An-Najah National University Faculty of Graduate Studies Wavelets Numerical Methods for Solving Differential Equations By Yousef Mustafa Yousef Ahmed Bsharat Supervisor Dr. Anwar Saleh This Thesis is Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Computational Mathematics, Faculty of Graduate Studies, An-Najah National University, Nablus, Palestine. 2015 III Dedication I dedicate this thesis To my beloved Palestine To my parents To my sisters To my family To my friends Who helped me, stood by me and encouraged me. IV Acknowledgement All praise be to almighty God, without whose mercy and clemency nothing would have been possible. I wish to express my appreciation to Dr. Anwar Saleh, my advisor, for giving me all the necessary support I needed to complete this work, without him this work would not have been accomplished. I would sincerely like to thank and deeply grateful to my external examiner Dr. Abdel Halim Ziqan and to internal examiner Dr. Samir Matar for their useful and valuable comments. Also, my great thanks are due to my family for their support, encouragement and great efforts for me. I would also like to acknowledge to all my teachers at An- Najah National University. VI Table of Contents No. Contents Page Dedication III Acknowledgement IV Declaration V Table of Contents VI List of Tables VII Abstract VIII Chapter One: Introduction 1 1.1 Overview 1 A brief history of the development of wavelets in solving 1.2 3 differential equations Chapter Two: Overview of numerical methods for 7 differential equations Numerical methods for solving ordinary differential 2.1 7 equations Finite difference methods for solving partial differential 2.2 17 equations Chapter Three: Wavelets and applications 20 3.1 Wavelet transform 20 3.2 Advantages of wavelet theory 28 3.3 Comparison of wavelet transform with Fourier transform 29 Chapter Four: Haar wavelet 31 4.1 Haar wavelet function 31 4.2 Properties of Haar wavelet 40 4.3 Wavelet collocation method 40 4.4 Haar wavelet transformation 41 4.5 Function approximation 43 4.6 Convergence analysis of Haar wavelets 44 4.7 Integration of Haar wavelets 45 4.8 The product operational matrix of the Haar wavelet 47 Chapter Five: Wavelet method for differential equations 52 5.1 The method of solution for differential equations 52 Haar wavelet method for solving second linear ordinary 5.2 54 differential equations Haar wavelet method for solving linear partial differential 5.3 82 equations 5.4 Conclusion 97 References 99 صخلملا ب VII List of Tables No. Title Pages Table:2.1 The results of Euler's method 10 Table:2.2 The results of Taylor methods of order one and four 11 Table:2.3 The results of RK2 and RK4 13 Table:2.4 The results of linear shooting 15 Table:2.5 The result of linear finite difference method 17 Table:2.6 The result of finite difference for wave-like equation. 19 Table:4.1 Index computations for Haar basis function 42 Table:5.1 The numerical solution of example 5.1 57 Table:5.2 Haar wavelet and 𝑅𝐾4 method, for example 5.2 60 Table:5.3 Error using Haar wavelet method and 𝑅𝐾4 for example 5.2 61 Table:5.4 The numerical solution of example 5.3 62 Table:5.5 Error using Haar wavelet and RK4 for system for example 5.3 66 Table:5.6 The numerical solution of example 5.4 67 Table:5.7 The numerical solution for example 5.5 76 Table:5.8 Error in example 5.5 77 Table:5.9 Numerical result of the linear Klein-Gordon equation 80 Table:5.10 Maximum error of the equation at different time 81 Table:5.11 Maximum absolute error of the equation 82 Table:5.12 Numerical result of the linear Klein-Gordon equation 86 Table:5.13 Maximum absolute error of the equation at different time 87 Table:5.14 The numerical solution for wave like equation at 𝑡 = 0.01 91 Table:5.15 The numerical solution for wave like equation at 𝑡 = 0.1 92 Table:5.16 The numerical solution for wave like equation at 𝑡 = 0.2 93 Table:5.17 Maximum error of the equation at different time 94 Table:5.18 The numerical solution for diffusion equation at time 𝑡 = 0.1 96 Table:5.19 The numerical solution for diffusion equation at time 𝑡 = 0.1 97 VIII Wavelets Numerical Methods for Solving Differential Equations By Yousef Mustafa Yousef Ahmed Bsharat Supervisor Dr. Anwar Saleh Abstract In this thesis, a computational study of the relatively new numerical methods of Haar wavelets for solving linear differential equations is used. A comparison between the new method and some classical methods for linear differential equations has been made. The aim is to show the efficiency of the presented method and its advantage over other method. The new method is simple and its numerical results are close or more accurate than some classical methods. 1 Chapter One Introduction 1.1. Overview Wavelet analysis is an exciting new method for solving difficult problems in mathematics, physics, and engineering [7]. Wavelets are functions that satisfy certain mathematical requirement and can be used to represent a function, such as the solution of ODE and PDE. Representation of functions is an old subject. Weierstrass theorem [3] guarantees the existence of polynomial that approximates any continuous function on any closed interval of ℝ to any level. In 1714 Taylor theorem [1,3] represents the polynomial that approximates a function using the set {(𝑥 − 𝑥 )𝑛}∞ as basis. In 1808, 0 𝑛=0 Joseph Fourier [6] used the set {𝑠𝑖𝑛(𝑛𝑥),𝑐𝑜𝑠(𝑛𝑥)}∞ as basis to represent 𝑛=1 functions of period 𝑝 = 2𝜋, then generalized to functions of any period 𝑝 = 2𝐿. Wavelet representation of functions uses the wavelet basis defined in chapter three. The wavelet decomposition analysis is used most often in wavelet signal processing [7]. It is used in signal compression as well as in signal identification. Wavelet transform of a function, as Fourier transform, is powerful tool for analyzing the components of stationary phenomena. However, the wavelet transform has the advantage of the ability of analyzing nonstationary phenomena where Fourier transform fails [7,8,28]. Wavelet transform, as one of the mathematical real or complex valued function, is one which has become widely used in various fields of 2 application, mainly medicine, communication, computer software and human related applications. In specific, wavelet can be found in scanning, disease diagnosis to help doctors do their job precisely in this human sensitive field. It can also help encode audio and video signals in the field of telecommunications. As well, there are other useful applications which can effectively help intelligence agencies recognize the tinniest details of human bodies for security purposes and in cases of terrorist acts, airplane collapse, ship wreckage, and other human verification uses. For example, the Federal Bureau of Investigation (FBI) of the United States uses wavelet application to identify and verify millions of people’s fingerprints. In the future, it is expected that mathematical wavelet technology will cover hundreds of applications and it will mainly focus human welfare and healthcare subjects to achieve the best results possible [7,8,9,30]. In chapter one, we give a brief history of wavelets and a review of literature on subject of wavelets. In chapter two, we review some classical numerical methods for ordinary and partial differential equations, for comparison with the Haar wavelet method used in this thesis. In chapters three wavelets and their applications are considered. In chapter four, the Haar wavelet method for solving differential equations is given. In chapter five, numerical examples of the Haar wavelet method are given and the results are compared with results from the classical methods introduced in chapter two. Conclusion is given in section 5.4