ZEROS OF ENTIRE SOLUTIONS TO COMPLEX LINEAR DIFFERENCE EQUATIONS

In this article, we study the complex oscillation problems of entire solutions to homogeneous and nonhomogeneous linear difference equations, and obtain some relations of the exponent of convergence of zeros and the order of growth of entire solutions to complex linear difference equations.

MEROMORPHIC SOLUTIONS OF A TYPE OF SYSTEM OF COMPLEX DIFFERENTIAL-DIFFERENCE EQUATIONS

Applying Nevanlinna theory of the value distribution of meromorphic functions, we mainly study the growth and some other properties of meromorphic solutions of the type of system of complex differential and difference equations of the following form {j=1∑nαj（z）f1（λj1）（z＋cj）=R2（z,f2（z））,j=1∑nβj（z）f2（λj2）（z＋cj）=R1（z,f1（z））. where λij （j = 1, 2,…, n; i = 1, 2） are finite non-negative integers, and cj （j = 1, 2,… , n） are distinct, nonzero complex numbers, αj（z）, βj（z） （j = 1,2,… ,n） are small functions relative to fi（z） （i =1, 2） respectively, Ri（z, f（z）） （i = 1, 2） are rational in fi（z） （i =1, 2） with coefficients which are small functions of fi（z） （i = 1, 2） respectively.

VALUE DISTRIBUTION OF MEROMORPHIC SOLUTIONS OF CERTAIN NON-LINEAR DIFFERENCE EQUATIONS

In this article, we consider the non-linear difference equation(f(z + 1)f(z)-1)(f(z)f(z-1)-1) =P(z, f(z))/Q(z, f(z)),where P(z, f(z)) and Q(z, f(z)) are relatively prime polynomials in f(z) with rational coefficients. For the above equation, the order of growth, the exponents of convergence of zeros and poles of its transcendental meromorphic solution f(z), and the exponents of convergence of poles of difference △f(z) and divided difference △f(z)/f(z)are estimated. Furthermore, we study the forms of rational solutions of the above equation.

Unconditionally Explicit Stable Difference Schemes for Solving Some Linear and Non-Linear Parabolic Differential Equation

We present the numerical method for solution of some linear and non-linear parabolic equation. Using idea [1], we will present the explicit unconditional stable scheme which has no restriction on the step size ratio k/h2 where k and h are step sizes for space and time respectively. We will also present numerical results to justify the present scheme.

A New Fourth Order Difference Approximation for the Solution of Three-dimensional Non-linear Biharmonic Equations Using Coupled Approach

This paper deals with a new higher order compact difference scheme, which is, O(h4) using coupled approach on the 19-point 3D stencil for the solution of three dimensional nonlinear biharmonic equations. At each internal grid point, the solution u(x,y,z) and its Laplacian Δ4u are obtained. The resulting stencil algo-rithm is presented and hence this new algorithm can be easily incorporated to solve many problems. The present discretization allows us to use the Dirichlet boundary conditions only and there is no need to discretize the derivative boundary conditions near the boundary. We also show that special treatment is required to handle the boundary conditions. Convergence analysis for a model problem is briefly discussed. The method is tested on three problems and compares very favourably with the corresponding second order approximation which we also discuss using coupled approach.

The Growth Order of Solutions of Systems Complex Difference Equations

In this paper, using Nevanlinna theory of the value distribution of meromorphic functions, the problem of growth order of solutions of a class of system of complex difference equations is investigated, some results are improved and generalized. More precisely,some results of the growth order of solutions of system of differential equations to difference equations are extended.

ON MEROMORPHIC SOLUTIONS OF RICCATI AND LINEAR DIFFERENCE EQUATIONS

In this paper, meromorphic solutions of Riccati and linear difference equations are investigated. The growth and Borel exceptional values of these solutions are discussed, and the growth, zeros and poles of differences of these solutions are also investigated. Furthermore, several examples are given showing that our results are best possible in certain senses.

On growth of meromorphic solutions of some kind of non-homogeneous linear difference equations

In this paper,we investigate the growth of meromorphic solutions of some kind of non-homogeneous linear difference equations with special meromorphic coefficients.When there are more than one coefficient having the same maximal order and the same maximal type,the estimates on the lower bound of the order of meromorphic solutions of the involved equations are obtained.Meanwhile,the above estimates are sharpened by combining the relative results of the corresponding homogeneous linear difference equations.

