Exact solutions to magnetogasdynamics using Lie point symmetries
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Exact solutions to magnetogasdynamics using Lie point symmetries B. Bira · T. Raja Sekhar
Received: 26 February 2012 / Accepted: 20 October 2012 / Published online: 8 November 2012 © Springer Science+Business Media Dordrecht 2012
Abstract 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. Keywords Magnetogasdynamics · Group theoretic method · Hyperbolic system · Exact solution · Weak discontinuities
1 Introduction Lie group of transformations has been extensively applied to the linear and nonlinear differential equaB. Bira · T. Raja Sekhar () Department of Mathematics, National Institute of Technology Rourkela, Rourkela-8, India e-mail: [email protected]
tions in the mathematical physics, engineering, applied mathematics, gasdynamics and mechanics to deal with symmetry reductions, similarity solutions and conservation laws. The method of Lie symmetry groups is the most important approach to obtain analytical solutions of nonlinear PDEs. The basic tool in the study is the use of the corresponding infinitesimal representations of Lie algebras. By an expanded Lie group of transformations of partial differential equations we mean a continuous group of transformations acting on the expanded space of variables which includes the equation parameters in addition to independent and dependent variables. One of the most powerful methods to determine particular solutions to PDEs is based upon the study of their invariance with respect to one parameter Lie group of point transformations (see, [1–7]). Indeed, with the help of symmetry generators of these equations, one can construct similarity variables which can reduce these equations to ordinary differential equations (ODEs); in some cases, it is possible to solve these ODEs exactly [8, 9]. Besides these similarity solutions, the symmetries admitted by given PDEs enable us to look for appropriate canonical variables which transform the original system to an equivalent one whose simple solutions provide nontrivial solutions of the original system (see, [10–15]). Using this procedure, Ames and Donato [16] obtained solutions for the problem of elastic-plastic deformation generated by a torque and analyzed the evolution of a weak discontinuity in a state characterized by invariant solutions. For nonlinear wave propagation i
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