These research represent the continuous transform with the wavelet of Morlet and Gauss families as a solution of Cauchy problem for the system of partial differential equations. The transformed function plays a role of initial value. This approach shows close connections with the algorithms of diffusion signal and image processing. Both sides, a mathematical background and a practical application are under study. The developing methods provide the high-accuracy and fast tools for the transform of high-irregular samples including non-equispaced ones. The approach is applied to the large variety of real-world signals, e.g. astrophysical, acoustical and genomic data, chaotic signals.
The main goal is to provide a simple and physically clear method for the estimation of main parameters of the complex kinetic structures' growth and anomalous spread. For example, the coarse-graining statistical model of Diffusion-Limited Aggregation is suggested for analytical calculating of fractal dimension and simple numerical simulation of some fractal properties. Another areas of applications is a mathematical epidemiology and related topics: the modeling of the contact infections spread and anomalous diffusion in complex networks.
One direction of research relates to the mathematical modeling of self-sustained oscillations and traveling waves in biophysical condensed matter, particularly the oscillating glycolitic reaction and wave pattern generated due to this process. Additionally, the development of algorithms for estimation of the phase transitions' parameters is under consideration. The processes of the surfactants' micellar aggregation and the critical point for non-polar liquids are the areas of application.
A new method for a multiscale computing of the Hankel transform is proposed. It is based on the reducing of this integral transform to the exact analytical representation as the series of the Bessel and Struve functions multiplied by the wavelet coefficients of the input function. The Haar and the wavelets based on the B-splines are considered.
The research of the ferroliquids has been completed in the collaboration with experimenters from the Laboratory of Magnetic Liquids of Kursk State Technical University. The main problems are connected with the description of the ferroliquid sealants' oscillations and the waves propagating in the closed tubes filled by ferroliquid. The second part of interests in magnetoacoustics was connected with the analytical solutions of 2D magnetoelastic equations describing the waves propagating through the solid conducting media.