非线性薛定谔方程的平均向量场方法

利用平均向量场方法(AVF)对非线性薛定谔方程进行求解, 在理论上得到了一个保非线性薛定谔方程描述的系统能量守恒的AVF格式, 再分别用非线性薛定谔方程的AVF格式和辛格式数值模拟孤立波的演化行为, 并比较两个格式是否保系统能量守恒特性. 数值结果表明, AVF格式也能很好地模拟孤立波的演化行为,并且比辛格式更能保持系统的能量守恒.     

Cahn-Hilliard方程的高阶保能量散逸性方法

能量散逸性是物理和力学中某些微分方程一项重要的物理特性.构造精确地保持微分方程能量散逸性的数值格式对模拟具有能量散逸性的微分方程具有重要的意义.本文利用四阶平均向量场方法和傅里叶谱方法构造了Cahn-Hilliard方程高阶保能量散逸性格式.数值结果表明高阶保能量散逸性格式能很好地模拟Cahn-Hilliard方程在不同初始条件下解的行为,并且很好地保持了Cahn-Hilliard方程的能量散逸特性.     

多辛Dirac方程的高阶整体保能量格式

基于四阶平均向量场方法和拟谱方法构造了Dirac方程的高阶整体保能量格式，利用构造的高阶整体保能量格式数值模拟方程孤立波的演化行为.数值模拟结果表明构造的高阶整体保能量格式可以很好地模拟Dirac方程孤立波的演化行为，并且可以精确地保持方程的整体能量守恒特性.     

带有弱奇性核的多项分数阶非线性随机微分方程解的适定性

本文主要研究一类带有多项分数阶Caputo导数的非线性随机微分方程初值问题的解的适定性.具体地,首先把多项分数阶随机微分方程等价地转化为随机Volterra积分方程;然后,给出了该随机积分方程的Euler-Maruyama (EM)格式;最后,借助于该EM格式,证明了多项分数阶随机微分方程的解的适定性.     

三阶Airy方程满足两个守恒律的非线性差分格式

近年来,学者们对发展型偏微分方程设计了一种能保持多个守恒律的数值方法,这类方法无论在解的精度还是长时间的数值模拟方面都表现出非常好的性质.将这类思想应用到三阶Airy方程,即三阶散射方程,对其设计了满足两个守恒律的非线性差分格式.该格式不仅计算数值解,同时计算数值能量,并且保证数值解和数值能量同时守恒.从数值结果可以看出,该格式在长时间的数值模拟中具有更好的保结构性质.     

非线性脉冲向量双曲型微分方程解的H-振动性

研究一类具脉冲时滞的非线性双曲型向量泛函微分方程解的H-振动性.方法是采用由Domslak引进的H-振动性的概念,将向量微分方程解的振动问题转化为纯量微分不等式正解和负解的不存在性问题.得到了解的H-振动性的若干判别准则.     

非线性向量泛函微分方程解的振动比较定理

本文借助交换与不等式等方法将非线性向量泛函微分方程解的振动性与相应数量微分不等式是否存在正解作比较,得到若干振动比较定理并给出了具体应用.     

一阶非线性时滞微分方程解的振动性

本文给出了某些含超线性项的一阶非线性时滞微分方程振动的充分条件，解决了现有文献的方法难以处理的某些非线性振动问题。另外，也给出了它们对于包括时滞Ｌｏｇｉｓｔｉｃ方程在内的一类非线性方程的应用。     

带阻尼项的非线性分数阶微分方程的振动性[英文]

本文研究一类带阻尼项的非线性分数阶微分方程的振动性问题. 利用Riemann-Liouville微积分、Riccati变换及不等式的方法, 获得带阻尼项的非线性微分方程振动性的充分条件, 推广了关于分数阶微分方程振动已有的结果.     

时滞抛物型与双曲型微分方程解的振动性的讨论

对一类线性以及非线性抛物型时滞微分方程的解在第一或第二边值条件下解振动的充分必要条件进行了讨论,给出了解振动的一些结论.并且对一类线性以及带强迫项的非线性双曲型时滞微分方程的解在第一或第二边值条件下解振动的充分必要条件进行了讨论,也给出了一些结论.     

Energy preserving integration of bi-Hamiltonian partial differential equations

The energy preserving average vector field (AVF) integrator is applied to evolutionary partial differential equations (PDEs) in bi-Hamiltonian form with nonconstant Poisson structures. Numerical results for the Korteweg de Vries (KdV) equation and for the Ito type coupled KdV equation confirm the long term preservation of the Hamiltonians and Casimir integrals, which is essential in simulating waves and solitons. Dispersive properties of the AVF integrator are investigated for the linearized equations to examine the nonlinear dynamics after discretization.     

