Stability and convergence of the wavelet-Galerkin method for the sideways heat equation

Abstract

- We consider an inverse heat conduction problem, the Sideways Heat Equation. This is a Cauchy problem for the heat equation in a quarter-plane, with data given along the line x = 1, where the solution is sought for 0 ≤ x < 1. The problem is ill-posed, in the sense that the solution (if it exists) does not depend continuously on the data. We discuss the stability and convergence properties of the wavelet- Galerkin method for solving the sideways heat equation. The wavelets are of Meyer type that have compact support in frequency space. Previous stability results for this method were suboptimal. We show that with additional assumptions concerning the smoothness of the solution, and concerning the definition of the wavelets, we can obtain almost optimal error estimates.

On regularization and error estimates for the backward heat conduction problem with time-dependent thermal diffusivity factor

“…This basis can be modified in a systematic way and can be applied in different regions of space with different resolutions. Therefore, wavelet methods have been introduced for solving the inverse and ill-posed parabolic partial differential equations (PDEs) [36][37][38][39]. Recently a wavelet regularization method was proposed by the authors for solving the Helmholtz equation [36].…”

Simplified Tikhonov Regularization for Two Kinds of Parabolic Equations

“…A lot of authors deal with the above illposed problems by different methods. Roughly speaking, the methods involved are simplified Tikhonov regularization [4], Fourier method [2,4], Wavelet Mayer method [2,15], numerical method [1,5,13,16,8].…”

The multi-resolution method applied to the sideways heat equation

“…(1) have been used by L. Elden and T. Reginska [1,[3][4][5], etc. They described a multi-resolution Galerkin method which is based on the Meyer MRA.…”

“…However, up to now, the theoretical results concerning the error estimates were unsatisfactory. In papers [1] and [3], the results about multi-resolution method were not better than about the Fourier method; in paper [1], although the authors imposed an additional assumption on the Meyer wavelet function, the error estimate was not optimal.…”