Asymptotic Approximation of the Solution of the Reaction-Diffusion-Advection Equation with a Nonlinear Advective Term

We consider a solution in a moving front form of the initial-boundary value problem for a singularly perturbed reaction-diffusion equation in a band with periodic conditions in one of the variables. Interest in solutions of the front type is associated with combustion problems or nonlinear acoustic waves. In the domain of the function which describes the moving front there is a subdomain where the function has a large gradient. This subdomain is called the internal transition layer. Boundary value problems with internal transition layers have a natural small parameter that is equal to the ratio of the transition layer width to the width of the region under consideration. The presence of a small parameter at the highest spatial derivative makes the problem singularly perturbed. The numerical solution of such problems meets certain difficulties connected with the choice of grids and initial conditions. To solve these problems the use of analytical methods is especially successful. Asymptotic analysis which uses Vasilieva’s algorithm was carried out in the paper. That made it possible to obtain an asymptotic approximation of the solution, which can be used as an initial condition for a numerical algorithm. We also determined the conditions for the existence of a front type solution. In addition, the analytical methods used in the paper make it possible to obtain in an explicit form the front motion equation approximation. This information can be used to develop mathematical models or numerical algorithms for solving boundary value problems for the reaction-diffusion-advection type equations.

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