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On the integrability and exact solutions of the nonlinear diffusion equation with a nonlinear source

Published online by Cambridge University Press:  17 February 2009

K. Vijayakumar
K. Vijayakumar
Affiliation:
Dept of Mathematics, Panjab University, Chandigarh 160014, India.
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Abstract

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The generalized diffusion equation with a nonlinear source term which encompasses the Fisher, Newell-Whitehead and Fitzhugh-Nagumo equations as particular forms and appears in a wide variety of physical and engineering applications has been analysed for its generalized symmetries (isovectors) via the isovector approach. This yields a new and exact solution to the generalized diffusion equation. Further applications of group theoretic techniques on the travelling wave reductions of the Fisher, Newell-Whitehead and Fitzhugh-Nagumo equations result in integrability conditions and Lie vector fields for these equations. The Lie group of transformations obtained from the exponential vector fields reduces these equations in generalized form to a standard second-order differential equation of nonlinear type, which for particular cases become the Weierstrass and Jacobi elliptic equations. A particular solution to the generalized case yields the exact solutions that have been obtained through different techniques. The group-theoretic integrability relations of the Fisher and Newell-Whitehead equations have been cross-checked through Painlevé analysis, which yields a new solution to the Fisher equation in a complex-valued function form.

Type
Research Article
Information
The ANZIAM Journal , Volume 39 , Issue 4 , April 1998 , pp. 513 - 527
Copyright
Copyright © Australian Mathematical Society 1998

References

[1]Ablowitz, M. J. and Clarkson, P. A., Solitons, Nonlinear Evolution Equations and Inverse Scattering (Cambridge University Press, Cambridge, 1991).CrossRefGoogle Scholar
[2]Ames, W. F. and Rogers, C., Nonlinear Boundary value Problems in Science and Engineering (Academic Press, New York, 1989).Google Scholar
[3]Arrigo, D. J., Hill, J. M. and Broadbridge, P., “Nonclassical symmetry reductions of the linear diffusion equation with a nonlinear source”, IMA J. Appl. Math. 52 (1994), 1–24.CrossRefGoogle Scholar
[4]Bhutani, O. P. and Vijayakumar, K., “On the isogroups of the generalised diffusion equation”, Int. J. Eng. Sci. 28 (1990) 375–387.CrossRefGoogle Scholar
[5]Bluman, and Kumei, G. W., Symmetries and Differential Equations (Springer, New York, 1989).CrossRefGoogle Scholar
[6]Cariello, F., and Tabor, M., “Painlevé expansions for nonintegrable evolution equations”, Physica D 39 (1989) 77–94.Google Scholar
[7]Dorodnitsyn, V. A., “On invariant solutions of the equation of non-linear heat conduction with a source”, USSR Comp. Math. Phys. 22 (1982) 115–122.CrossRefGoogle Scholar
[8]Edelen, D. G. B., Applied Exterior Calculus (Willey-Interscience, New York, 1985).Google Scholar
[9]Edelen, D. G. B. and Wang, J., Transformation Methods for Nonlinear Partial Differential Equations (World Scientific, Singapore, 1992).CrossRefGoogle Scholar
[10]Hereman, W. and Takaoka, M., “Solitary wave solutions of nonlinear evolution and wave equations using a direct method and MACSYMA”, J. Phys. A: Math Gen. 23 (1990) 4805–4822.CrossRefGoogle Scholar
[11]Hill, J. M., Differential Equations and Group Methods for Scientists and Engineers (CRC Press, London, 1992).Google Scholar
[12]Nucci, M. C. and Clarkson, P. A., “The nonclassical method is more general than the direct method for symmetry reductions. An example of the Fitzhugh-Nagumo equation”, Phys. Lett. A 164 (1992) 49–56.CrossRefGoogle Scholar
[13]Olver, P. J., Applications of Lie Groups to Differential Equations (Springer (GTM(106)), Berlin, 1986).CrossRefGoogle Scholar
[14]Ovsyannikov, , Group Analysis of Differential Equations (Trans. Ames, W. F.) (Academic Press, New York, 1982).Google Scholar
[15]Ramani, A., Grammaticos, B. and Bountis, T., “The Painleve property and singularity analysis of integrable and non-integrable systems”, Physics Reports 180 (1989) 159–245.CrossRefGoogle Scholar
[16]Sachdeva, P. L., Nonlinear Diffusive Waves (Cambridge University Press, Cambridge, 1987).CrossRefGoogle Scholar
[17]Wang, X. Y. “Exact and explicit solitary wave solutions for the generalised Fisher equation”, Phys. Lett. A 131 (1988) 277–279.CrossRefGoogle Scholar
[18]Wilhelmson, H., “Explosive instabilities of reaction-diffusion equations”, Physical Review A 36 (1987) 965–966.CrossRefGoogle Scholar