ON GLOBAL MEROMORPHIC SOLUTIONS OF SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS WITH MEROMORPHIC COEFFICIENTS

The main purpose of this article is to study the existence theories of global meromorphic solutions for some second-order linear differential equations with meromorphic coefficients, which perfect the solution theory of such equations.

INSTABILITY OF SOLUTION FOR THE FOURTH ORDER LINEAR DIFFERENTIAL EQUATION WITH VARIED COEFFICIENT

In this paper, we give some sufficient conditions of the instability for the fourth order linear differential equation with varied coefficient, at least one of the characteristic roots of which has positive real part, by means of Liapunov's second method.

Criteria for System of Three Second-Order Ordinary Differential Equations to Be Reduced to a Linear System via Restricted Class of Point Transformation

This paper is devoted to the study of the linearization problem of system of three second-order ordinary differential equations and . The necessary conditions for linearization by general point transformation and are found. The sufficient conditions for linearization by restricted class of point transformation and are obtained. Moreover, the procedure for obtaining the linearizing transformation is provided in explicit forms. Examples demonstrating the procedure of using the linearization theorems are presented.

A HIGH ACCURACY DIFFERENCE SCHEME FOR THE SINGULAR PERTURBATION PROBLEM OF THE SECOND-ORDER LINEAR ORDINARY DIFFERENTIAL EQUATION IN CONSERVATION FORM

In this paper, combining the idea of difference method and finite element method, we construct a difference scheme for a self-adjoint problem in conservation form. Its solution uniformly converges to that of the original differential equation problem with order h3.

EXISTENCE OF MEROMORPHIC SOLUTIONS OF SOME HIGHER ORDER LINEAR DIFFERENTIAL EQUATIONS

This article discusses the problems on the existence of meromorphic solutions of some higher order linear differential equations with meromorphic coefficients. Some nice results are obtained. And these results perfect the complex oscillation theory of meromorphic solutions of linear differential equations.

Equation governing the probability density evolution of multi-dimensional linear fractional differential systems subject to Gaussian white noise

Stochastic fractional differential systems are important and useful in the mathematics,physics,and engineering fields.However,the determination of their probabilistic responses is difficult due to their non-Markovian property.The recently developed globally-evolving-based generalized density evolution equation(GE-GDEE),which is a unified partial differential equation(PDE)governing the transient probability density function(PDF)of a generic path-continuous process,including non-Markovian ones,provides a feasible tool to solve this problem.In the paper,the GE-GDEE for multi-dimensional linear fractional differential systems subject to Gaussian white noise is established.In particular,it is proved that in the GE-GDEE corresponding to the state-quantities of interest,the intrinsic drift coefficient is a time-varying linear function,and can be analytically determined.In this sense,an alternative low-dimensional equivalent linear integer-order differential system with exact closed-form coefficients for the original highdimensional linear fractional differential system can be constructed such that their transient PDFs are identical.Specifically,for a multi-dimensional linear fractional differential system,if only one or two quantities are of interest,GE-GDEE is only in one or two dimensions,and the surrogate system would be a one-or two-dimensional linear integer-order system.Several examples are studied to assess the merit of the proposed method.Though presently the closed-form intrinsic drift coefficient is only available for linear stochastic fractional differential systems,the findings in the present paper provide a remarkable demonstration on the existence and eligibility of GE-GDEE for the case that the original high-dimensional system itself is non-Markovian,and provide insights for the physical-mechanism-informed determination of intrinsic drift and diffusion coefficients of GE-GDEE of more generic complex nonlinear systems.

ON THE DEPENDENT PROPERTY OF SOLUTIONS FOR HIGHER ORDER PERIODIC DIFFERENTIAL EQUATIONS

This article investigates the property of linearly dependence of solutions f（z） and f（z ＋ 2πi） for higher order linear differential equations with entire periodic coefficients.

LINEARIZED OSCILLATIONS FOR ODD-ORDERNEUTRAL DELAY DIFFERENTIAL EQUATIONSWITH ASYMPTOTIC PERIODIC COEFFICIENTS

In this paper, we prove that, under appropriate hypotheses, the odd-order nonlinear neutral delay differential equation has the same oscillatory character as the associated linear equation with periodic coefficients Our method is based on the establishment of a new comparison theorem for the oscillation of Eq.(E). Hence we prove that the parameter set such that Eq.(E) has a nonoscillatory solution is closed in certain metric space, and avoid the difficulty to set the necessary and sufficient condition for the oscillation of Eq.(E).

ITERATION OF POSITIVE SOLUTION FOR A SECOND-ORDER ORDINARY DIFFERENTIAL EQUATION WITH CHANGE OF SIGN

An iterative process of positive solution for BVP w'+h(t)f(w)=0, w(0)=w(1)= 0 is established, where h(t) is allowed to changes sign on [0,1]. The process starts from a simple function.

Linear quadratic optimal control of conditional McKean-Vlasov equation with random coefficients and applications

We consider the optimal control problem for a linear conditional McKeanVlasov equation with quadratic cost functional.The coefficients of the system and the weighting matrices in the cost functional are allowed to be adapted processes with respect to the common noise filtration.Semi closed-loop strategies are introduced,and following the dynamic programming approach in(Pham and Wei,Dynamic programming for optimal control of stochastic McKean-Vlasov dynamics,2016),we solve the problem and characterize time-consistent optimal control by means of a system of decoupled backward stochastic Riccati differential equations.We present several financial applications with explicit solutions,and revisit,in particular,optimal tracking problems with price impact,and the conditional mean-variance portfolio selection in an incomplete market model.

