Numerical Solution of Differential Equations Using Haar Wavelets

Abstract

The present dissertation entitled, “NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS USING HAAR WAVELETS”, embodies a brief account of investigations carried out by various authors on, Haar wavelets under the supervision of Dr.Vivek, Assistant Professor, School of Mathematics and Computer Applications, Thapar University, Patiala. The work presented in this dissertation has been divided into four chapters. The aim of this work is to study some results on Haar wavelets and to solve initial value problem and boundary value problem. In the very first chapter there is brief introduction about the differential equations and Haar wavelets. Differential equations arise in the mathematical modeling of many physical, chemical and biological phenomena and many more areas of science and engineering. Asymptotic and numerical are two principle approaches for solving differential equations. In the Present study, the numerical techniques have been used to approximate the solution. There are many numerical methods to find an approximate solution such as finite difference methods (FDM) ,finite element methods(FEM), finite volume methods(FVM), boundary element methods(BEM), etc. The present work focuses o Haar wavelet technique for solving differential equations. First chapter, the basic concepts related to Haar wavelets like wavelets, Haar Wavelets, Haar matrix, Haar transform etc. have been elaborated. In the second chapter a uniform Haar wavelets methods has been presented for solving initial value problems. Firstly the proposed method has been discussed in detail. The general expression of Haar wavelets basis functions have been presented with their properties. Then the algorithm for solving the initial value problem has been presented and explained in details with help of example wherever needed. In the last, the error estimates have been described. The third chapter focuses on the non-uniform Haar wavelet method. The non-uniform Haar wavelet method is preferred in situations where non-uniform or sharp transitions occur. One such type of problems are singularly perturbed problems where very sharp boundary layer arise as the singular perturbation parameter tends to zero. In this chapter non-uniform Haar wavelet method has been proposed for solving singularly perturbed problems. Because sharp change occurs in the solution and hence special meshes have been proposed to capture these sharp variation. In the last algorithm for non-uniform Haar wavelet method has been discussed in detail over then non-uniform meshes. The last chapter is concentrated on the application of the Haar wavlet algorithms presented in second and third chapter. The initial value problem has been solved using uniform Haar Wavelet method. The singularly perturbed boundary value problem has been solved using non-uniform Haar wavelet method. The computed solution has been compared with the exact solution. To conclude the Haar wavelet method produces good results even for very small number of collocation points. The method has added advantage over other methods because of its clear and simple structure and less computation cost.