Similarity Solutions for Strong Shocks in an Ideal Gas

The method of Lie group invariance is used to obtain a class of self-similar solutions to the problem of shocks in an inhomogeneous gaseous medium and to characterize analytically the state-dependent form of the medium ahead for which the problem is invariant and admits self-similar solutions. Different cases of possible solutions, known in the literature, with a power law, exponential, or logarithmic shock paths are recovered as special cases depending on the arbitrary constants occurring in the expression for the generators of the transformation. Particular cases of plane rising shocks in an exponential medium and collapse of an imploding shock are worked out in detail. Numerical calculations have been performed to obtain the similarity exponents and the profiles of the flow variables, and comparison is made with the known results.     

The similarity forms and invariant solutions of two-layer shallow-water equations

In this study symmetry group properties and general similarity forms of the two-layer shallow-water equations are discussed by Lie group theory. We represent that Lie group theory can be used as an effective approach for investigation of the self-similar solutions for the shallow-water equations with variable inflow as the generalization of dimensional analysis that was used so far for a regular approach in the literature. We also represent that the results obtained by dimensional analysis are just a special case of the results obtained by Lie group theory and it is possible to obtain the new similarity forms and the new variable inflow functions for the study of gravity currents in two-layer flow under shallow-water approximations based on Lie group theory. The symmetry groups of the system of nonlinear partial differential equations are found and the corresponding similarity and reduced forms are obtained. Some similarity solutions of the reduced equations are investigated. It is shown that reduced equations and similarity forms of the system depend on the group parameters. We show that an analytic similarity solution for the system of equations can be found for some special values of them. For other values of the group parameters, the similarity solutions of the two-layer shallow-water equations representing the gravity currents with a variable inflow are found by the numeric integration.     

Similarity analysis of modified shallow water equations and evolution of weak waves

In this paper, we obtain exact solutions to the nonlinear system of partial differential equations (PDEs), describing the one dimensional modified shallow water equations, using invariance group properties of the governing system. Lie group of point symmetries with commuting infinitesimal operators, are presented. The symmetry generators are used for constructing similarity variables which lead the governing system of PDEs to system of ordinary differential equations (ODEs); in some cases, it is possible to solve these equations exactly. A particular solution to the governing system, which exhibits space-time dependence, is used to study the evolutionary behavior of weak discontinuities.     

Symmetry analysis and exact solutions of magnetogasdynamic equations

Using the invariance group properties of the governing systemof partial differential equations (PDEs), admitting Lie groupof point transformations with commuting infinitesimal generators,we obtain exact solutions to the system of PDEs describing one-dimensionalunsteady planar and cylindrically symmetric motions in magnetogasdynamicsinvolving shock waves. Some appropriate canonical variablesare characterised that transform the equations at hand to anequivalent autonomous form, the constant solutions of whichcorrespond to non-constant solutions of the original system.The governing system of PDEs includes as a special case theEuler's equations of non-isentropic gasdynamics. It is interestingto remark that in the absence of magnetic field, one of theexact solutions obtained here is precisely the blast wave solutionobtained earlier using a different method of approach. A particularsolution to the governing system, which exhibits space–timedependence, is used to study the wave pattern that finally developswhen a magnetoacoustic wave impacts with a shock. The influenceof magnetic field strength on the evolutionary behaviour ofincident and reflected waves and the jump in shock acceleration,after collision, are studied.     

Lie symmetry analysis,optimal system,and generalized group invariant solutions of the (2 + 1)-dimensional Date–Jimbo–Kashiwara–Miwa equation

In this work, Lie group theoretic method is used to carry out the similarity reduction and solitary wave solutions of (2 + 1)-dimensional Date–Jimbo–Kashiwara–Miwa (DJKM) equation. The equation describes the propagation of nonlinear dispersive waves in inhomogeneous media. Under the invariance property of Lie groups, the infinitesimal generators for the governing equation have been obtained. Thereafter, commutator table, adjoint table, invariant functions, and one-dimensional optimal system of subalgebras are derived by using Lie point symmetries. The symmetry reductions and some group invariant solutions of the DJKM equation are obtained based on some subalgebras. The obtained solutions are new and more general than the rest while known results reported in the literature. In order to show the physical affirmation of the results, the obtained solutions are supplemented through numerical simulation. Thus, the solitary wave, doubly soliton, multi soliton, and dark soliton profiles of the solutions are traced to make this research physically meaningful.     

