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 group analysis and exact solutions of isentropic magnetogasdynamics

In this paper, we obtain exact solutions to the nonlinear system of partial differential equations (PDEs), describing the one dimensional unsteady simple flow of an isentropic, inviscid and perfectly conducting compressible fluid, subjected to a transverse magnetic field. Lie group of point transformations 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.     

Weak shock waves and its interaction with characteristic shocks in polyatomic gas

We present Lie symmetry analysis for investigating the shock‐wave structure of hyperbolic differential equations of polyatomic gases. With the application of symmetry analysis, we derive particular exact group invariant solutions for the governing system of partial differential equations (PDEs). In the next step, the evolutionary behavior of weak shock along with the characteristic shock and their interaction is investigated. Finally, the amplitudes of reflected wave, transmitted wave, and the jump in shock acceleration influenced by the incident wave after interaction are evaluated for the considered system of equations.     

Conservation laws for laminar axisymmetric jet flows with weak swirl

The conservation laws for laminar axisymmetric jet flows with weak swirl are studied here. The multiplier approach is used to derive the conservation laws for the system of three boundary layer equations for the velocity components governing flow in laminar axisymmetric jet flows with weak swirl. Conservation laws for the system of two partial differential equations for the stream function are also derived.     

Group theoretic method for unsteady free convection flow of a micropolar fluid along a vertical plate in a thermally stratified medium

The group theoretic approach is applied for solving the problem of unsteady natural convection flow of micropolar fluid along a vertical flat plate in a thermally stratified medium. The application of two-parameter transformation group reduces the number of independent variables in the governing system consisting of partial differential equations and a set of auxiliary conditions from three to only one independent variable, and consequently the system of governing partial differential equations with boundary conditions reduces to a system of ordinary differential equations with appropriate boundary conditions. Numerical solution of the velocity, microrotation and heat transfer have been obtained. The possible forms of the ambient temperature variation with position and time are derived.     

On the incompressible limits for the full magnetohydrodynamics flows

In this article the incompressible limits of weak solutions to the governing equations for magnetohydrodynamics flows on both bounded and unbounded domains are established. The governing equations for magnetohydrodynamic flows are expressed by the full Navier-Stokes system for compressible fluids enhanced by forces due to the presence of the magnetic field as well as the gravity and with an additional equation which describes the evolution of the magnetic field. The scaled analogues of the governing equations for magnetohydrodynamic flows involve the Mach number, Froude number and Alfven number. In the case of bounded domains the establishment of the singular limit relies on a detail analysis of the eigenvalues of the acoustic operator, whereas the case of unbounded domains is being treated by their suitable approximation by a family of bounded domains and the derivation of uniform bounds.     

Propagation of discontinuities along bicharacteristics in the unsteady flow of a relaxing gas

Growth and decay of weak discontinuities headed by wave front of arbitrary shape in three dimensions are investigated in an unsteady flow of a relaxing gas. The transport equations representing the rate of change of discontinuities in the normal derivatives of the flow variables are obtained and it is found that the nonlinearity in the governing equations plays an important role in the interplay of damping and steepening tendencies of the wave. An explicit criterion for the growth and decay of weak discontinuities along bicharacteristic curves in the characteristic manifold of the governing differential equations is given and special reference is made of diverging and converging waves under different thermodynamical situations. It is shown that there is a special case of a compressive converging wave, irrespective of the thermodynamical state whether weak or strong, in which the effects of thermodynamical influences and that of wave front curvature are unable to overcome the tendency of the wave to grow into a shock.     

Conditional Lie‐Bäcklund Symmetry of Evolution System and Application for Reaction‐Diffusion System

Conditional Lie‐Bäcklund symmetry (CLBS) method is developed to study system of evolution equations. It is shown that reducibility of a system of evolution equations to a system of ordinary differential equations can be fully characterized by the CLBS of the considered system. As an application of the approach, a class of two‐component nonlinear diffusion equations is studied. The governing system and the admitted CLBS can be identified. As a consequence, exact solutions defined on the polynomial, exponential, trigonometric, and mixed invariant subspaces are constructed due to the corresponding symmetry reductions.     

Global Existence Results for Oldroyd Fluids with Wall Slip

We investigate the system of nonlinear partial differential equations governing the unsteady motion of an incompressible viscoelastic fluid of Oldroyd type in a bounded domain under Navier’s slip boundary condition. We prove the existence of global weak solutions for the corresponding initial-boundary value problem without assuming that the model constants, body force or the initial values of the velocity and the stress tensor are small.     

Lie symmetries of a system arising in plasma physics

Lie group classification for a diffusion‐type system that has applications in plasma physics is derived. The classification depends on the values of 5 parameters that appear in the system. Similarity reductions are presented. Certain initial value problems are reduced to problems with the governing equations being ordinary differential equations. Examples of potential symmetries are also presented.     

