由相容性求任意阶非线性偏微分方程的决定方程

研究了求任意阶的非线性偏微分方程的决定方程,运用相容性方法,和运用传统的向量场及其延拓的方法求得的结果相同.但运用相容性方法不用再计算复杂的无穷小生成元的延拓的系数,这样在计算过程中既能提高计算的速度,又能提高计算的准确率,因此,这种方法比运用向量场及其延拓的方法更简便、快捷,并举例验证了这一事实.     

Nonclassical symmetries of evolutionary partial differential equations and compatibility

The determining equations for the nonclassical reductions of a general nth order evolutionary partial differential equations is considered. It is shown that requiring compatibility with a first order quasilinear partial differential equation, the determining equations are obtained. Burgers' equation and the KdV equation and generalizations serve as examples illustrating how compatibility leads quickly and easily to the determining equations for their nonclassical symmetries.     

On compact approximations of divergence differential equations

A method for the construction of compact difference schemes approximating divergence differential equations is proposed. The schemes have an arbitrarily prescribed order of approximation on general stencils. It is shown that the construction of such schemes for partial differential equations is based on special compact schemes approximating ordinary differential equations in several independent functions. Necessary and sufficient conditions on the coefficients of these schemes with high order of approximation are obtained. Examples of reconstruction of compact difference schemes for partial differential equations with these schemes are given. It is shown that such compact difference schemes have the same order of accuracy both for classical approximations on smooth solutions and weak approximations on discontinuous solutions.     

Behaviour at time t = 0 of the solutions of semi-linear evolution equations

We investigate the regularity at time t = 0 of the solutions of linear and semi-linear evolutions equations (including the Stokes and Navier-Stokes equations), Necessary and sufficient conditions on the data for an arbitrary order of regularity are given (the classical “compatibility conditions”). In the case of the Stokes and Navier-Stokes equations the compatibility conditions which we find are global conditions on the data. The presentation given here seems to improve and generalize the known results even in the simplest case of linear evolution equations.     

Deformation of surfaces in three-dimensional space induced by means of integrable systems

The correspondence between different versions of the Gauss–Weingarten equation is investigated. The compatibility condition for one version of the Gauss–Weingarten equation gives the Gauss–Mainardi–Codazzi system. A deformation of the surface is postulated which takes the same form as the original system but contains an evolution parameter. The compatibility condition of this new augmented system gives the deformed Gauss–Mainardi–Codazzi system. A Lax representation in terms of a spectral parameter associated with the deformed system is established. Several important examples of integrable equations based on the deformed system are then obtained. It is shown that the Gauss–Mainardi–Codazzi system can be obtained as a type of reduction of the self-dual Yang–Mills equations.     

Lax pairs and a new spectral method for linear and integrable nonlinear PDEs

A new transform method for solving initial-boundary value problems for linear and integrable nonlinear PDEs in two independent variables has been recently introduced in [1]. For linear PDEs this method involves: (a) formulating the given PDE as the compatibility condition of two linear equations which, by analogy with the nonlinear theory, we call a Lax pair; (b) formulating a classical mathematical problem, the so-called Riemann-Hilbert problem, by performing a simultaneous spectral analysis of both equations defining the Lax pair; (c) deriving certain global relations satisfied by the boundary values of the solution of the given PDE. Here this method is used to solve certain problems for the heat equation, the linearized Korteweg-deVries equation and the Laplace equation. Some of these problems illustrate that the new method can be effectively used for problems with complicated boundary conditions such as changing type as well as nonseparable boundary conditions. It is shown that for simple boundary conditions the global relations (c) can be analyzed using only algebraic manipulations, while for complicated boundary conditions, one needs to solve an additional Riemann-Hilbert problem. The relationship of this problem with the classical Wiener-Hopf technique is pointed out. The extension of the above results to integrable nonlinear equations is also discussed. In particular, the Korteweg-deVries equation in the quarter plane is linearized.     

A Simple Criterion for Involutivity

A simple criterion for the involutivity of a system of partialdifferential equations of polynomial type is proved. The criterioninvolves the equations themselves and does not require the systemto be in orthonomic form. It is proved that a system of partialdifferential equations is involutive if it is a differentialGröbner basis with respect to a total degree ordering,and if the compatibility conditions of the symbol equationsof the system consist of equations of degree one. An algorithmfor calculating these compatibility conditions is given.     

