This Technical Report aims to explain how to apply the discreteWavelet Balance Method (dWBM) for solving periodic differential equations using periodized Daubechies wavelets. The primary goal is to provide the user sufficient details to implement the technique without extensive mathematical background.
Chapter 2 of this document reviews the discrete Fourier transform using a multivariate regression approach, followed by a review of the discrete Harmonic Balance Method (dHBM) for approximating solutions to periodic differential equations. It is important the reader complete this chapter, even if they are familiar with the topics, since the approaches demonstrated in Chapter 2 with harmonic functions are used again in Chapters 5 and 6 using wavelet functions without reviewing those details.
Chapter 3 presents a brief overview of wavelets to identify key concepts and terminology used in the remainder of the document. Chapter 4 details how to compute the discrete scaling and wavelet functions (and their derivatives) for periodized Daubechies wavelets at various levels. Finally, Chapter 5 provides examples of the discrete wavelet transform and Chapter 6 provides a comparison between the dHBM and dWBM.
Jones, S. (2021). An engineer's guide to periodic discrete wavelet transforms and the periodic wavelet balance method. Faculty Publications - Mechanical Engineering, 512. https://scholar.rose-hulman.edu/mechanical_engineering_fac/512