Asymptotic solutions of a generalized Burgers equation
Authors:
V. Vanaja and P. L. Sachdev
Journal:
Quart. Appl. Math. 50 (1992), 627-640
MSC:
Primary 35Q53; Secondary 76R99, 76S05
DOI:
https://doi.org/10.1090/qam/1193660
MathSciNet review:
MR1193660
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Abstract: The travelling wave solutions of the generalized Burgers equation \[ \frac {{\partial u}}{{\partial t}} = \frac {\partial }{{\partial x}}\left [ {D\left ( u \right ) \frac {{\partial u}}{{\partial x}}} \right ] - \frac {\partial }{{\partial x}}\left [ {K\left ( u \right )} \right ]\] are related to the solution of the initial boundary value problems for the same equation, subject to initial boundary conditions relevant to the physical problem of infiltration of moisture into a homogeneous soil. The theoretical prediction of the emergence of the travelling wave solutions as intermediate asymptotics is confirmed by numerical solutions of the problem for some specific choices of the functions $D\left ( u \right )$ and $K\left ( u \right )$.
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L. A. Peletier, Asymptotic behaviour of temperature profiles of a class of non-linear heat conduction problems, Quart. J. Mech. Appl. Math. 23, 441–447 (1970)
J. R. Philip, Flow in porous media, Ann. Rev. Fluid Mech. 2, 177–204 (1970)
P. L. Sachdev, Nonlinear Diffusive Waves, Cambridge Univ. Press, 1987
J. F. Scott, The longtime asymptotics of solution to the generalized Burgers equation, Proc. Roy. Soc. London Ser. A 373, 443–456 (1981)
J. Serrin, Asymptotic behaviour of velocity profiles in the Prandtl boundary layer theory, Proc. Roy. Soc. London. Ser. A 299, 491–507 (1967)
D. Swartzendruber, The flow of water in unsaturated soils, Flow Through Porous Media (R. J. M. Dewiest, ed.), Academic Press, New York; 1969, pp. 215–292
C. J. Van Duyn and L. A. Peletier, Asymptotic behaviour of solutions of a non-linear diffusion equation, Arch. Rational Mech. Anal. 65, 363–377 (1977)
C. J. Van Duyn and L. A. Peletier, A class of similarity solutions of the nonlinear diffusion equation, Nonlinear Anal., Theory, Methods Appl. 1, 223–233 (1977)
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Article copyright:
© Copyright 1992
American Mathematical Society