based on an extremum principle for the Atangana-Baleanu fractional derivative that is given in the paper, too. The maximum principle is then applied to show that the initial-boundary-value problem for the linear and nonlinear time-fractional diffusion equations possesses at most one classical solution and this solution continuously depends on the initial and boundary conditions.