Group properties of the extended Green–Naghdi equations

The group analysis method is applied to the extended Green–Naghdi equations. The equations are studied in the Eulerian and Lagrangian coordinates. The complete group classification of the equations is provided. The derived Lie symmetries are used to reduce the equations to ordinary differential equations. For solving the ordinary differential equations the Runge–Kutta methods were applied. Comparisons between solutions of the Green–Naghdi equations and the extended Green–Naghdi equations are given.     

An examination of the homentropic Euler equations with averaged characteristics

This paper examines the properties of the homentropic Euler equations when the characteristics of the equations have been spatially averaged. The new equations are referred to as the characteristically averaged homentropic Euler (CAHE) equations. An existence and uniqueness proof for the modified equations is given. The speed of shocks for the CAHE equations are determined. The Riemann problem is examined and a general form of the solutions is presented. Finally, numerically simulations on the homentropic Euler and CAHE equations are conducted and the behaviors of the two sets of equations are compared.     

Structural theory of special functions

A block diagram is suggested for classifying differential equations whose solutions are special functions of mathematical physics. Three classes of these equations are identified: the hypergeometric, Heun, and Painlevé classes. The constituent types of equations are listed for each class. The confluence processes that transform one type into another are described. The interrelations between the equations belonging to different classes are indicated. For example, the Painlevé-class equations are equations of classical motion for Hamiltonians corresponding to Heun-class equations, and linearizing the Painlevé-class equations leads to hypergeometric-class equations. The “confluence principle” is stated, and an example of its application is given. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 1, pp. 3–19, April, 1999.     

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.     

Tensor absolute value equations

This paper is concerned with solving some structured multi-linear systems, which are called tensor absolute value equations. This kind of absolute value equations is closely related to tensor complementarity problems and is a generalization of the well-known absolute value equations in the matrix case. We prove that tensor absolute value equations are equivalent to some special structured tensor complementary problems. Some sufficient conditions are given to guarantee the existence of solutions for tensor absolute value equations. We also propose a Levenberg-Marquardt-type algorithm for solving some given tensor absolute value equations and preliminary numerical results are reported to indicate the efficiency of the proposed algorithm.     

More common errors in finding exact solutions of nonlinear differential equations: Part I

In the recent paper by Kudryashov [11] seven common errors in finding exact solutions of nonlinear differential equations were listed and discussed in detail. We indicate two more common errors concerning the similarity (equivalence with respect to point transformations) and linearizability of differential equations and then discuss the first of them. Classes of generalized KdV and mKdV equations with variable coefficients are used in order to clarify our conclusions. We investigate admissible point transformations in classes of generalized KdV equations, obtain the necessary and sufficient conditions of similarity of such equations to the standard KdV and mKdV equations and carried out the exhaustive group classification of a class of variable-coefficient KdV equations. Then a number of recent papers on such equations are commented using the above results. It is shown that exact solutions were constructed in these papers only for equations which are reduced by point transformations to the standard KdV and mKdV equations. Therefore, exact solutions of such equations can be obtained from known solutions of the standard KdV and mKdV equations in an easier way than by direct solving. The same statement is true for other equations which are equivalent to well-known equations with respect to point transformations.     

层状二维流动的基本方程式

在很多海洋、大气等二维流动问题中所用的动力学方程往往沿用推广后的河流水力学方程或"纳维-斯托克斯方程"其中把湍流阻力项写成这样的方程式和湍流阻力项用到实际问题上去,无疑是存在着极大的局限性,而将导致矛盾百出.本文则从雷诺方程出发,把所有的物理量沿深度加以平均,求出平均以后的物理量所满足的运动方程,连续方程和扩散方程.     

Existence of solutions for the MHD-Leray-alpha equations and their relations to the MHD equations

In this paper we derive some new equations and we call them MHD-Leray-alpha equations which are similar to the MHD equations. We put forward the concept of weak and strong solutions for the new equations. Whether the 3-dimensional MHD equations have a unique weak solution is unknown, however, there is a unique weak solution for the 3-dimensional MHD-Leray-alpha equations. The global existence of strong solution and the Gevrey class regularity for the new equations are also obtained. Furthermore, we prove that the solutions of the MHD-Leray-alpha equations converge to the solution of the MHD equations in the weak sense as the parameter ε in the new equations converges to zero.     

Factorization method for linear and quasilinear singularly perturbed boundary value problems for ordinary differential equations

For linear singularly perturbed boundary value problems, we come up with a method that reduces solving a differential problem to a discrete (difference) problem. Difference equations, which are an exact analog of differential equations, are constructed by the factorization method. Coefficients of difference equations are calculated by solving Cauchy problems for first-order differential equations. In this case nonlinear Ricatti equations with a small parameter are solved by asymptotic methods, and solving linear equations reduces to computing quadratures. A solution for quasilinear singularly perturbed equations is obtained by means of an implicit relaxation method. A solution to a linearized problem is calculated by analogy with a linear problem at each iterative step. The method is tested against solutions to the known Lagerstrom-Cole problem.     

The equations of compatibility of deformations

Nine equations of compatibility of deformations are obtained in which, unlike the classical Saint-Venant compatibility equations, only first derivatives with respect to the coordinates occur. It is proved that, of these nine equations, only six are independent. It is shown that the classical compatibility equations can be obtained from these equations.     

