Main current projects:
1. The Continuous Wavelet Transform its applications to biophysical problems
The principal activity comprises both the development of new mathematical and computational tools aimed to improve effectiveness of the continuous wavelet transform (preferably with the Morlet wavelet and Morlet-like complex wavelet functions) in problems of local spectral analysis, as well as an application of the developed methods to practical problems of biophysical signal and image processing (in the fields of neuroscience and vascular dynamics).
2. Fluctuation Theory-based Tait-like Equation of State.
A development of the isothermal equation of state for liquids under high pressures, which has a form close to the classic Tait equation but allows for determining its parameters using thermodynamic data measured (or calculated) along the coexistence curve (or under normal pressure) only, as well as its application to predictive calculation of thermodynamic properties (the density, the isothermal compressibility and the isobaric expansivity, the speed of sound) of organic liquids.
3. Computational problems of dynamical and stochastic processes.
Determining parameters and simulations for the problems related to anomalous diffusion, mathematical epidemiology and neuroscience.
Evaluation of the Hankel transforms using the Discrete Wavelet Transform.
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.
Magnetoacoustics of ferroliquids and solid conductors.
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.
Mean-field description of Kinetic Aggregate’s Growth and Disease Spread .
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.
Dynamics of glycolitic self-sustained oscillations and traveling waves
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 under influx and temperature influence within the concept of the generalized Rayleigh equation.