Use of the wavelet transform for the analysis of fractal signals and for the solution of partial differential equations.

Summary This paper is composed of two independent parts. In the first part, we are interested in the study of fractal objects using the continuous wavelet transform. In particular, we apply the multifractal formalism of singular measurements to signals. The new formalism obtained allows us to study the relative importance of the different types of singularities involved in a singular signal. Numerical applications on computer-generated signals as well as on experimental signals from fully developed turbulence experiments illustrate our remarks. In the second part of this paper, we present a space and time adaptive numerical scheme for the solution of partial differential equations. This scheme is based on the use of orthogonal wavelet bases. The multiresolution structure generated by such bases allows, in a natural way, to adapt the fineness of the spatial grid of a numerical scheme to the local regularity of the solution and thus to obtain a space adaptive scheme. The aim is to adapt not only the fineness of the spatial grid but also that of the temporal grid in order to concentrate the numerical effort in the regions of space where strong singularities appear. Numerical tests concerning the stability, the complexity and the accuracy are performed on the burgers equation.