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Sukanta Nayak (National Institute of Technology Rourkela, India) and S. Chakraverty (National Institute of Technology Rourkela, India)

Source Title: Handbook of Research on Generalized and Hybrid Set Structures and Applications for Soft Computing

Copyright: © 2016
|Pages: 16
DOI: 10.4018/978-1-4666-9798-0.ch021

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TopWavelet 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 {*a _{l}*:

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, *L*^{2}(*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.

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 .

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