A quarter-plane problem for the modified Burgers’ equation

Abstract

In this paper, we address an initial-boundary value problem for the modified Burgers’ equation. The normalized modified Burgers’ equation considered is given by ut + up ux − uxx = 0, 0 < x < ∞, t > 0, where x and t represent dimensionless distance and time, respectively, and p (>1) is a parameter. In particular, we consider the case when the initial and boundary conditions are given by u(x, 0) = ui for 0 < x < ∞ and u(0, t) = ub for t > 0, respectively. We initially focus attention on the case when ui = 0 and ub > 0. In this case, the method of matched asymptotic coordinate expansions is used to obtain the complete large-t asymptotic structure of the solution to this problem, which exhibits the formation of a permanent form travelling wave solution propagating with speed \documentclass[12pt]{minimal}\begin{document}$v=\frac{u_{b}^{p}}{p+1}\; (>0)$\end{document}v=ubpp+1(>0) and connecting u = 0 ahead of the wave-front to u = ub at the rear of the wave. Further, the asymptotic correction to the propagation speed is of \documentclass[12pt]{minimal}\begin{document}$O\left(t^{-3/2} \exp \left(-\frac{v^2}{4}t\right)\right)$\end{document}Ot−3/2exp−v24t as t → ∞, and the rate of convergence of the solution of the initial-boundary value problem to the travelling wave is \documentclass[12pt]{minimal}\begin{document}$O\left(t^{-3/2} \exp \left(-\frac{v^2}{4}t\right)\right)$\end{document}Ot−3/2exp−v24t as t → ∞. We conclude the paper with a discussion of the structure of the large-time solution to the initial-boundary value problem for general values of ub and ui (excluding the trivial case when ui = ub).