Implementation of the Wavelet-Galerkin method for boundary value problems

The objective of this work is to develop a systematic method of implementing the Wavelet-Galerkin method for approximating solutions of differential equations. The beginning of this project included understanding what a wavelet is, and then becoming familiar with some of the applications. The Wavelet-Galerkin method, as applied in this paper, does not use a wavelet at all. In actuality, it uses the wavelet\u27s scaling function. The distinction between the two will be given in the following sections of this paper. The sections of this thesis will include defining wavelets and their scaling functions. This will give the reader valued insight to wavelets and Discrete Wavelet Transforms (DWT). Following this will be a section defining the Galerkin method. The purpose of this section will be to give the reader an understanding of how weighted residual methods work, in particular, the Galerkin Method. Next will be a section on how Scaling functions will be implemented in the Galerkin method, forming the Wavelet-Galerkin Method. The focus of this investigation will deal with solutions to a basic homogeneous differential equation. The solution of this basic equation will be analyzed using three separate, distinct methods, and then the results will be compared. These methods include the Wavelet-Galerkin Method, the Galerkin Method using quadratic shape functions, and standard analytical means. Factors to be studied include computational time, effort, accuracy, and ease of implementing the method of solution. After a thorough comparison has been made, there will be a section to talk about possible applications of the Wavelet-Galerkin method and recommendations for future work. Predictions of what avenues to pursue in refining the Wavelet-Galerkin method will also be stated. And suggestions on how to make the method more accurate will be given