Weak solutions for Falk's model of shape memory alloys

In this paper we discuss the system of two partial differential equations governing the dynamics of phase transitions in shape memory alloys. We consider the one‐dimensional model proposed by Falk, in which a term containing a fourth‐derivative appears. The main purpose is to show the uniqueness for weak solutions of the problem by using the approximate dual equations for the system without growth condition for the free energy function. Copyright © 2000 John Wiley & Sons, Ltd.     

Conservation laws via the partial Lagrangian and group invariant solutions for radial and two-dimensional free jets

The conservation laws for Prandtl’s boundary layer equations for an incompressible fluid governing the flow in radial and two-dimensional jets are investigated. For both radial and two-dimensional jets the partial Lagrangian method is used to derive conservation laws for the system of two differential equations for the velocity components. The Lie point symmetries are calculated for both cases and a symmetry is associated with the conserved vector that is used to establish the conserved quantity for the jet. This associated symmetry is then used to derive the group invariant solution for the system governing the flow in the free jet.     

Pulselike Exact Solutions for a Model Describing Nerve Fibers

A quasi-linear nonhomogeneous first order hyperbolic system describing nerve pulse transmission is considered. By requiring the compatibility of the governing equations with quasi-linear differential constraints, exact solutions to the model in question are determined. Furthermore classes of material response functions amenable to the mathematical approach are characterized. Initial and/or boundary value problems of interest in nerve pulse propagation are solved. It is proved that the governing model admits solutions which describe a "space clamp" situation and the propagation of a localized action potential pulse along the nerve.     

Asymptotic Analysis of a Hyperbolic Mixed Problem in One-Dimensional Viscoelasticity

We present an asymptotic analysis of the boundary-generated, small-amplitude, high-frequency waves in a one-dimensional, semi-infinite, viscoelastic solid characterized by a single-integral constitutive functional. The equations governing the wave motion constitute a 2×2 system of hyperbolic Volterra integrodifferential equations. The method of analysis is based on a single-wave expansion of nonlinear geometric optics.     

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.     

Analysis of a system related to a model for phase transitions in dissipative isochoric thermoviscoelastic materials

We analyze a highly nonlinear system of partial differential equations related to a model solidification and/or melting of thermoviscoelastic isochoric materials with the possibility of motion of the material during the process. This system consists of an internal energy balance equation governing the evolution of temperature, coupled with an evolution equation for a phase field whose values describe the state of material and a balance equation for the linear moments governing the material displacements. For this system, under suitable dissipation conditions, we prove global existence and uniqueness of weak solutions. Copyright © 2016 John Wiley & Sons, Ltd.     

Symmetry groups and similarity solutions for the system of equations for a viscous compressible fluid

Here, using Lie group transformations, we consider the problem of finding similarity solutions to the system of partial differential equations (PDEs) governing one-dimensional unsteady motion of a compressible fluid in the presence of viscosity and thermal conduction, using the general form of the equation of state. The symmetry groups admitted by the governing system of PDEs are obtained, and the complete Lie algebra of infinitesimal symmetries is established. Indeed, with the use of the entailed similarity solution the problem is transformed to a system of ordinary differential equations(ODEs), which in general is nonlinear; in some cases, it is possible to solve these ODEs to determine some special exact solutions.     

On certain doubly non-uniformly and singular non-uniformly elliptic systems

We consider the steady state of the thermistor problem consisting of a coupled set of nonlinear elliptic equations governing the temperature and the electric potential. We study the existence of weak solutions under two kind of assumptions. The first one considers the case in which the two diffusion coefficients are not bounded below far from zero, arising to a doubly non-uniformly elliptic system. In the second one, we assume in addition that the thermal conductivity blows up for a finite value of the temperature, arising to a singular and non-uniformly coupled system.     

Reflected pulses from a refractive random medium at grazing incidence

A problem of acoustic pulse reflection by a one-dimensionalrefractive random medium is considered in the case of grazingangle incidence. The material parameters of the medium are assumedto vary with a random microscale and a deterministic macroscale.A system of stochastic equations for random scattering variablesis derived based upon the random modelling of three separatescales of variations. The statistical properties of the reflectedpulses are characterized by an asymptotic diffusion limit theoremof stochastic differential equations with multiple scales. Thetransport equations governing the limiting stochastic distributionsof the random reflection coefficient are obtained in the propagatingregime, which leads to the power spectral densities of the reflectedpressure and particle velocity fields.     

Global existence of solutions of a strongly coupled quasilinear parabolic system with applications to electrochemistry

This paper consists of two parts. In the first part, we proved the global existence of weak solutions of a strongly coupled quasilinear parabolic system in Rⁿ using weak compactness method. In the second part, we considered the electrochemistry model studied in Choi and Lui (J. Differential Equations 116 (1995) 306) where the Poisson equation governing the electric potential is replaced by a local electro-neutrality condition. In one space dimension, the equations for the model is of the form considered in the first part of this paper except that the coefficient matrix is discontinuous at places where all the charged ions vanish. We approximate the equations by nicer operators and pass to the limit to obtain global existence of weak solutions. The non-negativity of weak solutions and L²-stability of the steady-state solutions are also shown under additional hypotheses.