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Local transformations between some nonlinear diffusion equations

Published online by Cambridge University Press:  17 February 2009

J. R. King
J. R. King
Affiliation:
Dept. of Theoretical Mechanics, University of Nottingham, Nottingham, NG7 2RD, U.K.
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Abstract

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We derive local transformations mapping radially symmetric nonlinear diffusion equations with power law or exponential diffusivities into themselves or into other equations of a similar form. Both discrete and continuous transformations are considered. For the cases in which a continuous transformation exists, many additional forms of group-invariant solution may be constructed; some of these solutions may be written in closed form. Related invariance properties of some multidimensional diffusion equations are also exploited.

Type
Research Article
Information
The ANZIAM Journal , Volume 33 , Issue 3 , January 1992 , pp. 321 - 349
Copyright
Copyright © Australian Mathematical Society 1992

References

[1] Ames, W. F., Nonlinear partial differential equations in engineering. Volume 2. (Academic Press, New York, 1972).Google Scholar
[2] Branson, T. P. and Steeb, W.-H., “Symmetries of nonlinear diffusion equations”, J. Phys. A 16 (1983) 469–.472.CrossRefGoogle Scholar
[3] Bluman, G. W. and Cole, J. D., Similarity methods for differential equations. (Springer-Verlag, New York, 1974).CrossRefGoogle Scholar
[4] Clarkson, P. A. and Kruskal, M. D., “New similarity reductions of the Boussinesq equation”, J. Math. Phys. 30 (1989), 2201–2213.CrossRefGoogle Scholar
[5] Dorodnitsyn, V. A., Knyazeva, I. V. and Svirshchevskii, S. R., “Group properties of the heat-conduction equation with a source in the two- and three-dimensional cases”, Differential Equations 19 (1983) 901–908.Google Scholar
[6] Fujita, H., “The exact pattern of concentration-dependent diffusion in a semi-infinite medium, part I”, Text. Res. J. 22 (1952) 757–760.CrossRefGoogle Scholar
[7] Galaktionov, V. A., Dorodnitsyn, V. A., Elenin, G. G., Kurdyumov, S. P. and Samarskii, A. A., “A quasilinear heat equation with a source: peaking, localization, symmetry, exact solutions, asymptotics, structures”, J. Sov. Math. 41 (1988) 1222–1292.CrossRefGoogle Scholar
[8] Hill, D. L. and Hill, J. M., “Similarity solutions for nonlinear diffusion—further exact solutions”, J. Eng. Math. 24 (1990), 109–124.CrossRefGoogle Scholar
[9] Hill, J. M., Solution of differential equations by means of one-parameter groups. (Pitman, Boston, 1982).Google Scholar
[10] Hill, J. M., “Similarity solutions for nonlinear diffusion—a new integration procedure”, J. Eng. Math. 23 (1989) 141–155.CrossRefGoogle Scholar
[11] Kersten, P. H. M. and Gragert, P. K. H., “The Lie algebra of infinitesimal symmetries of nonlinear diffusion equations”, J. Phys. A 16 (1983) L685–L688.CrossRefGoogle Scholar
[12] King, J. R., “Exact similarity solutions to some nonlinear diffusion equations”, J. Phys. A 23 (1990) 3681–3697.CrossRefGoogle Scholar
[13] King, J. R., “Some non-local transformations between nonlinear diffusion equations”, J. Phys. A 23 (1990) 5441–5464.CrossRefGoogle Scholar
[14] Lacey, A. A., Ockendon, J. R. and Tayler, A. B., “‘Waiting-time’ solutions of a nonlinear diffusion equation”, SIAM J. Appl. Math. 42 (1982) 1252–1264.CrossRefGoogle Scholar
[15] Nariboli, G. A., “Self-similar solutions of some nonlinear equations”, Appl. Sci. Res. 22 (1970) 449–461.CrossRefGoogle Scholar
[16] Ovsiannikov, L. V., “Group relations of the equation of nonlinear conductivity”, Dokl. Akad. Nauk SSSR 125 (1959) 492–495.Google Scholar
[17] Stuart, J. T., “On finite amplitude oscillations in laminar mixing layers”, J. Fluid Mech. 29 (1967) 417–440.CrossRefGoogle Scholar
[18] Qing-jian, Yang, Xing-Zhen, Chen, Ke-jie, Zheng and Zhu-Liang, Pan, “Similarity solutions to three-dimensional nonlinear diffusion equations”, J. Phys. A 23 (1990) 265–269.CrossRefGoogle Scholar