Optimal System and Invariant Solutions on ((U_{yy}(t,s,y)－U_{t}(t,s,y))y－2sU_{sy}(t,s,y))y＋（s^{2}＋1)U_{ss}(t,s,y)＋2sU_{s}=0

Author(s)
Sopita Khamrod

ABSTRACT

The purpose of this paper is to find the invariant solutions of the reduction of the Navier-Stokes equations where s=z/y ((U_{yy}(t,s,y)－U_{t}(t,s,y))y－2sU_{sy}(t,s,y))y＋（s^{2}＋1)U_{ss}(t,s,y)＋2sU_{s}=0 This equation is constructed from the Navier-Stokes equations rising to a partially invariant solutions of the Navier-Stokes equations. Group classification of the admitted Lie algebras of this equation is obtained. Two-dimensional optimal system is constructed from classification of their subalgebras. All invariant solutions corresponding to these subalgebras are presented.

KEYWORDS

Optimal System; Invariant Solutions; Partially Invariant Solutions; Navier-Stokes Equations

Optimal System; Invariant Solutions; Partially Invariant Solutions; Navier-Stokes Equations

Cite this paper

S. Khamrod, "Optimal System and Invariant Solutions on ((U_{yy}(t,s,y)－U_{t}(t,s,y))y－2sU_{sy}(t,s,y))y＋（s^{2}＋1)U_{ss}(t,s,y)＋2sU_{s}=0," *Applied Mathematics*, Vol. 4 No. 8, 2013, pp. 1154-1162. doi: 10.4236/am.2013.48154.

S. Khamrod, "Optimal System and Invariant Solutions on ((U

References

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[12] A. F. Sidorov, V. P. Shapeev and N. N. Yanenko, “The Method of Differential Constraints and Its Applications in Gas Dynamics,” Nauka, Novosibirsk, 1984.

[13] S. V. Meleshko, “Classification of the Solutions with De generate Hodograph of the Gas Dynamics and Plasticity Equations,” Doctoral Thesis, Ekaterinburg, 1991.

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[15] L. V. Ovsiannikov, “Isobaric Motions of a Gas,” Differ ential Equations, Vol. 30, No. 10, 1994, pp. 1792-1799.

[16] A. P. Chupakhin, “On Barochronic motions of a Gas,” Doklady Rossijskoj Akademii Nauk, Vol. 352, No. 5, 1997, pp. 624-626.

[17] A. M. Grundland and L. Lalague, “Invariant and Partially Invariant Solutions of the Equations Describing a Non Stationary and Isotropic Flow for an Ideal and Compres sible fluid in (3+1) Dimensions,” Journal of Physics A: Mathematical and General, Vol. 29, No. 8, 1996, pp. 1723-1739. doi:10.1088/0305-4470/29/8/019

[18] L. V. Ovsiannikov, “Group Analysis of Differential Equa tions,” Nauka, Moscow, 1978.

[19] L. V. Ovsiannikov, “Regular and Irregular Partially Inva riant Solutions,” Doklady Academy of Sciences of USSR, Vol. 343, No. 2, 1995, pp. 156-159.

[20] L. V. Ovsiannikov, “Program SUBMODELS. Gas Dy namics,” Journal of Applied Mathematics and Mechanics, Vol. 58, No. 1, 1994, pp. 30-55.

[21] S. Khamrod, “Optimal System of Subalgebras for the Re duction of the Navier-Stokes Equations,” Journal of Ap plied Mathematics, Vol. 1, No. 4, 2013, pp. 124-134. doi:10.4236/am.2013.41022

[1] L. V. Ovsiannikov and A. P. Chupakhin, “Regular Par tially Invariant Submodels of the Equations of Gas Dy namics,” Journal of Applied Mechanics and Technics, Vol. 6, No. 60, 1996, pp. 990-999.

[2] S. Lie, “On General Theory of Partial Differential Equa tions of an Arbitrary Order,” German, Vol. 4, 1895, pp. 320-384.

