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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 ut+Φ(t)uux=Ψ(t)uxx,<x<,t>0,where x and t represent dimensionless distance and time, respectively, and Ψ(t), Φ(t) are given functions of t. In particular, we consider the case when the initial data have algebraic decay as |x|, with u(x,t)u+ as x and u(x,t)u as x. The constant states u+ and u(u+) are problem parameters. Two specific initial‐value problems are considered. In initial‐value problem 1 we consider the case when Φ(t)=et and Ψ(t)=1, while in initial‐value problem 2 we consider the case when Φ(t)=1 and Ψ(t)=et. 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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