Certain contributions to wavelet analysis and its applications

Abstract

This thesis is dedicated to presenting approximate solutions to several mathematical newlineproblems that are either unsolved or require the most accurate solutions with minimal newlinecomputational cost and time. The most accurate structure is defined accordingly. Chap newlineter 1 and Chapter 2 are presented for the introduction and the review of literature. newlineIn Chapter 3 of this study, we have presented a mathematical model representing the newlinespread of HIV and COVID-19 diseases. These two models were analyzed simultane newlineously to understand the exact transmission behavior, with the primary objective of this newlinestudy being to provide a more accurate, efficient, and less time-consuming solution to newlinethese models, described by a system of coupled nonlinear differential equations. The newlinemethod used to address these problems is primarily based on the operational matrix of newlineintegration of Fibonacci wavelets. The Fibonacci wavelet approximation, along with the newlineapplication of the operational matrix andthecollocation technique, results in a system of newlinealgebraic equations, which are further solved using the Newton-Raphson method or an newlineother suitable iterative method to finally express the Fibonacci wavelet solution for the newlinegiven mathematical models. Convergence analysis is also provided to demonstrate that newlinethe approximation we assume in the methodology uniformly converges to the function newlineitself. Figures and tables are included to compare and validate the results of the pro newlineposed methodology. All these operations are implemented and executed using the latest newlineversion of MATLAB software. newlineIn Chapters 4 and 5, we present the general form of a well-known singular differential newlineequation under various physical conditions, as well as the Fredholm integro-differential newlineequation for analysis. These two problems were assumed to be solved numerically using newlinean approximation process involving a new and specific type of wavelet called the Taylor newlinewavelet. The use of this wavelet is absolutely remarkable, and the outcomes are sur newlineprisingly accurate when applied to these pro