An accurate regularization technique for the backward heat conduction problem with time-dependent thermal diffusivity factor

TL;DR

This paper introduces a Meyer wavelet-based regularization method to accurately solve the ill-posed backward heat conduction problem with time-dependent diffusivity, demonstrating high accuracy and stability through numerical examples.

Contribution

A novel regularization technique using Meyer wavelets for the backward heat problem with time-dependent diffusivity, providing optimal stability and convergence estimates.

Findings

The method achieves high accuracy in one and two-dimensional examples.

The regularization technique is stable and convergent.

Numerical results outperform existing methods.

Abstract

In this work, an accurate regularization technique based on the Meyer wavelet method is developed to solve the ill-posed backward heat conduction problem with time-dependent thermal diffusivity factor in an infinite "strip". In principle, the extremely ill-posedness of the considered problem is caused by the amplified infinitely growth in the frequency components which lead to a blow-up in the representation of the solution. Using the Meyer wavelet technique, some new stable estimates are proposed in the H\"older and Logarithmic types which are optimal in the sense of given by Tautenhahn. The stability and convergence rate of the proposed regularization technique are proved. The good performance and the high-accuracy of this technique is demonstrated through various one and two dimensional examples. Numerical simulations and some comparative results are presented.