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On Substitution Principles in Ideal Magneto-Gasdynamics by Means of Lie Group Analysis

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Abstract

The equations governing the flow of an inviscid thermally non-conducting fluid of infinite electrical conductivity in the presence of a magnetic field and subject to no extraneous forces are considered within the framework of Lie group analysis. It is shown how to recover and extend some results, known in literature as substitution principles, by conveniently requiring the invariance of the basic governing equations under a one-parameter Lie group of point transformations. Moreover, a new substitution principle for the equations ruling the plane motion of a fluid with adiabatic index Γ = 2 subjected to a transverse magnetic field is given. Some applications of the results are also given.

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References

  1. Munk, M. and Prim, R., ‘On the multiplicity of steady gas flows having the same streamline pattern’, Proceedings of National Academy of Science U.S.A. 33, 1947, 137–143.

    Google Scholar 

  2. Prim, R., ‘Steady rotational flow of ideal gases’, Journal of Rational Mechanics and Analysis 1, 1952, 425–497.

    Google Scholar 

  3. Smith, P., ‘An extension of the substitution principle to certain unsteady gas flows’, Archives of Rational Mechanics and Analysis 15, 1964, 147–153.

    Google Scholar 

  4. Oliveri, F., ‘On the equations of ideal-gas dynamics with a separable equation of state: Lie group analysis and substitution principles’, International Journal of Non-Linear Mechanics 27, 1992, 773–784.

    Article  Google Scholar 

  5. Oliveri, F., ‘Galilean quasilinear systems of PDE's and the substitution principle’, in Nonlinear Hyperbolic Equations: Theory, and Computational Aspects, Notes on Numerical Fluid Mechanics, Vol. 43, A. Donato and F. Oliveri (eds.), Vieweg, Wien, 1993, pp. 457–464.

  6. Ames, W. F., Nonlinear Partial Differential Equations in Engineering, Vol. 2, Academic Press, New York, 1972.

    Google Scholar 

  7. Ovsiannikov, L. V., Group Analysis of Differential Equations, Academic Press, New York, 1982.

    Google Scholar 

  8. Ibragimov, N. H., Transformation Groups Applied to Mathematical Physics, D. Reidel, Dordrecht, 1985.

    Google Scholar 

  9. Olver P. J., Applications of Lie Groups to Differential Equations, Springer, New York, 1986.

    Google Scholar 

  10. Bluman, G. W. and Kumei, S., Symmetries and Differential Equations, Springer, New York, 1989.

    Google Scholar 

  11. Rogers, C. and Ames, W. F., Nonlinear Boundary Value Problem in Science and Engineering, Academic Press, New York, 1989.

    Google Scholar 

  12. Ibragimov, N. H., Handbook of Lie Group Analysis of Differential Equations (3 volumes), CRC Press, Boca Raton, Florida, 1994, 1995, 1996.

  13. Oliveri, F. and Speciale, M. P., ‘Exact solutions to the unsteady equations of perfect gases through Lie group analysis and substitution principles’, International Journal of Non-Linear Mechanics 37, 2002, 257–274.

    Article  Google Scholar 

  14. Smith, P., ‘The steady magnetodynamic flow of perfectly conducting fluids’, Journal of Mathematical Mechanics 12, 1963, 505–520.

    Google Scholar 

  15. Power, G. and Rogers, C., ‘Substitution principles in nonsteady magneto-gasdynamics’, Applied Science Research 21, 1969, 176–184.

    Article  Google Scholar 

  16. Hearn, A. C., Reduce 3.7 User's Manual, Rand Corporation, Santa Monica, California, 1999.

    Google Scholar 

  17. Oliveri, F., ‘Relie: A reduce program for calculating Lie symmetries’, Available at the URL http://mat520.unime.it/oliveri/.

  18. Olver, P. J. and Rosenau, P., ‘The construction of special solutions to partial differential equations’, Physics Letters A 114, 1986, 107–112.

    Article  Google Scholar 

  19. Olver, P. J. and Rosenau, P., ‘Group-invariant solutions of differential equations’, SIAM Journal of Applied Mathematics 47, 1987, 263–278.

    Article  Google Scholar 

  20. Fuchs, J. C. and Richter, E. W., ‘Similarity solutions for the two-dimensional non-stationary ideal MHD equations’, Journal of Physics A: Mathematics and General 20, 1987, 3135–3157.

    Article  Google Scholar 

  21. Oliveri, F. and Speciale, M. P., ‘Exact solutions to the ideal magneto and gasdynamics equations through Lie group analysis and substitution principles’, Preprint, 2005.

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  1. Department of Mathematics, University of Messina, Contrada Papardo Salita Sperone 31, 98166, Messina, Italy

    Francesco Oliveri

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  1. Francesco Oliveri
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Oliveri, F. On Substitution Principles in Ideal Magneto-Gasdynamics by Means of Lie Group Analysis. Nonlinear Dyn 42, 217–231 (2005). https://doi.org/10.1007/s11071-005-3584-3

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  • DOI: https://doi.org/10.1007/s11071-005-3584-3

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