输油管道油流静电带电数值计算

对输油管道内油品流动带电问题的数值计算进行了研究.紊流条件下的电荷输运方程是一个对流占优的对流扩散反应方程,采用算子分裂法,将该方程分解为纯对流方程、纯扩散方程和纯反应方程,分别采用特征线法和差分法求解.算例证明,该方法能准确描述管道内电荷分布,因而提供了一种获取冲流电流的可靠方法.     

The integrability of an extended fifth-order KdV equation with Riccati-type pseudopotential

The extended fifth-order KdV equation in fluids is investigated in this paper. Based on the concept of pseudopotential, a direct and unifying Riccati-type pseudopotential approach is employed to achieve Lax pair and singularity manifold equation of this equation. Moreover, this equation is classified into three categories: extended Caudrey–Dodd–Gibbon–Sawada–Kotera (CDGSK) equation, extended Lax equation and extended Kaup–Kuperschmidt (KK) equation. The corresponding singularity manifold equations and auto-Bäcklund transformations of these three equations are also obtained. Furthermore, the infinitely many conservation laws of the extended Lax equation are found using its Lax pair. All conserved densities and fluxes are given with explicit recursion formulas.     

A Note on the Relationship Between Solutions of Einstein, Ramanujan and Chazy Equations

The Einstein equation for the Friedmann-Robertson-Walker metric plays a fundamental role in cosmology. The direct search of the exact solutions of the Einstein equation even in this simple metric case is sometime a hard job. Therefore, it is useful to construct solutions of the Einstein equation using a known solutions of some other equations which are equivalent or related to the Einstein equation. In this work, we establish the relationship the Einstein equation with two other famous equations namely the Ramanujan equation and the Chazy equation. Both these two equations play an important role in the number theory. Using the known solutions of the Ramanujan and Chazy equations, we find the corresponding solutions of the Einstein equation.     

基于标架场理论的完整系统Boltzmann-Hamel方程简化方法研究

研究选取合适的准坐标简化完整系统Boltzmann-Hamel方程的问题.基于流形上的标架场理论,指出了定常构形空间中的准速度与标架场的联系,并从几何不变性的角度上导出了完整系统的Boltzmann-Hamel方程.证明了对于任意广义力为零的均匀构形空间、广义力不为零的零曲率构形空间,Boltzmann-Hamel方程均可以化简为可积分的形式,同时给出具体的简化方法并举例说明本方法的适用性.本文方法为寻找运动方程的解析解提供了一条新途径.     

Causal interpretation of the modified Klein-Gordon equation

A consistent causal interpretation of the Klein-Gordon equation treated as a field equation has been developed, and leads to a model of entities described by the Klein-Gordon equation, i.e., spinless, massive bosons, as objectively existing fields. The question arises, however, as to whether a causal interpretation based on a particle ontology of the Klein-Gordon equation is also possible. Our purpose in this article will be to indicate, by making what we believe is a best possible attempt at developing a particle interpretation of the Klein-Gordon equation, that such an interpretation is untenable. To resolve the nonpositive-definite probability density difficulties with the Klein-Gordon equation, we modify this equation by the introduction of an evolution parameter. We base our subsequent considerations on this modified Klein-Gordon equation. Partly to motivate the need for a relativistic causal interpretation and partly to give emphasis to aspects of the causal interpretation often overlooked, we begin our article with a brief historical survey of the causal interpretation.Other work commitments prevented publication of this article in the special issue ofFoundations of Physics in honor of Prof. J. P. Vigier. I would nevertheless like to dedicate this work to Prof. Vigier in recognition of this untiring contributions to the causal interpretation in particular and to the foundations of physics in general. I take this opportunity to thank Prof. Vigier for his help during my Royal Society fellowship spent at the Institut Henri Poincaré in the academic year 1988–1989.     