Integrating factors, adjoint equations and Lagrangians

Integrating factors and adjoint equations are determined for linear and non-linear differential equations of an arbitrary order. The new concept of an adjoint equation is used for construction of a Lagrangian for an arbitrary differential equation and for any system of differential equations where the number of equations is equal to the number of dependent variables. The method is illustrated by considering several equations traditionally regarded as equations without Lagrangians. Noether's theorem is applied to the Maxwell equations.     

Study of Volterra integro-differential equations arising in viscoelasticity theory

Volterra integrodifferential equations with unbounded operator coefficients in a Hilbert space that are operator models of integrodifferential equations arising in viscoelasticity theory are studied. These equations are shown to be well-posed in Sobolev spaces of vector functions, and spectral analysis is applied to the operator functions that are the symbols of the given equations.     

Mac-Millan方程对非线性非完整力学系统的推广

本文首先将Mac-Millan方程推广到最一般的非完整力学系统,得到非线性非完整系统的广义Mac-Millan方程.其次,证明广义Mac-Millan方程与广义Чаплыгин方程的等价性,最后给出一个例子.     

Perturbation of Volterra Difference Equations

A fixed point theorem is used to investigate nonlinear Volterra difference equations that are perturbed versions of linear equations. Sufficient conditions are established to ensure that the stability properties of linear Volterra difference equations are preserved under perturbation. The existence of asymptotically periodic solutions of perturbed Volterra difference equations is also proved.     

Modulation equations and Reynolds averaging for finite-amplitude non-linear waves in an incompressible fluid

A formal perturbation scheme is developed to determine originalmodulation equations for laminar finite-amplitude non-linearwaves in an incompressible fluid. Three idealized problems areanalysed. The modulation equations comprise conservation ofwaves, averaged conditions for conservation of mass, momentum,kinetic energy and angular momentum and the averaged projectionof the Navier–Stokes equations onto the vorticity vector.The last of these modulation equations, which is related tovortex stretching, only appears in 3D problems. The techniqueof Reynolds averaging is also employed to obtain equations forthe mean velocities and pressure. The Reynolds-averaged Navier–Stokesequations correspond to the modulation equations for conservationof mass and momentum. However, the Reynolds stress transportequations are shown to be inconsistent with the other necessarymodulation equations. In two further idealized problems, exactsolutions of the Navier–Stokes equations are obtainedby employing the modulation equations.     

非齐次Toda晶格的对称,精确解和可积性

研究非齐次Toda晶格，即一类非齐次非线性微分差分方程的对称与可积性。给出了这一类方程的Lie点对称，条件对称和精确解。给出这类方程与Toda晶格之间的可逆点变换，从而表明这一类方程是可积的。     

The Lagrange differential-algebraic equations

The Lagrange equations are considered in the case when the Lagrangen function is independent of some of the velocities, and the properties of the differential-algebraic equations arising are studied. It is proved that, when non-degeneracy conditions are satisfied, the equations arising reduce to differential Lagrange equations of lower dimension. The problem of the planar oscillations of an elastic pendulum is considered as an example.     

A method for the numerical solution of the Painlevé equations

A numerical method for solving the Cauchy problem for all the six Painlevé equations is proposed. The difficulty of solving these equations is that the unknown functions can have movable (that is, dependent on the initial data) singular points of the pole type. Moreover, the Painlevé III–VI equations may have singularities at points where the solution takes certain finite values. The positions of all these singularities are not a priori known and are determined in the process of solving the equation. The proposed method is based on the transition to auxiliary systems of differential equations in neighborhoods of the indicated points. The equations in these systems and their solutions have no singularities at the corresponding point and its neighborhood. Such auxiliary equations are derived for all Painlevé equations and for all types of singularities. Efficient criteria for transition to auxiliary systems are formulated, and numerical results illustrating the potentials of the method are presented.     

On RF-pairs, Bäcklund transformations and linearization of nonlinear equations

Some classes of nonlinear equations of mathematical physics are described that admit order reduction through the use of a hydrodynamic-type transformation, where the unknown function is taken as a new independent variable and an appropriate partial derivative is taken as the new dependent variable. RF-pairs and associated Bäcklund transformations are constructed for evolution equations of general form. The results obtained are used for order reduction of hydrodynamic equations (Navier-Stokes and boundary layer) and constructing exact solutions to these equations. A generalized Calogero equation and a number of other new linearizable nonlinear differential equations of the second, third and forth orders are considered. Some integro-differential equations are analyzed.     

A method for calculating the Painlevé transcendents

A numerical method for solving the Cauchy problem for all the six Painlevé equations is proposed. The difficulty of solving these equations is that the unknown functions can have movable (that is, dependent on the initial data) singular points of the pole type. Moreover, the Painlevé III–VI equations may have singularities at points where the solution takes certain finite values. The positions of all these singularities are not a priori known and are determined in the process of solving the equation. The proposed method is based on the transition to auxiliary systems of differential equations in neighborhoods of the indicated points. The equations in these systems and their solutions have no singularities at the corresponding point and its neighborhood. Such auxiliary equations are derived for all Painlevé equations and for all types of singularities. Efficient criteria for transition to auxiliary systems are formulated, and numerical results illustrating the potentials of the method are presented.