V.T. Volkov

Development of the asymptotic method of differential inequalities for investigation of periodic contrast structures in reaction-diffusion equations

The asymptotical method of differential inequalities is developed for a new class of periodic problems of reaction-diffusion type. The problem of the existence and Lyapunov stability of periodic solutions with internal transient layers in the case of balanced nonlinearity is studied.

Analytic-Numerical Approach to Solving Singularly Perturbed Parabolic Equations with the Use of Dynamic Adapted Meshes

The main objective of the paper is to present a new analytic-numerical approach toxa0singularly perturbed reaction-diffusion-advection models with solutions containing moving interior layersxa0(fronts). We describe some methods to generate the dynamic adapted meshes for an efficient numericalxa0solution of such problems. It is based on a priori information about the moving front propertiesxa0provided by the asymptotic analysis. In particular, for the mesh construction we take into accountxa0a priori asymptotic evaluation of the location and speed of the moving front, its width and structure.xa0Our algorithms significantly reduce the CPU time and enhance the stability of the numerical processxa0compared with classical approaches. The article is published in the authors’ wording.

On the formation of sharp transition layers in two-dimensional reaction-diffusion models

For a singularly perturbed parabolic equation in two dimensions, the formation of a solution with a sharp transition layer from a sufficiently general initial function is considered. An asymptotic analysis is used to estimate the time required for the formation of a contrast structure. Numerical results are presented.

Front formation and dynamics in one reaction-diffusion-advection model

The paper is concerned with the asymptotic behavior of a solution with an inner transition layer (front) in the reaction-diffusion-advection mathematical model to describe an in-situ combustion process.