About the Convergence of a Family of Initial Boundary Value Problems for a Fractional Diffusion Equation with Robin Conditions

TL;DR

This paper investigates the convergence of solutions for a family of fractional diffusion equations with Robin boundary conditions, showing they approach the Dirichlet problem as the heat transfer coefficient varies, using Fourier methods.

Contribution

It establishes existence, uniqueness, and convergence of solutions for fractional diffusion problems with Robin conditions, connecting them to Dirichlet boundary solutions.

Findings

Solutions exist and are unique for each Robin problem.

Solutions converge to the Dirichlet problem solution as the heat transfer coefficient increases.

Fourier approach effectively proves convergence and boundary condition transition.

Abstract

We consider a family of initial boundary value problems governed by a fractional diffusion equation with Caputo derivative in time, where the parameter is the Newton heat transfer coefficient linked to the Robin condition on the boundary. For each problem we prove existence and uniqueness of solution by a Fourier approach. This will enable us to also prove the convergence of the family of solutions to the solution of the limit problem, which is obtained by replacing the Robin boundary condition with a Dirichlet boundary condition.