### 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 u t + u p u x − u xx = 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) = u i for 0 < x < ∞ and u(0, t) = u b for t > 0, respectively. We initially focus attention on the case when u i = 0 and u b > 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 and connecting u = 0 ahead of the wave-front to u = u b at the rear of the wave. Further, the asymptotic correction to the propagation speed is of as t → ∞, and the rate of convergence of the solution of the initial-boundary value problem to the travelling wave is 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 u b and u i (excluding the trivial case when u i = u b ).

Received 11 August 2012
Accepted 26 August 2013
Published online 12 September 2013

Article outline:

I. INTRODUCTION
II. ASYMPTOTIC SOLUTION OF AS ** ***t → 0*
III. ASYMPTOTIC SOLUTION OF AS ** ***x → ∞*
IV. ASYMPTOTIC SOLUTION OF AS ** ***t → ∞*
V. NUMERICAL SOLUTION OF THE INITIAL-BOUNDARY VALUE PROBLEM
VI. DISCUSSION
A. ** ***p* is an even integer
B. ** ***p* is an odd integer
VII. CONCLUSION

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