A Theory of Quantized Fields Based on Orthogonal and Symplectic Clifford Algebras

The transition from a classical to quantum theory is investigated within the context of orthogonal and symplectic Clifford algebras, first for particles, and then for fields. It is shown that the generators of Clifford algebras have the role of quantum mechanical operators that satisfy the Heisenberg equations of motion. For quadratic Hamiltonians, the latter equations are obtained from the classical equations of motion, rewritten in terms of the phase space coordinates and the corresponding basis vectors. Then, assuming that such equations hold for arbitrary path, i.e., that coordinates and momenta are undetermined, we arrive at the equations that contains basis vectors and their time derivatives only. According to this approach, quantization of a classical theory, formulated in phase space, is replacement of the phase space variables with the corresponding basis vectors (operators). The basis vectors, transformed into the Witt basis, satisfy the bosonic or fermionic (anti)commutation relations, and can create spinor states of all minimal left ideals of the corresponding Clifford algebra. We consider some specific actions for point particles and fields, formulated in terms of commuting and/or anticommuting phase space variables, together with the corresponding symplectic or orthogonal basis vectors. Finally we discuss why such approach could be useful for grand unification and quantum gravity.     

Local Classical Solutions to the Equations of Relativistic Hydrodynamics

In this paper, we prove that the convexity of the negative thermodynamical entropy of the equations of relativistic hydrodynamics for ideal gas keeps its invariance under the Lorentz transformation if and only if the local sound speed is less than the light speed in vacuum. Then a symmetric form for the equations of relativistic hydrodynamics is presented and the local classical solution is obtained. Based on this, we prove that the nonrelativistic limit of the local classical solution to the relativistic hydrodynamics equations for relativistic gas is the local classical solution of the Euler equations for polytropic gas.     

A Differential Geometric Approach to the Helmholtz Equations and an Application to a Space with a Nontrivial Riemannian Metric

The necessary and sufficient conditions for variationality are obtained from the requirement that a differential two-form be closed. The classical Helmholtz equations are shown to follow from these equations. An application of these results to the case in which one of the functions in these equations is taken to be a Riemannian metric on a curved space is presented.     

A phenomenological and kinetic description of diffusion and heat transport in multicomponent gas mixtures and plasma

Different forms of expressing diffusion and heat fluxes in multicomponent mixtures, obtained by methods of non-equilibrium thermodynamics and the kinetic theory of gas mixtures, are analysed and compared. It is shown that an alternative representation of the linear relations of non-equilibrium thermodynamics is possible, which enables them to be written in a form similar to that of the well-known Stefan–Maxwell equations. A relation between the phenomenological coefficients of non-equilibrium thermodynamics and the corresponding transport coefficients obtained in kinetic theory is established, with a confirmation that the Onsager reciprocity relations are satisfied. It is shown that there is an advantage in writing the transport relations on the basis of the “forces in terms of fluxes” representation, compared with the classical “fluxes in terms of forces” representation, used in standard schemes of phenomenological non-equilibrium thermodynamics and the Chapman–Enskog method, traditional for kinetic theory. A generalization of the Stefan-Maxwell equations and the equation for the heat flux is considered, which takes into account the contribution to these equations of the time and space derivatives of the fluxes. The relaxation form of the equations obtained enable one to approach the analysis of the propagation of small heat and concentration perturbations in gas mixtures to be justified, which, within the framework of classical transport relations, propagate with infinitely high velocity. The results presented in this review enable one to determine the areas of effective application of different methods of describing diffusion and heat transfer in multicomponent gas mixtures when solving specific gas-dynamic problems.     

Maximum principle for functional equations in the space of discontinuous functions of three variables

The paper is devoted to the maximum principles for functional equations in the space of measurable essentially bounded functions. The necessary and sufficient conditions for validity of corresponding maximum principles are obtained in a form of theorems about functional inequalities similar to the classical theorems about differential inequalities of the Vallee Poussin type. Assertions about the strong maximum principle are proposed. All results are also true for difference equations, which can be considered as a particular case of functional equations. The problems of validity of the maximum principles are reduced to nonoscillation properties and disconjugacy of functional equations. Note that zeros and nonoscillation of a solution in a space of discontinuous functions are defined in this paper. It is demonstrated that nonoscillation properties of functional equations are connected with the spectral radius of a corresponding operator acting in the space of essentially bounded functions. Simple sufficient conditions of nonoscillation, disconjugacy and validity of the maximum principles are proposed. The known nonoscillation results for equation in space of functions of one variable follow as a particular cases of these assertions. It should be noted that corresponding coefficient tests obtained on this basis cannot be improved. Various applications to nonoscillation, disconjugacy and the maximum principles for partial differential equations are proposed.     

On using the more-accurate equations of thin coatings in the theory of axisymmetric contact problems for composite foundations

More-accurate equations describing the axisymmetric deformations of elastic, thin-walled elements (coatings) are derived using the asymptotic analysis of the solution to the first fundamental problem of the theory of elasticity for a layer. The notable difference distinguishing these relations from the classical, Kirchhoff-Love and Reissner-Timoshenko equations of flexure of plates, and their modifications /1/, is, that there are no concentrated forces at the edges of the stamp when the corresponding contact problems are solved. Moreover, the formulas obtained contain the equations of classical theory as a special case. The solutions obtained using various applied theories are compared with the corresponding solution obtained using the equations of the theory of elasticity, using the example of the axisymmetric contact problem of impressing a plane circular stamp into a layer lying on a Fuss-Winkler foundation. The characteristic parameters of the problem in question are computed by numerical methods.     