Instability of the self-similar flow into a cavity

In this paper we have investigated the instability of the self-similar flow behind the boundary of a collapsing cavity. The similarity solutions for the flow into a cavity in a fluid obeying a gas lawp = K_ρ ^γ, K = constant and 7 ≥ γ > 1 has been solved by Hunter, who finds that for the same value of γ there are two self-similar flows, one with accelerating cavity boundary and other with constant velocity cavity boundary. We find here that the first of these two flows is unstable. We arrive at this result only by studying the propagation of disturbances in the neighbourhood of the singular point.     

Lie group analysis of gravity currents

We consider shallow water theory to study the self-similar gravity currents that describe the motion of a heavy fluid flowing into another lighter ambient fluid. Gratton and Vigo investigated the shallow water theory representing the self-similar gravity currents by using dimensional analysis [J. Gratton, C. Vigo, Self-similarity gravity currents with variable inflow revisited: Plane currents, J. Fluid. Mech. 258 (1994) 77–104]. But in this study, the self-similarity solutions of the one-layer shallow-water equations representing gravity currents are investigated by using Lie group analysis and it is shown that Lie group analysis is the generalization of the dimensional analysis for investigating the self-similarity solutions of the one-layer shallow-water equations. Applying Lie group theory, reduced equations of the shallow water equations are found. Therefore, it becomes possible to obtain the similarity forms depending on the Lie group parameters and also the self-similarity solutions for the special values of these group parameters.     

Isothermal flow behind a shock wave propagating in the solar wind

A Parker-type blast wave, which is headed by a strong shock, driven out by a propelling contact surface, moving into an ambient solar wind having a strictly inverse square law radial decay in density, is studied. Assuming the self-similar flow behind the shock to be isothermal, approximate analytical and exact numerical solutions are obtained. There is a good agreement between the approximate analytical and exact numerical solutions. It is observed that the mathematical singularity in density at the contact surface is removed for the isothermal flow.     

Lie group analysis and propagation of weak discontinuity in one-dimensional ideal isentropic magnetogasdynamics

The aim of this paper is to carry out symmetry group analysis to obtain important classes of exact solutions from the given system of nonlinear partial differential equations (PDEs). Lie group analysis is employed to derive some exact solutions of one dimensional unsteady flow of an ideal isentropic, inviscid and perfectly conducting compressible fluid, subject to a transverse magnetic field for the magnetogasdynamics system. By using Lie group theory, the full one-parameter infinitesimal transformations group leaving the equations of motion invariant is derived. The symmetry generators are used for constructing similarity variables which leads the system of PDEs to a reduced system of ordinary differential equations; in some cases, it is possible to solve these equations exactly. Further, using the exact solution, we discuss the evolutionary behavior of weak discontinuity.     

Lie-group theoretic method for analyzing interaction of discontinuous waves in a relaxing gas

A Lie group of transformations method is used to establish self-similar solutions to the problem of shock wave propagation through a relaxing gas and its interaction with the weak discontinuity wave. The forms of the equilibrium value of the vibrational energy and the relaxation time, varying with the density and pressure are determined for which the system admits self-similar solutions. A particular solution to the problem has been found out and used to study the effects of specific heat ratio and ambient density exponent on the flow parameters. The coefficients of amplitudes of reflected and transmitted waves after the interaction are determined.     

Lie-group theoretic method for analyzing interaction of discontinuous waves in a relaxing gas

A Lie group of transformations method is used to establish self-similar solutions to the problem of shock wave propagation through a relaxing gas and its interaction with the weak discontinuity wave. The forms of the equilibrium value of the vibrational energy and the relaxation time, varying with the density and pressure are determined for which the system admits self-similar solutions. A particular solution to the problem has been found out and used to study the effects of specific heat ratio and ambient density exponent on the flow parameters. The coefficients of amplitudes of reflected and transmitted waves after the interaction are determined.     

Approximate analytical solutions for strong shocks with variable energy

Approximate analytical solutions are obtained for self-similar flows behind strong shocks with variable energy deposition or withdrawal at the wavefront in a perfect gas at rest with constant initial density. Numerical solutions are also obtained and the approximate solutions agree with these solutions. The effect of the adiabatic index on the solutions is investigated. The dependence of shock density ratio on the parameter characterizing the energy of the flow is studied. It is observed that the rate of deposition of energy at the wavefront decreases with increase of the parameter that specifies the total energy of the flow.     

Conservation laws and exact solutions of the Whitham-type equations

In this paper, we study conservation laws for some partial differential equations. It is shown that interesting conserved quantities arise from multipliers by using homotopy operator that is a powerful algorithmic tool. Furthermore, the invariance properties of the conserved flows with respect to the Lie point symmetry generators are investigated via the symmetry action on the multipliers. Furthermore, the similarity reductions and some exact solutions are provided.     

