In this paper, we consider an initial‐value problem for Burgers' equation with variable coefficients $${u}_{t}+\Phi \left(t\right)\phantom{\rule{0.16em}{0ex}}u{u}_{x}=\Psi \left(t\right)\phantom{\rule{0.16em}{0ex}}{u}_{xx},\phantom{\rule{1em}{0ex}}-\infty <x<\infty ,\phantom{\rule{1em}{0ex}}t>0,$$ where

*x*and*t*represent dimensionless distance and time, respectively, and $\Psi \left(t\right)$, $\Phi \left(t\right)$ are given functions of*t*. In particular, we consider the case when the initial data have algebraic decay as $\left|x\right|\to \infty $, with $u(x,t)\to {u}_{+}$ as $x\to \infty $ and $u(x,t)\to {u}_{-}$ as $x\to -\infty $. The constant states ${u}_{+}$ and ${u}_{-}\phantom{\rule{0.16em}{0ex}}(\ne {u}_{+})$ are problem parameters. Two specific initial‐value problems are considered. In initial‐value problem 1 we consider the case when $\Phi \left(t\right)={e}^{t}$ and $\Psi \left(t\right)=1$, while in initial‐value problem 2 we consider the case when $\Phi \left(t\right)=1$ and $\Psi \left(t\right)={e}^{t}$. The method of matched asymptotic coordinate expansions is used to obtain the large‐*t*asymptotic structure of the solution to both initial‐value problems over all parameter values.No References

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**Document Type:** Research Article

Publication date: February 1, 2016