Moving Front Solution of the Reaction-Diffusion Problem

Abstract

In this paper, we study the moving front solution of the reaction-diffusion initialboundary value problem with a small diffusion coefficient. Problems in such statements can be used to model physical processes associated with the propagation of autowave fronts, in particular, in biophysics or in combustion. The moving front solution is a function the distinctive feature of which is the presence in the domain of its definition of a subdomain where the function has a large gradient. This subdomain is called an internal transition layer. In the nonstationary case, the position of the transition layer varies with time which, as it is well known, complicates the numerical solution of the problem as well as the justification of the correctness of numerical calculations. In this case the analytical method is an essential component of the study. In the paper, asymptotic methods are applied for analytical investigation of the solution of the problem posed. In particular, an asymptotic approximation of the solution as an expansion in powers of a small parameter is constructed by the use of the Vasil’eva algorithm and the existence theorem is carried out using the asymptotic method of differential inequalities. The methods used also make it possible to obtain an equation describing the motion of the front. For this purpose a transition to local coordinates takes place in the region of the front localization. In the present paper, in comparison with earlier publications dealing with two-dimensional problems with internal transition layers the transition to local coordinates in the vicinity of the front has been modified, that led to the simplification of the algorithm of determining the equation of the curve motion.

Keywords

• reaction-diffusion problem
• two-dimensional moving front
• asymptotic representation
• small parameter
• asymptotic method of differential inequalities