Compact implicit integration factor methods for some complex-valued nonlinear equations

The compact implicit integration factor （cIIF） method is an efficient time discretization scheme for stiff nonlinear diffusion equations in two and three spatial dimensions. In the current work, we apply the cIIF method to some complex-valued nonlinear evolutionary equations such as the nonlinear SchrSdinger （NLS） equation and the complex Ginzburg-Landau （GL） equation. Detailed algorithm formulation and practical implementation of cIIF method are performed. The numerical results indicate that this method is very accurate and efficient.

端接复杂电路传输线网络的电磁耦合时域并行计算方法

针对端接复杂电路传输线(TL)网络的电磁耦合问题,仍缺乏高效的场路协同仿真技术。该文将传输线方程与时域有限差分(FDTD)方法、诺顿定理和置换定理以及NGSPICE软件相结合,并引入消息传递接口(MPI)并行技术,提出一种高效的时域混合并行算法(FDTDTL-NGSPICE)。首先,根据诺顿定理和置换定理,将传输线网络分解为传输线子系统和复杂电路子系统,并构建对应的等效电路模型。然后,使用FDTDTL并行算法计算传输线子系统沿线各点的电压和电流,并获取对应诺顿等效电路的电流源和等效导纳大小。最后,使用NGSPICE对复杂电路子系统进行传导干扰分析,获得复杂电路各元件上的瞬态响应,并将端口电压反馈给传输线子系统作为边界,实现传输线网络电磁耦合的场线路联合协同仿真。通过对3类典型场景的计算实例,分别使用时域混合并行算法和电磁仿真软件CST电缆工作室(CS)进行数值模拟并对比,验证所提算法的置信度。

Numerical simulation of the soliton solutions for a complex modified Korteweg—de Vries equation by a finite difference method

In this paper,a Crank-Nicolson-type finite difference method is proposed for computing the soliton solutions of a complex modifed Korteweg de Vries(MKdV)equation(which is equivalent to the Sasa-Satsuma equation)with the vanishing boundary condition.It is proved that such a numerical scheme has the second order accuracy both in space and time,and conserves the mass in the discrete level.Meanwhile,the resuling scheme is shown to be unconditionally stable via the von Nuemann analysis.In addition,an iterative method and the Thomas algorithm are used together to enhance the computational efficiency.In numerical experiments,this method is used to simulate the single-soliton propagation and two-soliton collisions in the complex MKdV equation.The numerical accuracy,mass conservation and linear stability are tested to assess the scheme's performance.

Fermat型复微分差分方程的整函数解

文章利用复微分方程理论和复差分方程理论研究了形式为(μf(z)+λf′(z))^(2)+f(z)^(2)=P(z)的复微分方程和形式为(μf(z)+λf′(z))^(2)+f(z+c)^(2)=P(z)的复微分-差分方程的任意整函数解的存在形式。首先,用Weierstrass因式分解定理将两个方程进行分解,计算出f(z)和μf(z)+λf′(z)的具体形式;其次,对因式分解后产生的指数h(z)进行讨论,分为h(z)为常数和h(z)为非常数整函数两种情形;最后,研究每一种情形下整函数解中各个变量之间的关系。文章得到了两个关于Fermat型方程的整函数解的存在形式,在一定范围内推广和改进了前人的结论。

Exact Solutions for Fractional Differential-Difference Equations by an Extended Riccati Sub-ODE Method

In this paper, an extended Riccati sub-ODE method is proposed to establish new exact solutions for fractional differential-difference equations in the sense of modified Riemann-Liouville derivative. By a fractional complex transformation, a given fractional differential-difference equation can be turned into another differential-difference equation of integer order. The validity of the method is illustrated by applying it to solve the fractional Hybrid lattice equation and the fractional relativistic Toda lattice system. As a result, some new exact solutions including hyperbolic function solutions, trigonometric function solutions and rational solutions are established.

