The Large-Time Solution of Burgers' Equation with Time- Dependent Coefficients. II. Algebraic Coefficients

In this paper, we consider an initial-value problem for Burgers' equation with variable coefficients ut+Φ(t)uux=Ψ(t)uxx,-∞0,where x and t represent dimensionless distance and time, respectively, while Ψ(t), Φ(t) are given continuous functions of t ( > 0). In particular, we consider the case when the initial data has 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. We focus attention on the case when Φ(t)=tδ (with δ>-1) and Ψ(t)=1. The method of matched asymptotic coordinate expansions is used to obtain the large-t asymptotic structure of the solution to the initial-value problem over all parameter values.