# On the Non-Linear Integral Equation Approach for an Inverse Boundary Value Problem for the Heat Equation

## TL;DR

This paper presents a novel non-linear integral equation approach for solving an inverse boundary problem in heat conduction, utilizing Laguerre transforms, boundary integral equations, and regularization techniques for accurate reconstructions.

## Contribution

It introduces a new method combining Laguerre transforms and non-linear boundary integral equations for inverse heat problems, with a developed iterative algorithm and stability analysis.

## Key findings

- The method achieves accurate boundary reconstructions.
- The approach demonstrates stability through Tikhonov regularization.
- Numerical results confirm the effectiveness of the proposed technique.

## Abstract

We consider the inverse problem of reconstructing the interior boundary curve of a doubly connected domain from the knowledge of the temperature and the thermal flux on the exterior boundary curve. The use of the Laguerre transform in time leads to a sequence of stationary inverse problems. Then, the application of the modified single-layer ansatz, reduces the problem to a sequence of systems of non-linear boundary integral equations. An iterative algorithm is developed for the numerical solution of the obtained integral equations. We find the Fr\'echet derivative of the corresponding integral operator and we show the unique solvability of the linearized equation. Full discretization is realized by a trigonometric quadrature method. Due to the inherited ill-possedness of the derived system of linear equations we apply the Tikhonov regularization. The numerical results show that the proposed method produces accurate and stable reconstructions.