"Wavelet Transforms of some Equations of Fluid Mechanics"
by
Lewalle, J.
ABSTRACT
This paper explores the application of wavelet transforms to equations
rather than 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 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 new form of the convolution of wavelet transforms.