ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.13(2012) No.3,pp.325-345 Wavelet Transform and Wavelet Based Numerical Methods: an Introduction Manoj Kumar, Sapna Pandit ∗ Department of Mathematics, Motilal Nehru National Institute of Technology, Allahabad-211004 (U.P.), India (Received 24 August 2011, accepted 21 October 2011) Abstract: Wavelet transformation is a new development in the area of applied mathematics. Wavelets are mathematical tools that cut data or functions or operators into different frequency components, and then study each component with a resolution matching to its scale. In this article, we have made a brief discussion on historical development of wavelets, basic definitions, formulations of wavelets and different numerical meth- ods based on Haar and Daubechies wavelets for the numerical solution of differential equations, integral and integro-differential equations. Keywords: wavelet transform; multi-resolution analysis; Daubechies wavelet; Haar wavelet; differential equation; integro-differential equation 1 Introduction Wavelets are already recognized as a powerful new mathematical tool in signal and image processing, time series analysis, geophysics, approximation theory and many other areas. First of all, wavelets were introduced in seismology to provide a time dimension to seismic analysis, where Fourier analysis fails. Fourier analysis is ideal for studying stationary data (data whose statistical properties are invariant over time) but it is not well suited for studying data with transient events that can not be statistically predicted from the data past. Wavelets were designed with such non stationary data in mind, There generality and strong results have quickly become useful to a number of disciplines. The wavelet transform has been perhaps the most exciting development in the decade to bring together researchers in several different field such as signal processing, quantum mechanics, image processing, communications, computer science and mathematics - to name a few as in [1]. Today wavelet is not only the workspace in computer imaging and animation; they are also used by the FBI to encode its data base of million fingerprints. In future, scientist may put wavelet analysis for diagnosing breast cancer, looking for heart abnormalities, predicting the weather, signal processing, data compression, smoothing and image compression, fingerprints verification, DNA analysis, protein analysis, Blood-pressure, heart-rate and ECG analysis, finance, internet traffic description, speech recognition, computer graphics, and many others [10, 11, 12, 14]. Some applications of wavelet transform are described at the end of this paper. Wavelet analysis provides additional freedom as compared to Fourier analysis since the choice of atoms of the transform deduced from the analyzing wavelet is left to the user. Wavelet theory involves representing general functions in terms of simpler building blocks at different scale and positions. The fundamental idea behind the wavelet transform is to analyze according to scale. The applications of wavelet theory in numerical methods for solving differential equations and integro-differential equations are roughly last two decades years old. In the early nineties, people were very optimistic because it seems that many nice properties of wavelets would automatically leads to efficient numerical method for differential equations. The reason for this optimism was the fact that many nonlinear partial differential equations (PDEs) have solution containing local phenomena (e.g. formation of shock, hurricanes) and interactions between several scales (e.g. turbulence, partic- ularly, atmospheric turbulence because there is motion on a continuous range of length scales). Such solutions can be well represented using wavelet bases because of its nice properties, for example compact support (locality in space) and vanishing moment (locality in scale). Before, the most common numerical methods used for numerical solution of physical, chemical and biological prob- lems were finite difference methods (FDM), finite volume methods (FVM), finite elements methods (FEM) and spectral * Corresponding author. E-mail address: sappu15maths@gmail.com Copyright c ⃝World Academic Press, World Academic Union IJNS.2012.06.15/612