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.     

Stability and boundedness of solutions of Stieltjes Differential Equations

Dynamical equations on time scales are formulated by means of Stieltjes differential equations, which, depending on the time integrator, include ordinary differential equations and difference equations as well as mixtures of both. Explicit conditions for the boundedness and stability of solutions are presented here for linear and nonlinear Stieltjes differential equations. In addition, the continuous dependence of solutions on the time integrator is established by means of a Gronwall-like inequality for equations with different time integrators.     

Study of the Solvability of Some Volterra-Type Integral and Integro-Differential Equations of Third Kind

For Volterra integral equations of the third kind and for Volterra-type integrodifferential equations of the third kind, theorems on the existence of solutions in Sobolev spaces (i.e., regular solutions) are proved. The proofs are based on the theory of boundary value problems for degenerate ordinary differential equations and on the theory of boundary value problems for parabolic equations with a changing evolution direction.     

一类对偶积分方程组正则化为Cauchy奇异积分方程组解法

本文基于Mellin变换法求解复杂更一般形式的对偶积分方程组．通过积分变换，由实数域化成复数域上的方程组，引入未知函数的积分变换，移动积分路径，应用Cauchy积分定理，实现退耦正则化为Cauchy奇异积分方程组，由此给出一般性解，并严格证明了对偶积分方程组退耦正则化为Cauchy奇异积分方程组与原对偶积分方程组等价性，以及对偶积分方程组解的存在性和唯一性．给出的解法和理论解，作为求解复杂对偶积分方程组一种有效解法，可供求解复杂的数学、物理、力学中的混合边值问题应用．     

Regularized shallow water equations and an efficient method for numerical simulation of shallow water flows

Regularized shallow water equations are derived as based on a regularization of the Navier-Stokes equations in the form of quasi-gasdynamic and quasi-hydrodynamic equations. Efficient finite-difference algorithms based on the regularized shallow water equations are proposed for the numerical simulation of shallow water flows. The capabilities of the model are examined by computing a test Riemann problem, the flow over an obstacle, and asymmetric dam break.     

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 use of splitting of the equations of motion in the numerical solution of problems of the dynamics of elastic systems

Linear elastic systems with a finite number of degrees of freedom, the initial equations of motion of which are constructed using the finite element method or other discretization methods, are considered. Since, in applied dynamics problems, the motions are usually investigated in a frequency range with an upper bound, the degrees of freedom of the initial system of equations are split into dynamic and quasi-dynamic degrees. Finally, the initial system of equations is split into a small number of differential equations for the dynamic degrees of freedom and into a system of algebraic equations for determining the quasi-static displacements, represented in the form of a matrix series. The number of terms of the series taken into account depends on the accuracy required.     

An extension of the Hamilton-Jacobi method

A method of solving the canonical Hamilton equations, based on a search for invariant manifolds, which are uniquely projected onto position space, is proposed. These manifolds are specified by covector fields, which satisfy a system of first-order partial differential equations, similar in their properties to Lamb's equations in the dynamic of an ideal fluid. If the complete integral of Lamb's equations is known, then, with certain additional assumptions, one can integrate the initial Hamilton equations explicitly. This method reduces to the well-known Hamilton-Jacobi method for gradient fields. Some new conditions for Hamilton's equations to be accurately integrable are indicated. The general results are applied to the problem of the motion of a variable body.     

Analysis of the Euler–Poisson equations by methods of power geometry and normal form

New approaches and methods for studying non-linear problems are applied to the classical problem of the motion of a heavy rigid body about a fixed point, i.e., to the system of Euler–Poisson equations. All the asymptotic expansions of the solutions of the Kowalewski equations, to which the Euler–Poisson equations reduce when certain constraints are imposed on the parameters, are found using power geometry. They form 24 families. Then all the exact solutions of the Kowalewski equations of a specific class (which includes almost all the known exact solutions) are found on the basis of these expansions. Five new families of such solutions are found. Instead of the conventional technique of studying the global integrability of the Euler–Poisson equations, studying their local integrability near stationary and periodic solutions is proposed. Normal forms are used for this purpose. Sets of real stationary solutions, in the vicinity of which these equations are locally integrable, are discovered using them. Other real stationary solutions, in the vicinity of which the Euler–Poisson equations are locally non-integrable, are also found. This is established using the theory of resonant normal forms developed and computer calculations of the coefficients of a normal form.     

Navier-Stokes方程五模类Lorenz方程组的动力学行为及数值仿真

本文对平面正方形区域上不可压缩的Navier-Stokes方程,进行傅立叶展开后,截断得到五模类Lorenz方程组.给出了该方程组定常解及其稳定性的讨论,证明了该方程组吸引子的存在性,并对其全局稳定性进行了分析和讨论,数值模拟了雷诺数在一定范围内变化时,类Lorenz方程组的动力学行为.     

