Applications Of Interpolating Wavelet In Numerical Computation

This thesis reviews the researches on the applications of interpolating wavelets in numerical computation, which include the general methods of using interpolating wavelet to solve partial differential equations and algorithms for integral equations of the second kind.The basic idea behind the wavelet based numerical methods is to represent a function in terms of basis functions, called wavelets, which are localized in both physical domain and frequency domain. Therefore wavelets are particularly suitable for approximating functions having singular points or regions of large variations. Compared with traditional numerical methods, the wavelet based methods generate adaptive grids so that , high-resolution computations are carried out only in those regions where large variations occur.Based on the rationalized Haar function method for solving the Fredholm integral equations, this thesis presents a new method for solving the Fredholm integral equations of the second kind. This numerical method converts the integral equation to a system of linear equations by using interpolating wavelet transform. Compared to standard methods such as Nystrom method, we get a sparse matrix after transformation because interpolating wavelets have compact support. This property reduces the amount of computation. Interpolating wavelets have better continuity in comparison to rationalized Haar wavelets, therefore are more suitable to approach smooth functions. In addition, it simplified sampling due to the cardinal property. Numerical examples show that the proposed method has high accuracy.