The general equations of analytical dynamics

It is shown that the generalized Poincaré and Chetayev equations, which represent the equations of motion of mechanical systems using a certain closed system of infinitesimal linear operators, are related to the fundamental equations of analytical dynamics. Equations are derived in quasi-coordinates for the case of redundant variables; it is shown that when an energy integral exists the operator X₀ = ∂/∂t satisfies the Chetayev cyclic-displacement conditions. Using the energy integral the order of the system of equations of motion is reduced, and generalized Jacobi-Whittaker equations are derived from the Chetayev equations. It is shown that the Poincaré-Chetayev equations are equivalent to a number of equations of motion of non-holonomic systems, in particular, the Maggi, Volterra, Kane, and so on, equations. On the basis of these, and also of other previously obtained results, the Poincaré and Chetayev equations in redundant variables, applicable both to holonomic and non-holonomic systems, can be regarded as general equations of classical dynamics, equivalent to the well-known fundamental forms of the equations of motion, a number of which follow as special cases from the Poincaré and Chetayev equations.     

The thermodynamically consistent equations of a thermoelastic saturated porous medium

A phenomenological model of a porous medium saturated with fluid is considered with in the framework of the hypothesis of interpenetrating continua. Assuming that there are no phase transitions, that the contribution of pulsations to the stress tensor and kinetic energy is small, and the components of the medium are in thermal equilibrium, mass, momentum and energy equations and a law of conservation of compatibility of the deformations and velocities are formulated. Using a representation of the force of interaction of the components in the form of the sum of equilibrium and dissipative components, a new form of inequality is obtained for the rate of entropy production. A definition of a thermoelastic saturated porous medium is given. The symmetry group of such a medium is considered as a set of two groups, corresponding to the symmetry of the skeleton and the fluid. It is shown that, in the class of thermoelastic porous media with an arbitrary type of symmetry of the skeleton, the saturating fluid can only be an ideal fluid, while the thermodynamic potentials and the porosity, stresses and entropies determined by them do not depend on the temperature gradient and the relative fluid velocity. It is found that the condition of incompressibility of only one of the components of the medium leads to the elimination of the porosity from the governing relations, rather than to kinematic limitations. The limitations imposed on the governing relations by the principle of thermodynamic consistency and the requirement of independence of the choice of the frame of reference are investigated. A form of the governing relations, necessary and sufficient to satisfy these principles, is obtained. It is shown that the Biot equations are one of the forms of thermodynamically consistent governing relations. A thermodynamic validation of the effective-stress tensor is given.     

Smoothness of solutions of functional differential equations

It is well known that the solutions of functional differential equations have jump-discontinuities in their derivatives, unless some rather restrictive compatibility conditions are imposed upon the initial function. In this paper a method for the calculation of the position and the value of these jumps is presented, and conditions that are sufficient for the solution to be smooth between the jump-discontinuities are given.     

A Quasisteady Stefan Problem with Curvature Correction and Kinetic Undercooling

A quasisteady Stefan problem with curvature correction and kinetic undercooling is considered. It is a problem with phase transition, in which not only the Stefan condition, but also the curvature correction and kinetic undercooling effect hold on the free boundary, and in phase regions elliptic equations are satisfied by the unknown temperature at each time. The existence and uniqueness of a local classical solution of this problem are obtained.     

Application of the Fourier Transform to the Solution of Singular Integral Equations

A method for solving a certain system of singular integral equations with constant coefficients is proposed. It is based on a procedure for reducing singular equations to equations with continuous difference kernel; the solution of the latter is constructed by using the classical Fourier transform in the class of absolutely integrable functions. Explicit expressions for the solution of the singular integral equations under consideration are obtained.     

Point vortices and classical orthogonal polynomials

Stationary equilibria of point vortices in the plane and on the cylinder in the presence of a background flow are studied. Vortex systems with an arbitrary choice of circulations are considered. Differential equations satisfied by generating polynomials of vortex configurations are derived. It is shown that these equations can be reduced to a single one. It is found that polynomials that are Wronskians of classical orthogonal polynomials solve the latter equation. As a consequence vortex equilibria at a certain choice of background flows can be described with the help of Wronskians of classical orthogonal polynomials.     

Determination of Lagrangians from Equations of Motion and Commutator Brackets

The question as to whether a given set of equations, which govern the dynamical evolution of a system, determine a Lagrangian is considered. This problem, which has come to be known as the inverse problem of the calculus of variations, is reviewed and theorems which contain systems of partial differential equations which determine a type of self-adjointness are developed. It is shown that, given a reasonable form for the classical correspondence, the usual quantum commutator brackets can be expressed in terms of classical quantities which satisfy a particular form of these equations.