Wavelet Solution Of Integral Equations

Wavelet analysis is a new promising mathematical branch. It is in the theory and application related to the harmonic analysis, functional analysis, numerical approximation, numerical solution of differential equations and integral equations and many other areas of mathematics. Wavelet analysis has a wide range of engineering applications due to its time-frequency localization and multi-scale analysis property. At present, wavelet analysis has become a popular and widely used subject in the areas of scientific research and engineering projects.Wavelet analysis is the development and perfection of the Fourier analysis. Since the development of the wavelet analysis is the basis to solve some practical problems, and then, it develops into a radioactive multi-disciplined theory, now it has become a hot field in the research internationally. Wavelet transforms complement the shortcomings of Fourier-based techniques because of their flexible time-frequency windows. Wavelets are widely applied in numerical analysis, signal processing, image processing and so on. Because of the high frequency component gradually refined using time domain or frequency domain sampling step, which can be focused on target any details. In this sense, it is praised as mathematical microscope and is expected to be an important frequently-used tool for scientific practitioners.As it is well-known, integral equations arise in various fields of science and engineering techniques and play a very important role in these fields. Methods for solving these equations thus become a key factor in such fields. For such equations, except some special cases, exact solutions are difficult to be derived by analytical methods. As a result, numerical methods or approximate methods remain much interest. It has drawn more and more mathematical researchers' attentions. Because the wavelets have the smooth and local compact properties, compared with traditional finite element method and finite difference method, it is a more useful method for solving integral equations. In this paper, we study the application of wavelet analysis to integral and differential equations. The main contributions of this dissertation are as follows. 1. Introduce foundational theories on wavelet and some basic integral equations. First the definition of the simplest wavelet, i.e., Haar wavelet and its basic properties are presented. Next the construction of the Haar wavelet integral operator matrix and the multiplication operator matrix from the Haar wavelet is introduced. Thirdly the Fredholm equations and Volterra equations are decomposed by the Haar wavelet and then are transformed by operator matrices into systems of nonlinear equations, which are numerically solved using collocation method. Finally, digital simulation diagrams and tables are given to visually compare numerical solutions and exact solutions. With the increase of the series index m, the obtained numerical solutions become more accurate. Therefore, we can increase m to get more accurate solutions.2. The definition and some basic properties of Legendre wavelets are given first. And then Legendre wavelet integral operator matrix and the product operator matrix are constructed through the Legendre wavelets. This time the Fredholm equations and Volterra equations are decomposed by the Legendre wavelet and then are transformed by operator matrices into systems of nonlinear equations, which are numerically solved using collocation method. As in the first part, comparisons of numerical and exact solutions are visually illustrated by means of digital simulation diagrams and tables. It is observed that the accuracy of the numerical solutions increases while increasing m and k. Consequently, larger m and k can yield more accurate solutions.And then gives the digital simulation diagram and tables are given a numerical solution and exact solutions of the comparison, the numerical solution obtained with the increase of m and k, and become more accurate, so to get a better solution, we can obtain a large m and k values.