Advanced Numerical and Semi-Analytical Methods for Differential Equations

16Wavelet Method

16.1 Introduction

Wavelet method has been proven to be an efficient tool in analyzing dynamic systems and differential equations arising in other science and engineering problems. A wave‐like oscillation having amplitude beginning at zero that monotonically increases and then decreases back to zero is referred to as wavelet. Wavelets also serve as a tool in analysis of transient, nonstationary, and time‐variate phenomena. A wavelet series is often represented in terms of complete square‐integrable function or set of orthonormal basis functions. The broad classification of wavelet classes is considered as discrete, continuous, and multiresolution‐based wavelets. Continuous wavelets are projected over continuous function space whereas discrete wavelets are often considered over discrete subset of upper half plane. While analyzing discrete wavelets, only a finite number of wavelet coefficients are taken into consideration which sometimes makes them numerically complex. In such cases, multiresolution‐based wavelets are preferred.

In mid‐1980s, orthonormal wavelets were initially studied by Meyer [1] over real line ℜ. Gradually, compactly supported orthogonal wavelets viz. Daubechies wavelets [2] were constructed by Ingrid Daubechies. Daubechies wavelets are sufficiently smooth, possess orthogonality, and have compact support. A detailed study of wavelets and their applications may be found in Refs. [3–6]. Chen et al. [7] studied the computation of differential‐integral ...