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# The Large‐Time Solution of Burgers' Equation with Time‐Dependent Coefficients. I. The Coefficients Are Exponential Functions

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In this paper, we consider an initial‐value problem for Burgers' equation with variable coefficients ${u}_{t}+\Phi \left(t\right)\phantom{\rule{0.16em}{0ex}}u{u}_{x}=\Psi \left(t\right)\phantom{\rule{0.16em}{0ex}}{u}_{xx},\phantom{\rule{1em}{0ex}}-\infty 0,$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 $|x|\to \infty$, with $u\left(x,t\right)\to {u}_{+}$ as $x\to \infty$ and $u\left(x,t\right)\to {u}_{-}$ as $x\to -\infty$. The constant states ${u}_{+}$ and ${u}_{-}\phantom{\rule{0.16em}{0ex}}\left(\ne {u}_{+}\right)$ 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.
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Document Type: Research Article

Publication date: February 1, 2016

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