Linear evolution equations on the half line with dynamic boundary conditions

TL;DR

This paper extends the classical heat equation problem on the half line by incorporating time-dependent dynamic Robin boundary conditions, providing a solution method and analyzing existence, uniqueness, and convergence.

Contribution

It introduces a novel approach using the Fokas transform for dynamic boundary conditions and generalizes to higher-order linear evolution equations.

Findings

Solution representation via Fokas transform

Convergence of approximate solutions

Existence and uniqueness of solutions

Abstract

The classical half line Robin problem for the heat equation may be solved via a spatial Fourier transform method. In this work, we study the problem in which the static Robin condition $bq(0,t)+q_x(0,t)=0$ is replaced with a dynamic Robin condition; $b=b(t)$ is allowed to vary in time. We present a solution representation, and justify its validity, via an extension of the Fokas transform method. We show how to reduce the problem to a variable coefficient fractional linear ordinary differential equation for the Dirichlet boundary value. We implement the fractional Frobenius method to solve this equation, and justify that the error in the approximate solution of the original problem converges appropriately. We also demonstrate an argument for existence and unicity of solutions to the original dynamic Robin problem for the heat equation. Finally, we extend these results to linear evolution equations of arbitrary spatial order on the half line, with arbitrary linear dynamic boundary conditions.