Application of the cut-off projection to solve a backward heat conduction problem in a two-slab composite system

TL;DR

This paper applies the cut-off projection method to stabilize and solve a one-dimensional backward heat conduction problem in a two-slab system, demonstrating theoretical stability, error estimates, and numerical validation.

Contribution

It introduces a regularized solution using the cut-off projection for a two-slab heat problem, with proven stability and error bounds, advancing inverse heat conduction methods.

Findings

The regularized solution is stable under noise.

Error estimates in $L^2$-norm are provided.

Numerical tests confirm the theoretical results.

Abstract

The main goal of this paper is applying the cut-off projection for solving one-dimensional backward heat conduction problem in a two-slab system with a perfect contact. In a constructive manner, we commence by demonstrating the Fourier-based solution that contains the drastic growth due to the high-frequency nature of the Fourier series. Such instability leads to the need of studying the projection method where the cut-off approach is derived consistently. In the theoretical framework, the first two objectives are to construct the regularized problem and prove its stability for each noise level. Our second interest is estimating the error in $L^2$-norm. Another supplementary objective is computing the eigen-elements. All in all, this paper can be considered as a preliminary attempt to solve the heating/cooling of a two-slab composite system backward in time. Several numerical tests are provided to corroborate the qualitative analysis.