Moving Front Solution of the Reaction-Diffusion Problem

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.

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