Contact Symmetry Algebras of Scalar Ordinary Differential Equations

Using Lie's classification of irreducible contact transformations in thecomplex plane, we show thata third-order scalar ordinary differential equation (ODE)admits an irreducible contact symmetry algebra if and only if it is transformableto q ⁽³⁾=0 via a local contact transformation. This result coupled with the classification of third-order ODEs with respect to point symmetriesprovide an explanation of symmetry breaking for third-order ODEs. Indeed, ingeneral, the point symmetry algebra of a third-order ODE is not asubalgebra of the seven-dimensional point symmetry algebra of q ⁽³⁾=0.However, the contact symmetry algebra of any third-order ODE, except forthird-order linear ODEs with four- and five-dimensional pointsymmetry algebras, is shown to be a subalgebra of the ten-dimensional contact symmetryalgebra of q ⁽³⁾==0. We also show that a fourth-orderscalar ODE cannot admit an irreducible contact symmetry algebra. Furthermore, weclassify completely scalar nth-order (n5) ODEs which admitnontrivial contact symmetry algebras.     

Periodic Solutions for Some Systems of Delay Differential Equations

In this paper, we give sufficient conditions for the existence of periodic orbits of some systems of delay differential equations with a unique delay having 3, 4 or n equations. Moreover, we provide examples of delay systems satisfying the different sets of sufficient conditions.     

Symmetries of Integro-Differential Equations: A Survey of Methods Illustrated by the Benny Equations

Classical Lie group theory provides a universal tool for calculatingsymmetry groups for systems of differential equations. However Lie'smethod is not as much effective in the case of integral orintegro-differential equations as well as in the case of infinitesystems of differential equations.This paper is aimed to survey the modern approaches to symmetriesof integro-differential equations. As an illustration, an infinitesymmetry Lie algebra is calculated for a system of integro-differentialequations, namely the well-known Benny equations. The crucial idea is tolook for symmetry generators in the form of canonical Lie–Bäcklundoperators.     

Lie Symmetry Analysis of Differential Equations in Finance

Lie group theory is applied to differential equations occurring as mathematical models in financial problems. We begin with the complete symmetry analysis of the one-dimensional Black–Scholes model and show that this equation is included in Sophus Lie's classification of linear second-order partial differential equations with two independent variables. Consequently, the Black–Scholes transformation of this model into the heat transfer equation follows directly from Lie's equivalence transformation formulas. Then we carry out the classification of the two-dimensional Jacobs–Jones model equations according to their symmetry groups. The classification provides a theoretical background for constructing exact (invariant) solutions, examples of which are presented.     

Exact Solutions of the Hydrodynamic Equations Derived from Partially Invariant Solutions

The paper proposes a heuristic approach to constructing exact solutions of the hydrodynamic equations based on the specificity of these equations. A number of systems of hydrodynamic equations possess the following structure: they contain a reduced system of n equations and an additional equation for an extra function w. In this case, the reduced system, in which w = 0, admits a Lie group G. Taking a certain partially invariant solution of the reduced system with respect to this group as a seed:rdquo; solution, we can find a solution of the entire system, in which the functional dependence of the invariant part of the seed solution on the invariants of the group G has the previous form. Implementation of the algorithm proposed is exemplified by constructing new exact solutions of the equations of rotationally symmetric motion of an ideal incompressible liquid and the equations of concentrational convection in a plane boundary layer and thermal convection in a rotating layer of a viscous liquid.     

On Solutions of Some Non-Linear Differential Equations Arising in Newtonian and Non-Newtonian Fluids

Some steady as well as unsteady solutions of the equations of motion for an incompressible Newtonian and non-Newtonian (second-grade) fluids are obtained by applying different methods including the Lie symmetry group method. The flows considered are axially symmetric with the swirling motion, and the governing equations for second-grade fluid flow have been modeled. Expressions for streamlines, velocity and vorticity components are constructed explicitly in each case. Exact analytical solutions in second-grade fluid are obtained and compared with the corresponding viscous solutions.     

Heteroclinic Connection of Periodic Solutions of Delay Differential Equations

For a certain class of delay equations with piecewise constant nonlinearities we prove the existence of a rapidly oscillating stable periodic solution and a rapidly oscillating unstable periodic solution. Introducing an appropriate Poincaré map, the dynamics of the system may essentially be reduced to a two dimensional map, the periodic solutions being represented by a stable and a hyperbolic fixed point. We show that the two dimensional map admits a one dimensional invariant manifold containing the two fixed points. It follows that the delay equations under consideration admit a one parameter family of rapidly oscillating heteroclinic solutions connecting the rapidly oscillating unstable periodic solution with the rapidly oscillating stable periodic solution.      

Solution of Ordinary Differential Equations with a Large Lie Symmetry Group

The group method of solving ODEs without any quadrature goes back to Lie. In order to apply it, the number of symmetries admitted by a given ODE has to be greater by one than the number of arbitraryconstants in the general solution of the equation. In this paper, we use thetechnique of canonical Lie–Bäcklund symmetries that makes theproof of the statement concerning integrals of ODEs more evident. The methodis extended to the solution of system of ODEs with a small parameter of arbitrary order.     

Invariant and Partially Invariant Solutions of the Green-Naghdi Equations

All invariant and partially invariant solutions of the Green-Naghdi equations are obtained that describe the second approximation of shallow water theory. It is proved that all nontrivial invariant solutions belong to one of the following types: Galilean-invariant, stationary, and self-similar solutions. The Galilean-invariant solutions are described by the solutions of the second Painleve equation, the stationary solutions by elliptic functions, and the self-similar solutions by the solutions of the system of ordinary differential equations of the fourth order. It is shown that all partially invariant solutions reduce to invariant solutions. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 6, pp. 26–35, November–December, 2005.     

