A quarter-plane problem for the modified Burgers’ equation

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 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 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 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 Ot−3/2exp−v24t as t → ∞, and the rate of convergence of the solution of the i...