Wavelet transforms of some equations of fluid mechanics

Abstract

This paper explores the application of wavelet transforms to equations rather than to data sets. An entire class of wavelets, obtained from recursive shifts and changes in scale of Gaussian filters, transforms Laplacians into first order derivatives in the scale factor. As a result, parabolic and elliptic equations are transformed into first-order wave equations or into ordinary differential equations. Examples are given for the diffusion, Burgers, Poisson and Navier-Stokes equations, which are formally integrated by the method of characteristics. It is also shown that the even-indexed Gaussian wavelets decompose a function into the local spectral contributions to its amplitude as well as to its variance. This gives a simpler inversion formula and a new form of the convolution of wavelet transforms.