The Fundamental Equations of Change in Statistical Ensembles and Biological Populations

A recent article in Nature Physics unified key results from thermodynamics, statistics, and information theory. The unification arose from a general equation for the rate of change in the information content of a system. The general equation describes the change in the moments of an observable quantity over a probability distribution. One term in the equation describes the change in the probability distribution. The other term describes the change in the observable values for a given state. We show the equivalence of this general equation for moment dynamics with the widely known Price equation from evolutionary theory, named after George Price. We introduce the Price equation from its biological roots, review a mathematically abstract form of the equation, and discuss the potential for this equation to unify diverse mathematical theories from different disciplines. The new work in Nature Physics and many applications in biology show that this equation also provides the basis for deriving many novel theoretical results within each discipline.     

A non-relativistic wave equation of second order in time for scalar fields

In this paper a generalization of the traditional non-relativistic Schrödinger equation is considered. It is a wave equation of second order in time and fourth order in the space coordinates for scalar fields. The equation has certain features, which make it a closer analogue of the Klein-Gordon equation than the traditional Schrödinger equation. However, the equation maintains the non-relativistic relation between energy and momentum.The implications of this generalized wave equation and the quantized field theory based on it are studied. The theory can be shown to be charge symmetric and allows to introduce anti-particles and pair creation. We compare the Green functions for this theory with those of conventional non-relativistic quantum theory.The theory allows to formulate a transformation for charge conjugation. The PCT-theorem is valid for it. The usual spin-statistics connection holds.     

A discussion of the relativistic equal-time equation

Ruan Tu-nan et al. [1] have proposed an equal-time equation for composite particles which is derived from Bethe-Salpeter (B-S) equation. Its advantage is that the kernel of this equation is a completely definite single rearrangement of the B-S irreducible kernel without any artificial assumptions. In this paper we shall give a further discussion of the properties of this equation. We discuss the behaviour of this equation as the mass of one of the two particles approaches the limitM ₂→∞ in the ladder approximation of single photon exchange. We show that up to orderO(α⁴) this equation is consistent with the Dirac equation. If the crossed two photon exchange diagrams are taken into account the difference between them is of orderO(α⁶).     

Derivation of Generalized Camassa-Holm Equations from Boussinesq-type Equations

In this paper we derive generalized forms of the Camassa-Holm (CH) equation from a Boussinesq-type equation using a two-parameter asymptotic expansion based on two small parameters characterizing nonlinear and dispersive effects and strictly following the arguments in the asymptotic derivation of the classical CH equation. The resulting equations generalize the CH equation in two different ways. The first generalization replaces the quadratic nonlinearity of the CH equation with a general power-type nonlinearity while the second one replaces the dispersive terms of the CH equation with fractional-type dispersive terms. In the absence of both higher-order nonlinearities and fractional-type dispersive effects, the generalized equations derived reduce to the classical CH equation that describes unidirectional propagation of shallow water waves. The generalized equations obtained are compared to similar equations available in the literature, and this leads to the observation that the present equations have not appeared in the literature.     

A General Solution of the Fokker-Planck Equation

The Fokker-Planck equation is useful to describe stochastic processes. Depending on the force acting in the system, the solution of this equation becomes complicated and approximate or numerical solutions are needed. The relation with the Schrödinger equation allows building a method to obtain solutions of the Fokker-Planck equation. However, this approach has been limited to the study of confined potentials, restricting its applicability. In this work, we suggest a general treatment for non-confining potentials through the use of series of functions based on the solution of the Schrödinger equation, with part of discrete spectrum and part of continuum spectrum. Two examples, the Rosen-Morse potential and a limited harmonic potential, are analyzed using the suggested approach.     

Applications of two reliable methods for solving a nonlinear conformable time-fractional equation

The current work presents analytical solutions of a nonlinear conformable time-fractional equation by using two different techniques. These are the modified simple equation method and the exponential rational function method. Based on the conformable fractional derivative and traveling wave transformation, the fractional partial differential equation is turned into the nonlinear non-fractional ordinary differential equation. Therefore, we implement the algorithms to this nonlinear non-fractional ordinary differential equation. To the best of our knowledge, the exact solutions obtained in this paper might be very useful in various areas of applied mathematics in interpreting some physical phenomena.     

