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The Large‐Time Solution of Burgers' Equation with Time‐ Dependent Coefficients. II. Algebraic Coefficients
Author(s) -
Sulaiman F. B. Ali,
Leach J. A.,
Needham D. J.
Publication year - 2016
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/sapm.12130
Subject(s) - dimensionless quantity , mathematics , algebraic number , burgers' equation , constant (computer programming) , initial value problem , mathematical analysis , variable (mathematics) , focus (optics) , algebraic solution , partial differential equation , physics , differential equation , thermodynamics , differential algebraic equation , ordinary differential equation , computer science , optics , programming language
In this paper, we consider an initial‐value problem for Burgers' equation with variable coefficientsu t + Φ ( t )u u x = Ψ ( t )u x x , − ∞ < x < ∞ , t > 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 + andu −( ≠ u + )are problem parameters. We focus attention on the case when Φ ( t ) = t δ(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.