Identification of source terms in wave equation with dynamic boundary conditions

TL;DR

This paper addresses an inverse problem for the wave equation with dynamic boundary conditions, focusing on reconstructing source terms using analytical and numerical methods, including gradient formulas and conjugate gradient algorithms.

Contribution

It introduces a new approach for source term identification in wave equations with dynamic boundaries, including theoretical analysis and numerical implementation.

Findings

Fréchet differentiability of the Tikhonov functional established

Gradient formula derived via an adjoint problem

Numerical experiments demonstrate effective source reconstruction

Abstract

This paper studies an inverse hyperbolic problem for the wave equation with dynamic boundary conditions. It consists of determining some forcing terms from the final overdetermination of the displacement. First, the Fr\'echet differentiability of the Tikhonov functional is studied, and a gradient formula is obtained via the solution of an associated adjoint problem. Then, the Lipschitz continuity of the gradient is proved. Furthermore, the existence and the uniqueness for the minimization problem are discussed. Finally, some numerical experiments for the reconstruction of an internal wave force are implemented via a conjugate gradient algorithm.