Walsh and wavelet methods for differential equations on the Cantor group

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:430

Article

Lebedeva, E.^1,2  Skopina, M.³ 

[1] St Petersburg State Univ, Math & Mech Fac, St Petersburg 198504, Russia

[2] St Petersburg State Polytech Univ, Inst Appl Math & Mech, St Petersburg 195251, Russia

[3] St Petersburg State Univ, Fac Appl Math & Control Proc, Univ Sky Prospekt 35, St Petersburg 198504, Russia

关键词: Cantor dyadic group;    Gibbs derivative;    Pseudo differential operators;    Walsh functions;    Haar wavelet;    Differential equation;   

DOI  :  10.1016/j.jmaa.2015.05.005

来源: Elsevier

【 摘 要 】

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. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

Free