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Large time asymptotics for solutions to a nonhomogeneous Burgers equation.pdf

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Large time asymptotics for solutions to a nonhomogeneous Burgers equation.pdf

Appl. Math. Mech. -Engl. Ed. 31(9), 1189–1196 (2010) Applied Mathematics DOI 10.1007/s10483-010-1352-9 and Mechanics c Shanghai University and Springer-Verlag Berlin Heidelberg 2010 (English Edition) Large time asymptotics for solutions to a nonhomogeneous Burgers equation∗ S. R. CHIDELLA, M. K. YADAV (Department of Mathematics, Indian Institute of Technology Madras, Chennai 600036, India) (Communicated by Xing-ming GUO) Abstract In this article, the solutions to a nonhomogeneous Burgers equation subject to bounded and compactly supported initial proﬁles are constructed. In an interesting study, Kloosterziel (Journal of Engineering Mathematics 24, 213–236 (1990)) represented a solution to an initial value problem (IVP) for the heat equation, with an initial data in a class of rapidly decaying functions, as a series of self-similar solutions to the heat equation. This approach quickly revealed the large time behaviour for the solution to the IVP. Inspired by Kloosterziel’s approach, the solution to the nonhomogeneous Burgers equation is expressed in terms of the self-similar solutions to the heat equation. The large time behaviour of the solutions to the nonhomogeneous Burgers equation is obtained. Key words nonhomogeneous Burgers equation, Hermite polynomials, self-similar solutions Chinese Library Classiﬁcation O175.2 2000 Mathematics Subject Classiﬁcation 35B40, 35C06, 35K15, 35K55 1 Introduction Nonhomogene