Abstract
Complex systems described by nonlinear partial differential equations of parabolic type or large-scale systems of ordinary differential equations with switching right-side are considered. The reduction method is applied to the corresponding problem for the system of ordinary differential equations without switching. A parametric family of time-periodic and anisotropic on spatial variables exact solutions of the reaction-diffusion system is constructed. The stability conditions of a large-scale system with switching are obtained, which consist in checking the stability of the reduced system without switching. The conditions for the existence of the first integrals for the reduced system of ordinary differential equations expressed by a combination of power and logarithmic functions are found. For the cases of two-dimensional and three-dimensional reduced systems, these conditions are written in the form of polynomial equations relating the system parameters.
Highlights
Diffusion processes in multicomponent medium with interacting components are described by systems of nonlinear partial differential equations of parabolic type (PDE PT) [1]
Since non-linear PDE systems are complex objects to study, the reduction method is usually applied to systems of ordinary differential equations (ODE) to construct exact solutions [1]
We consider a reaction-diffusion system modeled by PDE PT with power nonlinearities characterizing the reaction of the mixture components
Summary
Diffusion processes in multicomponent medium with interacting components are described by systems of nonlinear partial differential equations of parabolic type (PDE PT) [1]. Since non-linear PDE systems are complex objects to study, the reduction method is usually applied to systems of ordinary differential equations (ODE) to construct exact solutions [1]. Example 2.1 Consider the system like (2.1) of four equations in the case of four spatial coordinates This system has the following parametric family of exact, periodic in time and anisotropic in spatial variables solutions: uk(t, x) = W (x) + φk(t) , k = 1, 4, where. Questions of existence and construction of periodic solutions of reaction-diffusion systems are of interest for chemical technology and are studied in a number of papers [13], [14], [15]
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