New Steady and Self-Similar Solutions of the Euler Equations

Exact steady and self-similar solutions of the Euler equations are considered, which possess the property of partial invariance with respect to a certain six-parameter Lie group. New examples of vortex motion of a swirled liquid in curved channels are presented. A classification is given for self-similar solutions of the reduced system with two independent variables, which admits a three-parameter group of extensions, whereas the initial system of the Euler equations possesses a two-parameter group.     

Local Symmetry Group in the General Theory of Elastic Shells

We establish the local symmetry group of the dynamically and kinematically exact theory of elastic shells. The group consists of an ordered triple of tensors which make the shell strain energy density invariant under change of the reference placement. Definitions of the fluid shell, the solid shell, and the membrane shell are introduced in terms of members of the symmetry group. Within solid shells we discuss in more detail the isotropic, hemitropic, and orthotropic shells and corresponding invariant properties of the strain energy density. For the physically linear shells, when the density becomes a quadratic function of the shell strain and bending tensors, reduced representations of the density are established for orthotropic, cubic-symmetric, and isotropic shells. The reduced representations contain much less independent material constants to be found from experiments.     

Sharp Estimates of Bounded Solutions to Some Second-order Forced Dissipative Equations

Résumé A l’aide d’inégalités différentielles, on établit une estimation proche de l’optimalité pour la norme dans de l’unique solution bornée de u′′ + cu′ + Au = f(t) lorsque A = A ^* ≥ λ I est un opérateur borné ou non sur un espace de Hilbert réel H, V = D(A ^1/2) et λ, c sont des constantes positives, tandis que . By using differential inequalities, a close-to-optimal bound of the unique bounded solution of u′′ + cu′ + Au = f(t) is obtained whenever A = A ^* ≥ λ I is a bounded or unbounded linear operator on a real Hilbert space H, V = D(A ^1/2) and λ, c are positive constants, while .

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The Analysis of Some Characteristic Equations Arising in Population and Epidemic Models

Epidemic models with a general infective period distribution are formulated as functional differential equations, as are population models with a general life span distribution. The analysis of the local stability properties of equilibria of such models leads to a characteristic equation involving the Laplace transform of the infective period (or life span) distribution. We obtain conditions under which all roots of the characteristic equation are in the left half plane, implying asymptotic stability of equilibrium, for every infective period distribution. We also consider the converse problem of describing when instability can occur for specific infective period distributions.     

Some Results on the <Emphasis Type="BoldItalic">m</Emphasis>-Laplace Equations with Two Growth Terms

We prove the existence of positive radial solutions of the following equation:

Stability and Boundedness Results for Solutions of Certain Third Order Nonlinear Vector Differential Equations

In this paper, we investigate the asymptotic stability of the zero solution and boundedness of all solutions of a certain third order nonlinear ordinary vector differential equation. The results are proved using Lyapunov’s second (or direct method). Our results include and improve some well known results existing in the literature.     

Study on Exact Analytical Solutions for Two Systems of Nonlinear Evolution Equations

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Global Weak Solutions to Equations of Motion for Magnetic Fluids

We study the differential system governing the flow of an incompressible ferrofluid under the action of a magnetic field. The system consists of the Navier–Stokes equations, the angular momentum equation, the magnetization equation, and the magnetostatic equations. We prove, by using the Galerkin method, a global in time existence of weak solutions with finite energy of an initial boundary-value problem and establish the long-time behavior of such solutions. The main difficulty is due to the singularity of the gradient magnetic force.      

On the Backwards Behavior of the Solutions of the 2D Periodic Viscous Camassa–Holm Equations

The global behavior of the periodic 2D viscous Camassa–Holm equations is studied. The set of initial data for which the solution exists for all negative times and grows backwards with an exponential rate no greater then some given value is observed and proved to possess some interesting richness and density properties.     

On Symmetries and Anisotropies of Classical and Micropolar Linear Elasticities: A New Method Based upon a Complex Vector Basis and Some Systematic Results

A complete and unified study of symmetries and anisotropies of classical and micropolar elasticity tensors is presented by virtue of a novel method based on a well-chosen complex vector basis and algebra of complex tensors. It is proved that every elasticity tensor has nothing but 1-fold, 2-fold, 3-fold, 4-fold and ∞-fold symmetry axes. From this fact it follows that the crystallographic symmetries plus the isotropic symmetry are complete in describing the symmetries of any kind of classical elasticity tensors and micropolar elasticity tensors. Further, it is proved that for each given integer m>>2 every classical Green elasticity tensor with an m-fold symmetry axis must have at least m elastic symmetry planes intersecting each other at this symmetry axis. From this fact and the aforementioned fact it follows that for all possible material symmetry groups, there exist only eight distinct symmetry classes for classical Green elasticity tensors, which correspond to the isotropy group and the seven crystal classes S ₂, C _2h , D _2h , D _3d , D _4h , D _6h and O _h , while it is shown that there exist twelve distinct symmetry classes for any other kind of elasticity tensors, including the classical Cauchy elasticity tensor and the micropolar elasticity tensors, which correspond to the eight subgroup classes just mentioned and the four crystal classes S ₆, C _4h , C_6h and T _h . From these results, it turns out that all possible elasticity symmetry groups are nothing but the full orthogonal group, the transverse isotropy groups C _{∞ h} and D _{∞ h} , and the nine centrosymmetric crystallographic point groups except C _6h and D _6h . This revised version was published online in August 2006 with corrections to the Cover Date.