Large-time asymptotics to solutions of a generalized Burgers equation with linear damping on half-line

Abstract

In this article, we investigate an initial-boundary value problem posed for generalized Burgers equation (GBE) with linear damping via the method of matched asymptotic expansions. Asymptotic solutions are constructed for different sub-regions of the domain $x > 0,~ t > 0$. A special solution is derived, and it describes the large-time asymptotic behavior of the solutions of the GBE for certain parametric ranges. We also observe that a stationary solution of the GBE describes the large-time behavior of solutions for certain parametric ranges. The existence and uniqueness of the relevant stationary solution are proved using a shooting argument. A numerical study is presented comparing the numerical solutions (obtained by the method of lines) with the asymptotic solutions constructed.