, Volume 48, Issue 5, pp 1023–1029 | Cite as

Exact solutions to magnetogasdynamics using Lie point symmetries



In the present work, we find some exact solutions to the first order quasilinear hyperbolic system of partial differential equations (PDEs), governing the one dimensional unsteady flow of inviscid and perfectly conducting compressible fluid, subjected to a transverse magnetic field. For this, Lie group analysis is used to identify a finite number of generators that leave the given system of PDEs invariant. Out of these generators, two commuting generators are constructed involving some arbitrary constants. With the help of canonical variables associated with these two generators, the assigned system of PDEs is reduced to an autonomous system whose simple solutions provide nontrivial solutions of the original system. Using this exact solution, we discuss the evolutionary behavior of weak discontinuities.


Magnetogasdynamics Group theoretic method Hyperbolic system Exact solution Weak discontinuities 



Research support from, Ministry of Minority Affairs through UGC, Government of India (Ref. F1-17.1/2010/MANF-CHR-ORI-1839/(SA-III/Website)) and National Board for Higher Mathematics, Department of Atomic Energy, Government of India (Ref. No. 2/48(1)/2011/-R&D II/4715), gratefully acknowledged by the first and second authors, respectively.


  1. 1.
    Bluman GW, Kumei S (1989) Symmetries and differential equations. Springer, New York MATHCrossRefGoogle Scholar
  2. 2.
    Ovsiannikkov LV (1982) Group analysis of differential equations. Academic Press, New York Google Scholar
  3. 3.
    Olver PJ (1986) Applications of Lie groups to differential equations. Springer, New York MATHCrossRefGoogle Scholar
  4. 4.
    Bluman GW, Kumei S (1974) Similarity methods for differential equations. Springer, New York MATHCrossRefGoogle Scholar
  5. 5.
    Rezvan F, Yasar E, Ozer T (2011) Group properties and conservation laws for nonlocal shallow water wave equation. Appl Math Comput 218(3):974–979 MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Sahin D, Antar N, Ozer T (2010) Lie group analysis of gravity currents. Nonlinear Anal, Real World Appl 11(2):978–994 MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Jena J (2009) Lie group transformations for self-similar shocks in a gas with dust particles. Math Methods Appl Sci 32(16):2035–2049 MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Raja Sekhar T, Sharma VD (2008) Similarity solutions for three dimensional Euler equations using Lie group analysis. Appl Math Comput 196(1):147–157 MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Raja Sekhar T, Sharma VD (2012) Similarity analysis of modified shallow water equations and evolution of weak waves. Commun Nonlinear Sci Numer Simul 17(2):630–636 MathSciNetADSMATHCrossRefGoogle Scholar
  10. 10.
    Sharma VD, Radha R (2008) Exact solutions of Euler equations of ideal gasdynamics via Lie group analysis. Z Angew Math Phys 59(6):1029–1038 MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Donato A, Oliveri F (1993) Reduction to autonomous form by group analysis and exact solutions of axisymmetric MHD equations. Math Comput Model 18(10):83–90 MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Oliveri F, Speciale MP (2005) Exact solutions to the ideal gas magnetogasdynamics equations through Lie group analysis and substitution principles. J Phys A 38(40):8803–8820 MathSciNetADSMATHCrossRefGoogle Scholar
  13. 13.
    Pandey M, Radha R, Sharma VD (2008) Symmetry analysis and exact solutions of magnetogasdynamic equations. Q J Mech Appl Math 61(3):291–310 MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Oliveri F, Speciale MP (2002) Exact solutions to the unsteady equations of perfect gases through lie group analysis and substitution principles. Int J Non-Linear Mech 37(2):257–274 MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Donato A, Oliveri F (1995) When non-autonomous equations are equivalent to autonomous ones. Appl Anal 58(3–4):313–323 MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Ames WF, Donato A (1988) On the evolution of weak discontinuities in a state characterized by invariant solutions. Int J Non-Linear Mech 23(2):167–174 MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Sharma VD (2010) Quasilinear hyperbolic systems, compressible flows, and waves. CRC Press, Boca Raton MATHCrossRefGoogle Scholar
  18. 18.
    Singh LP, Husain A, Singh M (2011) A self-similar solution of exponential shock waves in non-ideal magnetogasdynamics. Meccanica 46(2):437–445 MathSciNetCrossRefGoogle Scholar
  19. 19.
    Donato A, Ruggeri T (2000) Similarity solutions and strong shocks in extended thermodynamics of rarefied gas. J Math Anal Appl 251(1):395–405 MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Nath G (2012) Self-similar solution of cylindrical shock wave propagation in a rotational axisymmetric mixture of a non-ideal gas and small solid particles. Meccanica 47(7):1815–1817 MathSciNetCrossRefGoogle Scholar
  21. 21.
    Vishwakarma JP, Nath G (2009) A self-similar solution of a shock propagation in a mixture of a non-ideal gas and small solid particles. Meccanica 44(3):239–254 MATHCrossRefGoogle Scholar
  22. 22.
    Raja Sekhar T, Sharma VD (2010) Evolution of weak discontinuities in shallow water equations. Appl Math Lett 23(3):327–330 MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Raja Sekhar T, Bira B (2012) Wave features and group analysis for axisymmetric flow of shallow water equations. Int J Nonlinear Sci 14(1):23–30 MathSciNetGoogle Scholar
  24. 24.
    Raja Sekhar T, Sharma VD (2010) Riemann problem and elementary wave interactions in isentropic magnetogasdynamics. Nonlinear Anal, Real World Appl 11(2):619–636 MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Raja Sekhar T, Sharma VD (2012) Solution to the Riemann problem in a one-dimensional magnetogasdynamic flow. Int J Comput Math 89(2):200–216 MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Boillat G, Ruggeri T (1979) On evolution law of weak discontinuities for hyperbolic quasilinear systems. Wave Motion 1(2):149–152 MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  1. 1.Department of MathematicsNational Institute of Technology RourkelaRourkela-8India

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