Abstract
Recently, Wavelet Method (WM) is becoming a powerful tool to solve various types of differential equations. In this method orthogonal wavelet functions are used as shape functions which are easier to compute. Till date WM has been used for crisp problems that is where the variables and parameters in the differential equations are considered as crisp. But generally in real world problems, every system contains uncertainty and this makes the corresponding mathematical model as uncertain. The uncertain and imprecise parameters make the system complex. Here the uncertainty has been managed by considering the parameters as interval. Accordingly, WM method is modified and Interval Wavelet Method (IWM) has been proposed to solve ODEs.
Top1. Introduction
Wavelet method is a powerful technique to investigate science and engineering problems. There are two types of wavelets such as discrete and continuous wavelets. Here discrete type of wavelet is considered. Discrete orthogonal wavelets are family of functions with compact support which form a basis on a bounded domain. The orthogonal wavelet family may be defined by a set of L filter coefficients {al: l=0,1,…,L–1}, where L is an even integer. Two fundamental scale equations in wavelet theory are defined as (Chen, Hwang, & Shih, 1996)
where φ(
x) and Ψ(
x) are the scaling and wavelet functions with fundamental support over finite intervals [0,
L–1] and [1–
L/2,
L/2], respectively. These equations are used to determine the value of the scaling and wavelet functions at dyadic points
x=
n/2
J,
n=0,1,…
The scaling functions at resolution level J may be defined as follows
.
In this context, (Daubechies, 1992) constructed a family of orthonomal bases of compactly supported wavelets for the space of square-integrable functions, L2(R). Due to the fact that they possess several useful properties, such as orthogonality, compact support, exact representation of polynomials to a certain degree, and ability to represent functions at different levels of resolution, Daubechies' wavelets have gained great interest in the numerical solutions of ordinary and partial differential equations. (Beylkin, 1992) described the exact and explicit expressions of differential operators in orthonormal bases of compactly supported wavelets as well as the representations of Hilbert transform which are applied to multidimensional convolution operators. Further various finite integrals whose integrands are product of Daubechies compactly supported wavelets and their derivatives are evaluated by (Chen et al., 1996). (Avudainayagam & Vani, 2000) used wavelet bases to the solution of integro-differential equations and two simple nonlinear integro-differential equations are investigated.
Haar wavelet is a special type of Daubechies wavelets. Haar family of wavelet is used by (Ülo Lepik & Tamme, 2004) to obtain the numerical solution of linear integral equations. The numerical solutions of five different integral equations with their exact solution have been compared in that paper. Then (Ü. Lepik, 2005) used Haar wavelet technique to solve ordinary and partial differential equations. Whereas (Mehra, 2009) discussed some computational aspects of wavelets and various wavelet methods. Further, (Ü Lepik, 2012) analysed free and forced vibrations of cracked Euler-Bernoulli beams by using Haar wavelet method.
Key Terms in this Chapter
Haar Wavelet: It is a sequence of scaled and shrink square-shaped functions which together form a wavelet family or basis.
Wavelet: A wavelet may be defined as a wave-like oscillation with amplitude that begins at zero, increases, and then decreases back to zero, that is the value over the whole domain is zero.
Discrete Orthogonal Wavelets: These are family of functions with compact support which form a basis on a bounded domain.
Interval: It may be written represented in the following manner: where and are lower and upper values of the interval respectively.
Width of the Interval: It is defined as the difference of the interval .