Direct linearization of nonlinear difference-difference equations

Starting from the linear integral equation for the solutions of the Korteweg-de Vries (KdV) equation, we obtain the direct linearization of a general nonlinear difference-difference equation. In a continuum limit this equation reduces to a general integrable differential-difference equation which contains e.g. the Toda equation and the discrete KdV and MKdV as special cases.     

Harmonic oscillations of elastic continua and detection of gravitational waves

In this article it is shown that the equation derived by weber from the equation of geodesic deviation — the equation that constitutes the basis for the theoretical studies concerning the detection of gravitational waves — can be obtained as an approximation to an equation deduced from Cauchy's equation which governs an elastic continuum. This is achieved within the frame of the formalism of the theory of relativistic continua developed by A. C. Eringen and his collaborators and the present author. The use of piezoelectric crystals in order to measure the stresses that result from incident gravitational waves is also examined within the framework provided by this theory.     

Double Reduction and Exact Solutions of Zakharov-Kuznetsov Modified Equal width Equation with Power Law Nonlinearity via Conservation Laws

The conservation laws for the (1+2)-dimensional Zakharov-Kuznetsov modified equal width (ZK-MEW) equation with power law nonlinearity are constructed by using Noether's approach through an interesting method of increasing the order of this equation. With the aid of an obtained conservation law, the generalized double reduction theorem is applied to this equation. It can be shown that the reduced equation is a second order nonlinear ODE. Finally, some exact solutions for a particular case of this equation are obtained after solving the reduced equation.     

Application of the complex point source method to the Schrödinger equation

The paraxial wave equation is a reduced form of the Helmholtz equation. Its solutions can be directly obtained from the solutions of the Helmholtz equation by using the method of complex point source. We applied the same logic to quantum mechanics, because the Schrödinger equation is parabolic in nature as the paraxial wave equation. We defined a differential equation, which is analogous to the Helmholtz equation for quantum mechanics and derived the solutions of the Schrödinger equation by taking into account the solutions of this equation with the method of complex point source. The method is applied to the problem of diffraction of matter waves by a shutter.     

Auto-Bäcklund transformations for a matrix partial differential equation

We derive auto-Bäcklund transformations, analogous to those of the matrix second Painlevé equation, for a matrix partial differential equation. We also then use these auto-Bäcklund transformations to derive matrix equations involving shifts in a discrete variable, a process analogous to the use of the auto-Bäcklund transformations of the matrix second Painlevé equation to derive a discrete matrix first Painlevé equation. The equations thus derived then include amongst other examples a semidiscrete matrix equation which can be considered to be an extension of this discrete matrix first Painlevé equation. The application of this technique to the auto-Bäcklund transformations of the scalar case of our partial differential equation has not been considered before, and so the results obtained here in this scalar case are also new. Other equations obtained here using this technique include a scalar semidiscrete equation which arises in the case of the second Painlevé equation, and which does not seem to have been thus derived previously.     

Painlevé Property and Allowed Transformations of the Variable-Coefficient Zakharov-Kuznetsov Equation

In this letter, we discuss the Painlevé property and allowed transformation for the variablecoefficient Zakharov-Kuznetsov equation which governs nonlinear ion-acoustic wqves in a magnetized plasma. The general solution of the singular manifold equation and stability solution of the VCZK equation are obtained. To prove some information of the integrability of the ZK equation, we prove the constraints that the variable-coefficient functions for the equation to possess the Painlevé property are not equivalent in order that the equations may be transformed into the constant coefficient equation. So we confirm that the ZK equation is not integrable.     

Existence of traveling wave solutions of a high-order nonlinear acoustic wave equation

In this Letter, we present an analytical study of a high-order acoustic wave equation in one dimension, and reformulate a previously given equation in terms of an expansion of the acoustic Mach number. We search for non-trivial traveling wave solutions to this equation, and also discuss the accuracy of acoustic wave equations in terms of the range of Mach numbers for which they are valid.