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On self-similar solutions of Barenblatt's nonlinear filtration equation

Published online by Cambridge University Press:  26 September 2008

Julian D. Cole  and
Barbara A. Wagner
Julian D. Cole
Affiliation:
Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180, USA
Barbara A. Wagner
Affiliation:
Department of Mathematics, Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, Newark, NJ 07102, USA
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Abstract

We derive the asymptotic form of the self-similar solutions of the second kind of the Cauchy problem for Barenblatt's nonlinear filtration equation by perturbing the Lie group of the underlying linear problem. We also show that the decay rate, appearing in the similarity solutions, can be found by a simple inspection of the corresponding energy dissipation law.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 7 , Issue 2 , April 1996 , pp. 151 - 167
Copyright
Copyright © Cambridge University Press 1996

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References

[1] Barenblatt, G. I. & SivashinskII, G. I. 1969 Self-similar solutions of the second kind. PMM 33, 861–870.Google Scholar
[2] Goldenfeld, N., Martin, O., Oono, Y. & Liu, F. 1990 Anomalous dimensions and the renormalization group in a nonlinear diffusion process. Phys. Rev. Lett. 64, 1361–1364.CrossRefGoogle Scholar
[3] Bluman, G. W. & Cole, J. D. 1974 Similarity Methods for Differential Equations. Applied Mathematics series, 13. Springer-Verlag.CrossRefGoogle Scholar
[4] Bluman, G. W. & Kumei, S. 1989 Symmetries and Differential Equation. Applied Mathematics series, 81. Springer-Verlag.CrossRefGoogle Scholar
[5] Kamenomostskaya, S. L. 1957 On a particular problem of filtration theory. Dokl. Akad. Nauk SSSR 116(1).Google Scholar
[6] Kamin, S., Peletier, L. A. & Vasquez, J. L. 1991 On the Barenblatt Equation of Elasto-Plastic Filtration. IMA Preprint Series, No. 817.Google Scholar
[7] Aronson, D. G. & Vasquez, J. L. 1991 Anomalous Exponents in Nonlinear Diffusion. IMA Preprint Series.Google Scholar
[8] Barenblatt, G. I. 1993 Intermediate asymptotics, scaling laws and renormalization group in continuum mechanics. Meccanica 28, 177–183.CrossRefGoogle Scholar
[9] Auramowitz, M. & Stegun, I. A. 1972 Handbook of Mathematical functions. Dover Publications, New York.Google Scholar