Walsh and wavelet methods for differential equations on the Cantor group

Ordinary and partial differential equations for unknown functions defined on the Cantor dyadic group are studied. We consider two types of equations: related to the Gibbs derivatives and to the fractional pseudo differential operators. Since the Cantor group is an ultrametric space, pseudo differential operators have spacial properties and are of interest for some applications to models of complex systems, e.g., ultrametric diffusion models in biophysics. We find solutions to the equations in classes of distributions and analyse under what assumptions these solutions are regular functions with some “good” properties. Haar wavelets are used to solve pseudo differential equation. It is very important that the Haar MRA coincides with the Shannon MRA on the Cantor group. To analyse solutions, specific computational method based on the multiresolution structure of the Haar basis was developed.