On stabilization of solutions to incomplete second-order integrodifferential equations

We consider abstract incomplete linear second-order integrodifferential equations in a Hilbert space. Operator coefficients of the equations are unbounded selfadjoint nonnegative operators. These equations arise naturally in viscoelasticity and hydroelasticity. We prove a theorem on asymptotic stability of strong solutions of the equations.     

Fundamental solutions in the linear theory of consolidation for elastic solids with double porosity

This paper concerns the theory of consolidation for elastic solids with double porosity, and the governing fully coupled linear quasi-static equations are considered. The system of these equations is based on the equilibrium equations for a solid, conservation of fluid mass, the effective stress concept, and Darcy’s law for material with double porosity. Two levels of spatial cases of consolidation theory for a solid with double porosity are considered: equations of steady vibrations and equations of equilibrium. The fundamental solutions of these equations are constructed by means of elementary functions. Finally, the basic properties of these solutions are established.     

Decomposition of a hierarchy of nonlinear evolution equations

The generalized Hamiltonian structures for a hierarchy of nonlinear evolution equations are established with the aid of the trace identity. Using the nonlinearization approach, the hierarchy of nonlinear evolution equations is decomposed into a class of new finite-dimensional Hamiltonian systems. The generating function of integrals and their generator are presented, based on which the finite-dimensional Hamiltonian systems are proved to be completely integrable in the Liouville sense. As an application, solutions for the hierarchy of nonlinear evolution equations are reduced to solving the compatible Hamiltonian systems of ordinary differential equations.     

The Theory of Linear G-Difference Equations

We introduce the notion of difference equations defined on a structured set. The symmetry group of the structure determines the set of difference operators. All main notions in the theory of difference equations are introduced as invariants of the action of the symmetry group. Linear equations are modules over the skew group algebra, solutions are morphisms relating a given equation to other equations, symmetries of an equation are module endomorphisms, and conserved structures are invariants in the tensor algebra of the given equation.We show that the equations and their solutions can be described through representations of the isotropy group of the symmetry group of the underlying set. We relate our notion of difference equation and solutions to systems of classical difference equations and their solutions and show that out notions include these as a special case.     

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.     

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.     

Derivation of Macroscale Equations for a Class of Physical Applications

A macroscale formulation is constructed from a system of partialdifferential equations which govern the microscale dependent variables. Theconstruction is based upon transformations of equations on second ordercontact manifolds under conditions of integrability. Necessary conditions onthe structure of the macroscale equations are obtained under the requirementthat the solutions of the macroscale equations satisfy, in some approximatesense, the equations associated with the microscale. This approach offers amethod whereby one can construct only those macroscale equations that can bevalidated by a condition of consistency based on the model error. Themethodology is employed to construct a turbulence closure model forincompressible flow. It is shown that the large eddy viscosity, whichsatifies contemporary tests based on Galilean invariance, fails theconsistency condition defined here.     

Short time uniqueness results for solutions of nonlocal and non-monotone geometric equations

We describe a method to show short time uniqueness results for viscosity solutions of general nonlocal and non-monotone second-order geometric equations arising in front propagation problems. Our method is based on some lower gradient bounds for the solution. These estimates are crucial to obtain regularity properties of the front, which allow to deal with nonlocal terms in the equations. Applications to short time uniqueness results for the initial value problems for dislocation type equations, asymptotic equations of a FitzHugh–Nagumo type system and equations depending on the Lebesgue measure of the fronts are presented.     

复微分-差分方程组的超越整函数解

利用亚纯函数的Nevanlinna值分布理论, 我们主要研究了一类复微分-差分方程和一类复微分-差分方程组的有限级超越整函数解的存在形式, 得到两个有趣的结论. 将复微分(差分)方程的一些结论推广到复微分-差分方程(组)中.     

Dynamic motion of whirling rods with Coriolis effect

Longitudinal vibrations coupled with transverse vibrations of whirling rods are investigated. It is known that longitudinal and transverse vibrations are governed by second and fourth order differential equations, respectively. Due to the Coriolis effect, a system of equations that governs the longitudinal and transverse displacements will be constructed by coupling these two equations together. Solutions of the equations assume small oscillations of vibration being superimposed on the steady state of the whirling rod. Exact and approximate solutions are obtained from the proposed governing equations, where the approximate solutions on displacements and natural frequencies are acquired by neglecting the Coriolis effect. A proposed numerical scheme known as complete function collocation method is implemented to solve the governing equations coupled with longitudinal and transverse displacements. The approximate results on both longitudinal and transverse natural frequencies show that natural frequencies are decreasing while the angular velocity of the rod is increasing. Exact and numerical results on both longitudinal and transverse natural frequencies show that there are no predictable trends whether natural frequencies are increasing or decreasing while the angular velocity of the rod is increasing.