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On certain new and exact solutions of the Emden-Fowler equation and Emden equation via invariant variational principles and group invariance

  • O. P. Bhutani (a1) and K. Vijayakumar (a2)
Abstract

After formulating the alternate potential principle for the nonlinear differential equation corresponding to the generalised Emden-Fowler equation, the invariance identities of Rund [14] involving the Lagrangian and the generators of the infinitesimal Lie group are used for writing down the first integrals of the said equation via the Noether theorem. Further, for physical realisable forms of the parameters involved and through repeated application of invariance under the transformation obtained, a number of exact solutions are arrived at both for the Emden-Fowler equation and classical Emden equations. A comparative study with Bluman-Cole and scale-invariant techniques reveals quite a number of remarkable features of the techniques used here.

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      On certain new and exact solutions of the Emden-Fowler equation and Emden equation via invariant variational principles and group invariance
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References
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[1] Bluman, G. W. and Cole, J. D., Similarity methods for differential equations (Springer-Verlag, 1974).
[2] Bhutani, O. P. and Mital, P., “On the first integral of the nonlinear shallow-membrane equation via Noether's theorem”, Int. J. Engng. Sci. 23 (1985) 353357.
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[4] Bhutani, O. P. and Mital, P., J. Meteor. Soc. Japan, Ser. II 64 (1986) 593598.
[5] Bhutani, O. P., Vijayakumar, K., Mital, P. and Chandrasekharan, G., “On invariant solutions of the generalised Korteweg-de-Vries equation in a variable medium-I”, Int. J. Engng. Sci. 27 (1989) 921929.
[6] Bhutani, O. P., Vijayakumar, K., Mital, P. and Chandrasekaran, G., “On invariant solutions of the generalised Korteweg-de Vries-Burger type equations-II”, Int. J. Engng. Sci. 27 (1989) 931941.
[7] Chan, C. Y. and Hon, Y. C., “A constructive solution for a generalised Thomas-Fermi theory of ionised atoms”, Quart. J. Appl. Math. 45 (1987) 591599.
[8] Chandrasekhar, S., Principles of stellar dynamics (University of Chicago Press, Chicago, 1942).
[9] Kamke, E., Differential gleichungen losungs method en und losungen (Chelsa Publishing Company, 1948).
[10] Logan, J. D., Invariant variational principles (Academic Press, 1977).
[11] Murphy, G. M., Ordinary differential equations and their solutions (D. Van Nostrand Company Inc., 1960).
[12] Olver, P. J., Applications of Lie Groups to differential equations (Springer-Verlag, 1986).
[13] Rosenau, P., “A note on integration of the Emden-Fowler equation”, Int. J. Nonlinear Mechanics, 19 (1984) 303308.
[14] Rund, H., Hamilton-Jacobi theory in the calculus of variations (Princeton, New Jersey, 1966).
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The ANZIAM Journal
  • ISSN: 1446-1811
  • EISSN: 1446-8735
  • URL: /core/journals/anziam-journal
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