Application Of Wavelet In Operator Equation And Inverse Problem

Wavelet analysis, as one of the most exciting topics to emerge from mathematical research, has a wide range of engineering and mathematical applications. Wavelet transforms complement the shortcomings of Fourier-based techniques because of their flexible time-frequency windows. Wavelets are widely applied in signal processing, image processing, communication, numerical analysis and so on.In this paper ,we describe in detail the basis theory of wavelet and study the applications of wavelet in the first kind singular integral equation and inverse heat conduction problems (IHCP) . Firstly, Wavelet-Galerkin algorithm for solving the first kind of singular integral equation with the Hilbert kernel is proposed, we use the characteristic of periodic wavelet on L~2([0,1]) and Hilbert kernel to solve and make stiff matrix lower dimensions and become sparser through thresholding,thus the cost of computation is reduced. Because of the singularity of Hilbert kernel we use Tikhonov regularization method to solve the system of stiff equation. At last the convergence and numerical result of approximate solution are given. Secondly, an approach of regularization based on Fourier is presented for sideways heat equation; we give the theory proof and error estimate. We solve the gap in reference between the theory error estimate and algorithms. The algorithm not only keeps parts of the high-frequency components but also has the same computing cost and accurate. Lastly, we give the regularization approaches by use the tools of Fourier, wavelet and Meyer window function, and then give the error estimates and numerical implements to test the validity of algorithms.