基于RKADE-SS-FDTD方法的无条件稳定高阶CFS-PML算法

为了提升完全匹配层(PML)算法的吸收效果,提出了一种用于截断非磁化等离子体的高阶复频率偏移PML(HO-CFS-PML)算法。该算法将龙格-库塔辅助微分方程(RKADE)算法和分裂步时域有限差分(SS-FDTD)算法相结合,来对非磁化等离子体进行仿真。在计算域边界处,引入了由辅助微分方程(ADE)算法实现的HO-CFS-PML用于截断开放区域中的等离子体,并通过一个数值算例进行模拟。结果表明:所提出的HO-RKADE-SS-CFS-PML算法在柯朗-弗里德里希斯-列维(CFL)数(CFLN)为8时,相比于传统CFS-PML算法,时间减少率可以达到79.2%,证明了该算法能够有效地消除柯朗-弗里德里希斯-列维(CFL)稳定性条件的约束,大大节省了计算时间;此外,HO-RKADE-SS-CFS-PML算法的最大相对反射误差可以达到-115 dB,明显小于其他算法的相对反射误差,并且随着CFLN的增大,该算法不会像RKADE-ADI-CFS-PML算法那样产生明显的误差增大现象,证明本文算法在大时间步长下拥有比其他算法更好的吸收电磁波的能力。

复杂地表起伏多重变加密网格有限差分法波动方程数值模拟

常规变加密网格有限差分波动方程数值模拟方法采用水平分层加密网格,该网格剖分策略在适应地形起伏和近地表速度结构变化特征方面效果较差。针对该问题,提出一种起伏多重变加密网格有限差分波动方程数值模拟方法。该方法根据地形起伏和近地表低速层到高速层的速度分布特征进行网格加密;采用不同网格中的变系数差分格式离散声波方程,在保证波场模拟精度的同时兼顾计算效率;同时,为了进一步保障地表附近波场模拟的精度,地表附近最细网格中的差分格式不做降阶处理,针对位于地表以上的虚像点的波场值,提出一种融入自由地表边界条件的法向虚像外推法。算例分析验证该算法对速度模型不同区域进行的网格多次加密显著提高了计算效率,以黄土塬实际模型为例,耗时为常规1 m×1 m网格耗时的43.3%,并可达到和细网格基本一致的模拟精度,模拟误差控制在10-12范围内,同时表现出很好的近地表散射压制和边界吸收效果,且算法能稳定地适应实际复杂地表介质。

一类非线性复微分差分方程超越整函数解的性质

本文主要运用亚纯函数值分布理论及其差分模拟和Hadamard分解定理,研究了一类非线性复微分差分方程超越整函数解的增长级和零点收敛指数问题,并分析了该类方程的超越整函数解是指数单项式形式解时的情形。

A Novel Numerical Method of O(h^(4))for Three-Dimensional Non-Linear Triharmonic Equations

In this article,we present two new novel finite difference approximations of order two and four,respectively,for the three dimensional non-linear triharmonic partial differential equations on a compact stencil where the values of u,δ^(2)u/δn^(2)andδ^(4)u/δn^(4)are prescribed on the boundary.We introduce new ideas to handle the boundary conditions and there is no need to discretize the derivative boundary conditions.We require only 7-and 19-grid points on the compact cell for the second and fourth order approximation,respectively.The Laplacian and the biharmonic of the solution are obtained as by-product of the methods.We require only system of three equations to obtain the solution.Numerical results are provided to illustrate the usefulness of the proposed methods.

复域差分和差分方程的研究

介绍了近十年来复域差分、q-差分、差分方程及q-差分方程的主要研究成果,其中包括亚纯函数对数导数引理的差分模拟;Clunie引理和Mohon'ko引理的差分模拟;慢增长亚纯函数的差分、均差分的零点和不动点的性质;差分多项式的值分布性质;差分Riccati方程与差分Painlevé方程亚纯解的性质;复域q-差分及q-差分方程的解析性质.

整体拱坝的仿真与可行温控措施

结合某拱坝的仿真计算讨论了中小拱坝不分缝分块,利用一个低温季节浇筑的可能性及应采取的温控措施.结果表明,碾压混凝土冬季施工方案、碾压混凝土加河水冷却和常态混凝土外掺MgO方案都不能完全解决温度应力问题而避免裂缝的发生.常态混凝土、掺适量MgO另加河水水管冷却可望使中小混凝土拱坝在一个冬季完工而不发生危害性裂缝.