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Journal of Mathematical Physics / Volume 28 / Issue 5

J. Math. Phys. 28, 997 (1987); http://dx.doi.org/10.1063/1.527520 (8 pages)

Generalized Burgers equations and Euler–Painlevé transcendents. II

P. L. Sachdev and K. R. C. Nair

Department of Applied Mathematics, Indian Institute of Science, Bangalore‐560012, India

(Received 25 April 1986; accepted 18 November 1986)

It was proposed earlier [P. L. Sachdev, K. R. C. Nair, and V. G. Tikekar, J. Math. Phys. 27, 1506 (1986)] that the Euler–Painlevé equation yy″+ay′2+ f(x)yy′+g(x) y2+by′+c=0 represents the generalized Burgers equations (GBE’s) in the same manner as Painlevé equations do the KdV type. The GBE was treated with a damping term in some detail. In this paper another GBE ut+uaux+Ju/2t =(gd/2)uxx (the nonplanar Burgers equation) is considered. It is found that its self‐similar form is again governed by the Euler–Painlevé equation. The ranges of the parameter α for which solutions of the connection problem to the self‐similar equation exist are obtained numerically and confirmed via some integral relations derived from the ODE’s. Special exact analytic solutions for the nonplanar Burgers equation are also obtained. These generalize the well‐known single hump solutions for the Burgers equation to other geometries J=1,2; the nonlinear convection term, however, is not quadratic in these cases. This study fortifies the conjecture regarding the importance of the Euler–Painlevé equation with respect to GBE’s.

KEYWORDS and PACS

Keywords

BURGERS EQUATION, DAMPING, ANALYTICAL SOLUTION, NUMERICAL SOLUTION, NONLINEAR PROBLEMS, CONVECTION, DIFFUSION, SYMMETRY, STRESS−STRAIN RELATION

PACS

  • 02.30.Hq

    Ordinary differential equations

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.527520

PUBLICATION DATA

ISSN

0022-2488 (print)  
1089-7658 (online)

For access to fully linked references, you need to log in.
    A. A. Karabutov and O. V. Rudenko, Sov. Phys. Tech. Phys. 20, 920 (1975).



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