均布载荷作用下波纹扁壳的非线性振动

应用轴对称旋转扁壳的非线性大挠度动力学方程,研究了波纹扁壳在均布载荷作用下的非线性受迫振动问题．采用格林函数方法,将扁壳的非线性偏微分方程组化为非线性积分微分方程组．再使用展开法求出格林函数,即将格林函数展开为特征函数的级数形式,积分微分方程就成为具有退化核的形式,从而容易得到关于时间的非线性常微分方程组．针对单模态振形,得到了谐和激励作用下的幅频响应．作为算例,研究了正弦波纹扁球壳的非线性受迫振动现象．该文的解答可供波纹壳的设计参考．     

Numerical solutions of distributed order fractional differential equations in the time domain using the Müntz–Legendre wavelets approach

In this paper, a numerical method is presented to obtain and analyze the behavior of numerical solutions of distributed order fractional differential equations of the general form in the time domain with the Caputo fractional derivative. The suggested method is based on the Müntz–Legendre wavelet approximation. We derive a new operational vector for the Riemann–Liouville fractional integral of the Müntz–Legendre wavelets by using the Laplace transform method. Applying this operational vector and collocation method in our approach, the problem can be reduced to a system of linear and nonlinear algebraic equations. The arising system can be solved by the Newton method. Discussion on the error bound and convergence analysis for the proposed method is presented. Finally, seven test problems are considered to compare our results with other well‐known methods used for solving these problems. The results in the tabulated tables highlighted that the proposed method is an efficient mathematical tool for analyzing distributed order fractional differential equations of the general form.     

The singularly perturbed problem of vector integro-differential equations

The singularly perturbed boundary value problem of scalar integro-differential equations has been studied extensively by the differential inequality method . However, it does not seem possible to carry this method over to a corresponding nonlinear vector integro-differential equation. Therefore , for n-dimensional vector integro-differential equations the problem has not been solved fully. Here, we study this nonlinear vector problem and obtain some results. The approach in this paper is to transform the appropriate integro-differential equations into a canonical or diagonalized system of two first-order equations.     

小垂度粘弹性索非线性响应及振动主动控制

研究具有初始应力的小垂度粘弹性索的非线性动态响应及振动主动控制。在假定索材料的本构关系为一般微分本构类型的基础上,建立小垂度粘弹性索的运动微分方程;应用Galerkin方法将其转化为可用Runge-Kutta数值积分方法求解的一系列三阶非线性常微分方程。在仅考虑面内的横向振动及忽略非线性的情况下得到了连续状态空间中的状态方程,将状态方程离散为差分方程形式,并用矩阵指数来逐步近似状态转移矩阵;基于二次性能指标的最小化得到了最优的控制力与状态向量。最后通过数值仿真研究说明了粘性参数对索动态响应的影响。     

A mixed method for free and forced vibration of rectangular plates

This paper presents a very first combined application of Ritz method and differential quadrature (DQ) method to vibration problem of rectangular plates. In this study, the spatial partial derivatives with respect to a coordinate direction are first discretized using the Ritz method. The resulting system of partial differential equations and the related boundary conditions are then discretized in strong form using the DQ method. The mixed method combines the simplicity of the Ritz method and high accuracy and efficiency of the DQ method. The results are obtained for various types of boundary conditions. Comparisons are made with existing analytical and numerical solutions in the literature. Numerical results prove that the present method is very suitable for the problem considered due to its simplicity, efficiency, and high accuracy.     

On the construction of solutions of terminal problems for multidimensional affine systems in quasicanonical form

We consider the construction of solutions of terminal problems for multidimensional affine systems. We show that the terminal problem for a regular system in quasicanonical form can be reduced to a boundary value problem for a system of ordinary differential equations of lower order with right-hand side depending on a vector parameter. We prove a sufficient condition for the existence of a solution of the above-mentioned boundary value problem. A method for constructing a numerical solution is developed.     

Three dimensional elastodynamics of 2D quasicrystals: The derivation of the time-dependent fundamental solution

The time-dependent differential equations of elasticity for 2D quasicrystals with general structure of anisotropy (dodecagonal, octagonal, decagonal, pentagonal, hexagonal, triclinic) are considered in the paper. These equations are written in the form of a vector partial differential equation of the second order with symmetric matrix coefficients. The fundamental solution (matrix) is defined for this vector partial differential equation. A new method of the numerical computation of values of the fundamental solution is suggested. This method consists of the following: the Fourier transform with respect to space variables is applied to vector equation for the fundamental solution. The obtained vector ordinary differential equation has matrix coefficients depending on Fourier parameters. Using the matrix computations a solution of the vector ordinary differential equation is numerically computed. Finally, applying the inverse Fourier transform numerically we find the values of the fundamental solution. Computational examples confirm the robustness of the suggested method for 2D quasicrystals with arbitrary type of anisotropy.     

Large amplitude vibration of heated corrugated circular plates with shallow sinusoidal corrugations

This paper presents a nonlinear free vibration analysis of corrugated circular plates with shallow sinusoidal corrugations under uniformly static ambient temperature. Based on the nonlinear bending theory of thin shallow shells, the governing equations for corrugated plates are established from Hamilton’s principle. These partial differential equations are reduced to corresponding ordinary ones by elimination of the time variable with Kantorovich method following an assumed harmonic time mode. The resulting equations, which form a nonlinear two-point boundary value problem in spatial variable, are then solved numerically by shooting method, and the temperature-dependent characteristic relations of frequency vs. amplitude for nonlinear vibration of heated corrugated plates are obtained successfully. The comparison with available published results shows that the numerical analysis here is of good reliability. A detailed parametric study is conducted involving the dependency of nonlinear frequency on the depth and density of corrugations along with the temperature change. Effects of these variables on the trend of nonlinearity are plotted and discussed.