导数空间另一类D‘Alembert原理有及任意阶非完整系统的新？…

作为笔者在文献（１，２，３）中提出的“导数空间”和导数空间中“相当动能”概念及“将非完整系统变为形式上的完整系统处理”思想的具体应用，本文导出了又一类新型的万有Ｄ’Ａｌｅｍｂｅｒｔ原理及适用于任意阶非完整系统的新的Ｍａｇｇｉ方程，并给出了应用此方程的算例。     

事件空间中单面非Chetaev型非完整系统的Noether定理

研究事件空间中单面非Ｃｈｅｔａｅｖ型非完整系统的Ｎｏｅｔｈｅｒ定理,首先给出了系统的Ｄ’Ａｌｅｍｂｅｒｔ－Ｌａｇｒａｎｇｅ原理;其次基于该原理在无限小变换下的不变性,研究了非Ｃｈｅｔａｅｖ型非完整系统的Ｎｏｅｔｈｅｒ定理及逆定理;最后举例说明结果的应用。     

变质量非完整系统打击运动的Kane方程

本文建立变质量非完整系统Routh形式的Kane方程,并由此导出变质量完整系统和非完整系统打击运动的Kane方程,其次指出Lagrange形式的打击运动方程与Kane方程的等价性。最后举例说明新方程的应用。     

带一类第一积分的非完整系统的Szebehely问题

研究非完整系统动力学的一类逆问题·给出非完整系统的运动方程及其显式，考虑一类仅受齐次非完整约束的力学系统的Ｓｚｅｂｅｈｅｌｙ问题，研究已知一类第一积分的一般非完整系统的情形·最后举例说明其应用·     

Poincare动力学方程Whittaker降阶法

Ｗｈｉｔｔａｋｅｒ降阶法是利用能量积分将一个完整动力系统的Ｌａｇｒａｎｇｅ运动方程降阶。本文论述了据李群理论构造的Ｐｏｉｎｃａｒｅ方程所描述的非完整保守系统的相应结果。     

一般力学中三类变量的广义变分原理

应用对合变换,将两类变量的广义变分原理的驻值条件变换为三类变量的基本方程.按照广义力和广义位移之间的对应关系,将各基本方程乘上相应的虚量,代数相加,然后积分,进而建立了完整系统的三类变量的广义变分原理.应用这种凑合法,建立了非完整系统的三类变量的广义变分原理.作为例子,将一般力学中的三类变量的广义变分原理和两类变量的广义变分原理推广应用于弹性动力学中.最后,讨论了有关的问题.     

Mac-Millan方程的推广

将动力学原理及Appell-Четаев定义推广到非惯性系,由此导出非惯性系中的非线性非完整系统的Mac-Millan方程.     

变质量高阶非ЧeTaeB型非完整约束系统动力学

本文给出了高了阶非ЧｅＴａｅＢ约束加在广义虚位移上的限制条件，建立了变质量高阶非ЧｅＴａｅＢ型非线性非完整系统的Ｒｏｕｔｈ方程，ЧａПЛЫГИН方程、Ｎｉｅｌｓｅｎ方程，给出了高阶非ЧｅＴａｅＢ型约束系统“ｄ”与“δ”之间的产换关系，建立了其积分变分原理，并得到了变质量高阶非ЧｅＴａｅＢ型约束系统的广义Ｎｏｅｔｈｅｒ守恒律。     

非完整系统在Gauss白噪声下的扰动

本文研究非完整系统在Gauss白噪声下的扰动,证明解过程的一次矩方程与无扰动情形下的方程一致,二次矩方程不含ε项,但包含ε2项,从而得出两个命题.最后,举例说明结果的应用.     

高阶非完整系统运动方程的一类积分

本文给出高阶非完整系统运动方程的一类积分及其存在条件,包括1阶积分(广义能量积分),2阶积分和p(p>2)阶积分,所有这些积分都可按系统的Lagrange函数来构造.举例说明本文方法的应用.     

Traveling wave solutions for fractional partial differential equations arising in mathematical physics by an improved fractional Jacobi elliptic equation method

In this paper, combining with a new generalized ansätz and the fractional Jacobi elliptic equation, an improved fractional Jacobi elliptic equation method is proposed for seeking exact solutions of space‐time fractional partial differential equations. The fractional derivative used here is the modified Riemann‐Liouville derivative. For illustrating the validity of this method, we apply it to solve the space‐time fractional Fokas equation and the the space‐time fractional BBM equation. As a result, some new general exact solutions expressed in various forms including the solitary wave solutions, the periodic wave solutions, and Jacobi elliptic functions solutions for the two equations are found with the aid of mathematical software Maple. Copyright © 2016 John Wiley & Sons, Ltd.     

含一阶导数的二阶微分方程组多点边值问题正解的存在性

讨论了一类含一阶导数的二阶常微分方程组多点边值问题正解的存在性.利用一个新的不动点定理,得到上述问题具有一个正解的充分条件.     

