Identification of source terms in heat equation with dynamic boundary conditions

TL;DR

This paper addresses an inverse heat problem with dynamic boundary conditions, proposing a weak solution approach, analyzing a cost functional, and demonstrating numerical reconstruction of heat sources in a 1-D setting.

Contribution

It introduces a novel method for identifying source terms in heat equations with dynamic boundaries, including theoretical analysis and numerical validation.

Findings

Gradient formula of the cost functional derived

Existence and uniqueness of quasi-solution established

Numerical results demonstrate algorithm efficiency

Abstract

We study an inverse parabolic problem of identifying two source terms in heat equation with dynamic boundary conditions from a final time overdetermination data. Using a weak solution approach by Hasanov, the associated cost functional is analyzed, especially a gradient formula of the functional is proved and given in terms of the solution of an adjoint problem. Next, the existence and uniqueness of a quasi-solution are also investigated. Finally, the numerical reconstruction of some heat sources in a 1-D equation is presented to show the efficiency of the proposed algorithm.