Numerical Investigation of Some Nonlinear Initial and Boundary Value Problems through HAAR Wavelets

Abstract

Wavelet analysis has attracted much attention for providing a unified framework for many diversified problems that are of interest to different fields, mostly in mathematical sciences, physics and electrical engineering. The purpose of this thesis is to develop methods and techniques based on wavelets, more precisely the Haar wavelets to solve nonlinear initial and boundary value problems encountered in the chosen application areas and related fields. The computational aspects of these methods are especially attractive, mainly because of their simple expressions in terms of the Haar wavelet family and the theory of MRA, generating the faster converging algorithms. The problems are mainly dealt with in two phases: first, the nonlinear differential equations are transformed into equivalent systems of the wavelet family and then, in the second phase, the systems are discretized using the collocation method and the results are optimized using the N-R method for nonlinear system of equations.

The thesis contains seven chapters. The first chapter begins with a brief introduction of differential equations, their classifications, real-life applications and a review of the existing techniques for dealing with a variety of differential equations. The chapter further covers the formal introduction of wavelets, their useful properties including MRA and the Haar wavelet system on the domain [0, 1]. In addition, a review of literature related to the Haar wavelet based methodologies is also provided for various classes of differential equations and their coupled systems. In chapters 2-3, the Haar wavelet collocation method is investigated for a class of two-point second and higher order singular initial and boundary value problems. The applicability of HWCM is further extended in chapter 4 to a class of coupled system of nonlinear singular initial value problems. In chapter 5, a special class of multi-point boundary value problems known as three-point BVPs is solved using the Haar wavelet collocation method. Chapter 6 investigates the Higher order Haar wavelet method (HOHWM), for a class of nonlinear boundary value problems and chapter 7 extends its application to a singular class of linear and nonlinear boundary value problems of second order. Finally, concluding remarks and future scope are included in the thesis.