Dynamical systems method (DSM) for nonlinear equations in Banach spaces

The DSM (dynamical systems method) is justified for nonlinear operator equations in a Banach space. The main assumption is on the spectral properties of the Frèchet derivative of the operator at a suitable point. A singular perturbation problem related to the original equation is studied.     

NUMERICAL SOLUTION OF THE SPACE FRACTIONAL DIFFERENTIAL EQUATION

In this paper, a space fractional differential equation is considered. The equation is obtained from the parabolic equation containing advection, diffusion and reaction terms by replacing the second order derivative in space by a fractional derivative in space of order. An implicit finite difference approximation for this equation is presented. The stability and convergence of the finite difference approximation are proved. A fractional-order method of lines is also presented. Finally, some numerical results are given.     

Transformations between Nonlocal and Local Integrable Equations

Recently, a number of nonlocal integrable equations, such as the ‐symmetric nonlinear Schrödinger (NLS) equation and ‐symmetric Davey–Stewartson equations, were proposed and studied. Here, we show that many of such nonlocal integrable equations can be converted to local integrable equations through simple variable transformations. Examples include these nonlocal NLS and Davey–Stewartson equations, a nonlocal derivative NLS equation, the reverse space‐time complex‐modified Korteweg–de Vries (CMKdV) equation, and many others. These transformations not only establish immediately the integrability of these nonlocal equations, but also allow us to construct their Lax pairs and analytical solutions from those of the local equations. These transformations can also be used to derive new nonlocal integrable equations. As applications of these transformations, we use them to derive rogue wave solutions for the partially ‐symmetric Davey–Stewartson equations and the nonlocal derivative NLS equation. In addition, we use them to derive multisoliton and quasi‐periodic solutions in the reverse space‐time CMKdV equation. Furthermore, we use them to construct many new nonlocal integrable equations such as nonlocal short pulse equations, nonlocal nonlinear diffusion equations, and nonlocal Sasa–Satsuma equations.     

Numerical methods for fractional partial differential equations with Riesz space fractional derivatives

In this paper, we consider the numerical solution of a fractional partial differential equation with Riesz space fractional derivatives (FPDE-RSFD) on a finite domain. Two types of FPDE-RSFD are considered: the Riesz fractional diffusion equation (RFDE) and the Riesz fractional advection–dispersion equation (RFADE). The RFDE is obtained from the standard diffusion equation by replacing the second-order space derivative with the Riesz fractional derivative of order α(1,2]. The RFADE is obtained from the standard advection–dispersion equation by replacing the first-order and second-order space derivatives with the Riesz fractional derivatives of order β(0,1) and of order α(1,2], respectively. Firstly, analytic solutions of both the RFDE and RFADE are derived. Secondly, three numerical methods are provided to deal with the Riesz space fractional derivatives, namely, the L1/L2-approximation method, the standard/shifted Grünwald method, and the matrix transform method (MTM). Thirdly, the RFDE and RFADE are transformed into a system of ordinary differential equations, which is then solved by the method of lines. Finally, numerical results are given, which demonstrate the effectiveness and convergence of the three numerical methods.     

Space-time fractional Schrödinger equation with time-independent potentials

We develop a space-time fractional Schrödinger equation containing Caputo fractional derivative and the quantum Riesz fractional operator from a space fractional Schrödinger equation in this paper. By use of the new equation we study the time evolution behaviors of the space-time fractional quantum system in the time-independent potential fields and two cases that the order of the time fractional derivative is between zero and one and between one and two are discussed respectively. The space-time fractional Schrödinger equation with time-independent potentials is divided into a space equation and a time one. A general solution, which is composed of oscillatory terms and decay ones, is obtained. We investigate the time limits of the total probability and the energy levels of particles when time goes to infinity and find that the limit values not only depend on the order of the time derivative, but also on the sign (positive or negative) of the eigenvalues of the space equation. We also find that the limit value of the total probability can be greater or less than one, which means the space-time fractional Schrödinger equation describes the quantum system where the probability is not conservative and particles may be extracted from or absorbed by the potentials. Additionally, the non-Markovian time evolution laws of the space-time fractional quantum system are discussed. The formula of the time evolution of the mechanical quantities is derived and we prove that there is no conservative quantities in the space-time fractional quantum system. We also get a Mittag-Leffler type of time evolution operator of wave functions and then establish a Heisenberg equation containing fractional operators.     

具有缓变系数的线性时滞系统的稳定性

关于滞后型泛函微分方程的稳定性，本文给出了更一般的判别准则，推广了Ｐａзｙмихин的结果。并将所得结论用于考察具有缓变系数的线性时滞系统的稳定性，放宽了通常的对系统系数矩阵特征值的限制，给出了直接由系统右端系数计算的简明的稳定性判据。