Datsko B. Mathematical modeling of selforganization phenomena by evolutionary nonlinear systems of equations with integer and fractional derivatives

The thesis is devoted to mathematical modeling of self-organization phenomena in evolutionary nonlinear systems. The work presents mathematical models and methods that make it possible to obtain new results in nonlinear differential equations with integer and fractional derivatives and theoretically explain and explore the variety of nonlinear nonequilibrium phenomena in physical, chemical and biological systems. In the study of nonlinear systems with classical derivatives the main attention is paid to the development of applied mathematical models. In particular, the matematical models for nonequilibrium electron-hole plasma are formulated. The analysis of possible instabilities and conditions for stratification of concentration of electric current carrier, its temperature and electric field is conducted. It is shown that the mathematical models are described by reaction-diffusion equations and that makes it possible to explore stratification phenomena in semiconductor structures. The new stationary spatially inhomogeneous solutions of autosoliton type of giant amplitude are studied, which made it possible to expand the understanding of attractors in the stable nonlinear nonequilibrium systems, and formulate the theoretical basis for the processes of microplasma instability. Based on reaction-diffusion mathematical models the formation of spatially inhomogeneous stationary and running pulses in semiconductor and low-temperature gas plasma is studied. The mathematical modeling of the self-organization phenomena during laser irradiation of semiconductor materials is performed. Based on mathematical models of reaction-diffusion mechanism of local melting, and based on the free boundary problem the process of complex structure formation during melting is studied. A method for numerical solution of quasi-stationary Stefan problem is developed. A new mathematical model of the heat propagation in living tissues described by the system of differential equations of reaction-diffusion, taking into account the fractal structure of the vascular network, is considered. The mathematical models with fractional derivatives and their applications are considered. The general principles of a mathematical model of spatial and temporal operators of fractional order in media with anomalous diffusion and heriditary attribute are described. The theory of linear stability for nonlinear evolution systems with ordinary and partial fractional derivatives is developed. The theoretical analysis of possible instabilities in reaction-diffusion systems with derivatives of rational order and obtained conditions for various types of instabilities depending on system parameters and the fractional order derivative is performed. A new type of bifurcations of spatially homogeneous solutions in reaction-diffusion systems with fractional derivatives is revealed. The new types of nonlinear solutions for the basic reaction-diffusion systems with fractional derivatives and new types of limit cycles for fractional systems of ordinary differential equations of classical nonlinearity are obtained. The characteristic equation for reaction-diffusion systems with rational derivatives of different order is studied. The dynamics of monostable and bistable systems with anomalous diffusion is performed. The main types of nonlinear dynamic systems with fractional derivatives are studied. The numerical and semi analytical methods for nonlinear systems for evolutionary equations with spatial and time fractional derivatives are proposed. The self-organization phenomena considered in this thesis are general in nature and can be used to investigate a wide class of nonlinear active systems.