[3] L. V. Ovsiannikov, “Partially Invariant Solutions of the Equations Admitting a Group,” Proceedings of the 11th International Congress of Applied Mechanics, Springer Verlag, Berlin, 1964, pp. 868-870.

[4] V. O. Bytev, “Group Properties of Navier-Stokes Equa tions,” Chislennye Metody Mehaniki Sploshnoi Sredy (Novosibirsk), Vol. 3, No. 3, 1972, pp. 13-17.

[5] B. J. Cantwell, “Introduction to Symmetry Analysis,” Cambridge University Press, Cambridge, 2002.

[6] B. J. Cantwell, “Similarity Transformations for the Two Dimensional, Unsteady, Stream-Function Equation,” Jour nal of Fluid Mechanics, Vol. 85, No. 2, 1978, pp. 257-271. doi:10.1017/S0022112078000634

[7] S. P. Lloyd, “The Infinitesimal Group of the Navier Stokes Equations,” Acta Mathematica, Vol. 38, No. 1-2, 1981, pp. 85-98.

[8] R. E. Boisvert, W. F. Ames and U. N. Srivastava, “Group Properties and New Solutions of Navier-Stokes Equa tions,” Journal of Engineering Mathematics, Vol. 17, No. 3, 1983, pp. 203-221. doi:10.1007/BF00036717

[9] A. Grauel and W. H. Steeb, “Similarity Solutions of the Euler Equation and the Navier-Stokes Equations in Two space Dimensions,” International Journal of Theoretical Physics, Vol. 24, No. 3, 1985, pp. 255-265. doi:10.1007/BF00669790

[10] N. H. Ibragimov and G. Unal, “Equivalence Transforma tions of Navier-Stokes Equation,” Bulletin of the Techni cal University of Istanbul, Vol. 47, No. 1-2 1994, pp. 203-207.

[11] R. O. Popovych, “On Lie Reduction of the Navier-Stokes Equations,” Non-Linear Mathematical Physics, Vol. 2, No. 3-4, 1995, pp. 301-311. doi:10.2991/jnmp.1995.2.3-4.10

[12] A. F. Sidorov, V. P. Shapeev and N. N. Yanenko, “The Method of Differential Constraints and Its Applications in Gas Dynamics,” Nauka, Novosibirsk, 1984.

[13] S. V. Meleshko, “Classification of the Solutions with De generate Hodograph of the Gas Dynamics and Plasticity Equations,” Doctoral Thesis, Ekaterinburg, 1991.

[14] S. V. Meleshko, “One Class of Partial Invariant Solutions of Plane Gas Flows,” Differential Equations, Vol. 30, No. 10, 1994, pp. 1690-1693.

[15] L. V. Ovsiannikov, “Isobaric Motions of a Gas,” Differ ential Equations, Vol. 30, No. 10, 1994, pp. 1792-1799.

[16] A. P. Chupakhin, “On Barochronic motions of a Gas,” Doklady Rossijskoj Akademii Nauk, Vol. 352, No. 5, 1997, pp. 624-626.

[17] A. M. Grundland and L. Lalague, “Invariant and Partially Invariant Solutions of the Equations Describing a Non Stationary and Isotropic Flow for an Ideal and Compres sible fluid in (3+1) Dimensions,” Journal of Physics A: Mathematical and General, Vol. 29, No. 8, 1996, pp. 1723-1739. doi:10.1088/0305-4470/29/8/019

[18] L. V. Ovsiannikov, “Group Analysis of Differential Equa tions,” Nauka, Moscow, 1978.

[19] L. V. Ovsiannikov, “Regular and Irregular Partially Inva riant Solutions,” Doklady Academy of Sciences of USSR, Vol. 343, No. 2, 1995, pp. 156-159.

[20] L. V. Ovsiannikov, “Program SUBMODELS. Gas Dy namics,” Journal of Applied Mathematics and Mechanics, Vol. 58, No. 1, 1994, pp. 30-55.

[21] S. Khamrod, “Optimal System of Subalgebras for the Re duction of the Navier-Stokes Equations,” Journal of Ap plied Mathematics, Vol. 1, No. 4, 2013, pp. 124-134. doi:10.4236/am.2013.41022