The Large‐Time Solution of Burgers' Equation with Time‐ Dependent Coefficients. II. Algebraic Coefficients. Issue 3 (2nd June 2016)

• Record Type: Journal Article
• Authors: Sulaiman, F. B. Ali; Leach, J. A.; Needham, D. J.
• Is Part Of: Studies in applied mathematics. Volume 137:Issue 3(2016)
• Page Start: 273
• Page End: 305
• Publication Date: 2016-06-02
• Subjects: Mathematics -- Periodicals; Mathématiques -- Périodiques; Mathématique; Mathématique appliquée; Ressource Internet (Descripteur de forme); Périodique électronique (Descripteur de forme); 519
• DOI: 10.1111/sapm.12130
• Languages: English
• ISSNs: 0022-2526

Abstract

Abstract : In this paper, we consider an initial‐value problem for Burgers' equation with variable coefficients u 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 + and u − ( ≠ 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.