publication . Article . Other literature type . Preprint . 2007

On the hierarchy of partially invariant submodels of differential equations

S. V. Golovin;
Open Access
  • Published: 15 Nov 2007 Journal: Journal of Physics A: Mathematical and Theoretical, volume 41, page 265,501 (issn: 1751-8113, eissn: 1751-8121, Copyright policy)
  • Publisher: IOP Publishing
It is noticed, that partially invariant solution (PIS) of differential equations in many cases can be represented as an invariant reduction of some PIS of the higher rank. This introduce a hierarchic structure in the set of all PISs of a given system of differential equations. By using this structure one can significantly decrease an amount of calculations required in enumeration of all PISs for a given system of partially differential equations. An equivalence of the two-step and the direct ways of construction of PISs is proved. In this framework the complete classification of regular partially invariant solutions of ideal MHD equations is given.
free text keywords: Modelling and Simulation, Statistics and Probability, Mathematical Physics, General Physics and Astronomy, Statistical and Nonlinear Physics, 76M60, 58J70, 76W05, 35C05, Algebra, Differential equation, Invariant (mathematics), Mathematical analysis, System of differential equations, Regular solution, Magnetohydrodynamics, Hierarchy, Mathematics, Enumeration, Equivalence (measure theory)
Related Organizations
18 references, page 1 of 2

[13] Chupakhin A P 1997 On barochronous gas motions Dokl. of Russ. Ac. of Sci. 352(5)

[14] Chupakhin A P 1998 Barochronous gas motions. General properties and submodel of types (1,2) and (1,1) (Preprint: Lavrentyev Institute of Hydrodynamics, Novosibirsk. No. 4) (in Russian).

[15] Ovsiannikov L V 2003 Symmetry of barochronous gas motions Sib. Math. J. 44(5) 857-866

[16] Pukhnachev V V 2000 An integrable model of nonstationary rotationally symmetrical motion of ideal incompressible fluid Nonlinear Dynamics 22 101-109

[17] Thailert K 2006 One class of regular partially invariant solutions of the Navier-Stokes equations Nonlinear Dynamics 43(4) 343-364

[18] Pommaret J F 1978 Systems of partial differential equations and Lie pseudogroups. (Gordon and Vreach Science Publishers, Inc.: New York)

[19] Ovsyannikov L V 1998 Hierarchy of invariant submodels of differential equations Dokl. Math. 58(1) 127-129

[20] Olver P J 1986 Applications of Lie groups to differential equations. (New York: Springer-Verlag)

[21] Pavlenko A S 2005 Symmetries and solutions of equations of two-dimensional motions of politropic gas Siberian Electronic Mathematical Reports ( 2 291-307

[22] Kulikovskij A G and Lyubimov G A 1965 Magnetohydrodynamics, (Addison-Wesley: Reading)

[23] Landau L D and Lifshitz E M 1984 Electrodynamics of Continuum Media (Pergamon, Oxford)

[24] Fuchs J C 1991 Symmetry groups and similarity solutions of MHD equations J. Math. Phys. 32 1703-1708

[25] Ovsyannikov L V 1994 The “PODMODELI” program. Gas dynamics. J. Appl. Math. Mech. 58(4) 601-627

[26] Grundland A M and Lalague L 1994 Lie subgroups of the symmetry group of the equations describing a nonstationary and isentropic flow: Invariant and partially invariant solutions. Can. J. Phys. 72(7-8) 362-374 [OpenAIRE]

[27] Ovsyannikov L V 2001 Lectures on the fundamentals of gas dynamics. (Moscow-Izhevsk: Institute for Computer Studies) (in Russian)

18 references, page 1 of 2
Any information missing or wrong?Report an Issue