Asymptotic step-like solutions of the singularly perturbed Burgers equation

Abstract

This paper deals with a problem of asymptotic step-like solutions of the Burgers equation with variable coefficients and a small parameter. By means of the nonlinear Wentzel–Kramers–Brillouin method, the algorithm of constructing these asymptotic solutions is proposed and statements on justification of the algorithm are proved. The obtained results are illustrated by an example, for which the first asymptotic step-like approximation is explicitly found. The asymptotic solution is global and has a form of the shock wave type function. There are also given graphs of these approximate solutions for certain numerical parameters.