Wavelet-Galerkin method for numerical solution of partial differential equation

Abstract

In recent years wavelets are given much attention in many branches of science and technology due to its comprehensive mathematical power and application potential. The advantage of wavelet techniques over finite difference or element method is well known.The objective of this work is to implementing the Wavelet-Galerkin method for approximating solutions of differential equations. In this paper, we elaborate the wavelet techniques and apply the Galerkin method procedure to analyse one dimensional wave equation as a test problem using fictitious boundary approach. The sections of this thesis will include defining wavelets and their scaling functions.This will give the reader valued insight about wavelets.Following this will be a section defining the Daubechies wavelet and its scaling function. This section comprises some subsection about computing the scaling function and its derivative and integral.The purpose of this section will be to give the reader an understanding how scaling function and its derivative are computed. Next will be a section on multiresolution analysis and its properties. The next section give information about the 2-term connection coefficients. The main focus of this work will be to solve the one dimensional wave equation using fictitious boundary approach and made a comparison between the exact and approximate solution which gives the accuracy and efficiency of this method.