Wavelet-Galerkin technique for solving certain numerical differential equations and inverse Ill-posed problems

Abstract

Due to advantage of wavelets over numerical and other approximation techniques and edge over Fourier analysis, the wavelet theory has penetrated a number of branches of science, technology and medicine. Wavelet is basically a wave pattern whose graph oscillates only over a short distance and dumps very fast. It is used as tools that cut up data, functions or operators into different frequencies and then study each frequency component with a resolution matching its scale. MATLAB computer programs have been formulated/ implemented wherever required. The outcomes of problems have been demonstrated/analysed either analytically and/or graphically. More than hundred strong and relevant references have been used. Chapter I ( Overview of Wavelets ) is devoted to the historical development of wavelet theory and basic fundamentals needed such as components of Hilbert space, Linear algebra and Fourier analysis. It also deals with (1) mathematical concept of wavelet and its properties, family of wavelets, examples, continuous wavelet transforms and wavelet series, (2) multiresolution analysis and convergence property, (3) some of important basic theorems in Fourier and wavelet domains, and (4) probability densities and generalized moments. Chapter II ( Filter and Connection Coefficients ) brings forth usual and alternate ways of derivations of filter coefficients, scaling function and connection coefficients using scaling equation. We also propose evaluation of filter coefficients based on multiple double shift orthogonality and normalization and convergence analysis. 2- and 3- terms improper connection coefficients are also evaluated for various values of j and N, as are required for solving ODEs and PDEs. Proper connection coefficients are also introduced. Chapter III ( Galerkin based Wavelet Methods for Solutions of Differential Equations ) attempts to evaluate various wavelet-Galerkin techniques for ODEs and PDEs. Finite difference based wavelet-Galerkin method for ODEs has been developed.