Analytic solution of self-similar strong shocks in an exponential medium for zero temperature gradient

The self-similar one-dimensional propagation of a strong shock wave in a medium with an exponentially decreasing density is studied. The flow behind the shock is assumed to be spatially isothermal rather than adiabatic to simulate the conditions of large radiative transfer behind the shock. The solution in closed form is obtained. An analytic expression for the similarity exponent has also been obtained.     

Self-similar time-varying flows of an ideal gas with a change in the adiabatic exponent in a “reflected” shock wave

Self-similar one-dimensional time-varying problems are considered under the assumption that there is a change in the adiabatic exponent in a shock wave (SW) running (“reflected”) from a centre or axis of symmetry (later from a centre of symmetry, CS) or from a plane. The medium is an ideal (inviscid and non-heat-conducting) perfect gas with constant heat capacities. In problems with strong SW, the change in the adiabatic exponent in a gas approximately simulates physicochemical processes such as dissociation and ionization and, in the problem of the collapse of a spherical cavity in a liquid, the conversion of liquid into vapour. In both cases, the adiabatic exponent decreases on passing across a reflected SW. Problems of the collapse of a spherical cavity, the reflection of a strong SW from a centre of symmetry and a simpler problem with a self-similarity index of one are examined. When it is assumed that there is an increase in the adiabatic exponent, the self-similar solutions of the first two problems are rejected due to the decrease in entropy from the instant when the SW is reflected. When it is assumed that there is a decrease in the adiabatic exponent, the solutions of these problems only become unsuitable after a finite time has elapsed for the same reason. Up to this time when the decrease in the adiabatic exponent has not reached a certain threshold, the structure of the self-similar solution does not undergo qualitative changes. When the above-mentioned threshold is exceeded, a self-similar solution is possible if a cylindrical or spherical piston expands according to a special law from the instant of SW reflection from the CS. When there is no piston, the flow behind the reflected wave becomes non-self- similar. In the case of the deceleration of a plane flow, conditions are possible with the joining of SW from different sides to a centred rarefaction wave.     

Exact solutions of Euler equations of ideal gasdynamics via Lie group analysis

In this paper, we explicitly characterize a class of solutions to the first order quasilinear system of partial differential equations (PDEs), governing one dimensional unsteady planar and radially symmetric flows of an adiabatic gas involving shock waves. 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 non trivial solutions of the original system. It is interesting to remark that one of the special solutions obtained here, using this approach, is precisely the blast wave solution known in the literature.      

Symmetry analysis of the option pricing model with dividend yield from financial markets

In this work, the option pricing Black–Scholes model with dividend yield is investigated via Lie symmetry analysis. As a result, the complete Lie symmetry group and infinitesimal generators of the one-dimensional Black–Scholes equation are derived. On the basis of these infinitesimal generators, the similarity variables and newly explicit solutions of the Black–Scholes equation are obtained by solving the corresponding characteristic equations. Finally, figures for an explicit solution with different dividend yields are presented to demonstrate the novel properties.     

Computational Experiments with Nilpotent Lie Algebras

Results of computer experiments on the study of properties of generic Lie subalgebras with two generators in the Lie algebra of nilpotent matrices whose order does not exceed 10 are presented. The calculations carried out have made it possible to formulate several statements (so-called virtual theorems) on properties of the Lie subalgebras in question. The dimensions of the lower and upper central series and of the series of commutator subalgebras and the characteristic nilpotency property of the Lie subalgebras constructed here and of generic Lie subalgebras of codimension 1 in these Lie subalgebras are studied.

     

Lie symmetry analysis of electron–electromagnetic wave interaction under condition of the anomalous Doppler effect

Lie symmetry analysis is applied for a problem of interaction of electron cyclotron oscillators with a slow electromagnetic wave under condition of the anomalous Doppler effect. This analysis reveals scaling invariance of the system and existence of self-similar solutions which describe amplification of a short electromagnetic pulse with its subsequent compression. The results of theoretical analysis are confirmed by numerical simulations.     

Lie group symmetry analysis of transport in porous media with variable transmissivity

We determine the Lie group symmetries of the coupled partial differential equations governing a novel problem for the transient flow of a fluid containing a solidifiable gel, through a hydraulically isotropic porous medium. Assuming that the permeability (K^∗) of the porous medium is a function of the gel concentration (c^∗), we determine a number of exact solutions corresponding to the cases where the concentration-dependent permeability is either arbitrary or has a power law variation or is a constant. Each case admits a number of distinct Lie symmetries and the solutions corresponding to the optimal systems are determined. Some typical concentration and pressure profiles are illustrated and a specific moving boundary problem is solved and the concentration and pressure profiles are displayed.