High Accuracy Arithmetic Average Discretization for Non-Linear Two Point Boundary Value Problems with a Source Function in Integral Form

In this article, we report the derivation of high accuracy finite difference method based on arithmetic average discretization for the solution of Un=F(x,u,u′)+∫K(x,s)ds , 0 x s < 1 subject to natural boundary conditions on a non-uniform mesh. The proposed variable mesh approximation is directly applicable to the integro-differential equation with singular coefficients. We need not require any special discretization to obtain the solution near the singular point. The convergence analysis of a difference scheme for the diffusion convection equation is briefly discussed. The presented variable mesh strategy is applicable when the internal grid points of the solution space are both even and odd in number as compared to the method discussed by authors in their previous work in which the internal grid points are strictly odd in number. The advantage of using this new variable mesh strategy is highlighted computationally.

变分量子线性求解算法在高速飞行器定常绕流数值模拟中的应用

CFD方法在用于三维大规模复杂流动的空气动力学数值模拟时,面临着计算成本高、模拟时间长等瓶颈问题。近年来,量子计算为航空航天CFD领域带来了新的解决思路,通过利用量子比特的叠加态和纠缠特性,理论上相比于经典计算机能够实现对数级的存储缩减和指数级的效率加速,在处理大规模空气动力学模拟等任务上具有巨大的潜力。聚焦于量子计算技术在航空航天CFD领域的应用探索,针对高速定常流动问题,采用变分量子线性求解器(variational quantum linear solver,VQLS)辅助求解CFD时间离散环节得到的高维线性方程组,进而发展了可以在含噪声中等规模量子(noisy intermediate-scale quantum,NISQ)器件上实现的VQLS-CFD耦合方法。通过三维双椭球模型和探测器火星科学实验室模型的超声速数值模拟测试,验证了VQLS-CFD耦合方法可以实现大规模复杂流动的鲁棒及准确模拟,并产生与试验数据吻合度较高的计算结果,最终获得高可信度的气动预测结果。然而,介绍的量子计算流体力学(quantum computational fluid dynamics,QCFD)方法在计算效率方面尚未达到真正意义上的加速效果,这一现象的突破将依赖量子计算机硬件的持续发展与QCFD算法的不断优化与成熟。

变系数Burgers方程的一些新精确解

利用齐次平衡原则 ,导出了变系数Burgers方程的B¨acklund变换 (BT) ;并由该B¨acklund变换 。

基于PDE的图像去噪和反差增强同步算法

针对反差较低且包含噪声污染的灰度图像,设计分段线性拉伸函数,引入TV下降流,建立新的偏微分方程(PDE)数学模型。该模型通过设置参数λ灵活控制去噪和反差增强的程度,实现2种灰度图像处理手段的同步进行。对比实验表明,该方法可有效缓解传统处理方法存在的问题,在抑制噪声的同时增强图像的反差。

磁浮列车的动力稳定性分析与Liapunov指数

针对弹性轨道上的磁悬浮列车线性动力控制系统，给出了采用Ｌｉａｐｕｎｏｖ特性指数判别动力系统稳定性的判据：当动力系统的全部Ｌｉａｐｕｎｏｖ特性指数小于零时，动力控制系统是稳定的；而当动力系统中有一Ｌｉａｐｕｎｏｖ特性指数大于零时，动力系统就成为不稳定的．由于Ｌｉａｐｕｎｏｖ特性指数可以由数值积分方式方便地给出，这一判断方法对于不便搜索参数稳定区域的高维动力系统，与寻求周期变系数线性常微分方程动力系统的Ｆｌｏｑｕｅｔ理论的变换矩阵特征值相比，具有计算量小和搜索方法简单等优点．数值算例表明本文方法是有效的．

一类非线性热传导方程的线性化解法

提出了用一阶线性常微分方程及其解构造非线性偏微分方程精确解的线性化解法.利用该方法求出(3+1)维和(1+1)维的Kolmogorov-Petrovskii-Piskunov型方程的精确解,其中包括扭状孤立波和代数孤立波解,并把(1+1)维的Kolmogorov-Petrovskii-Piskunov型方程推广到具有任意次非线性项的一般热传导方程,得到了其解.

基于变分的保持特征的有噪图像放大算法

针对图像放大后出现边缘模糊的现象,该文考察了典型边缘的特征,利用整体变分思想,提出了一种基于变分模型的有噪图像放大算法。利用导出的偏微分方程和图像边缘的局部梯度信息,该文算法实施一种自适应非线性滤波处理,能够较好地保持图像边缘锐度,同时有效地去除图像噪声。为了保持图像的特征和细节,文章利用图像的方向导数局部地调整正则系数.理论分析和实验结果皆表明此文算法的有效性。

二阶线性常微分方程的两点边值问题的新解法

基于变分原理,将二阶线性常微分方程的两点边值问题转化为等价的变分问题（即泛函极值问题）,利用两点三次Hermite插值构造一个逼近可行函数的近似函数,从而将问题转化为一个多元单目标优化问题,最后运用粒子群优化算法求解该优化问题,由此求得二阶线性常微分方程的两点边值问题的近似解.数值实验表明该方法优于传统的里兹法和有限差分方法.

一种二阶变系数线性微分方程的求解方法

在知道二阶变系数线性齐次微分方程一个特解的情况下,通过常数变易法,将二阶变系数线性非齐次微分方程转化为一阶线性微分方程,从而给出运算量较小的二阶变系数线性非齐次微分方程通解的一般公式,也给出了用刘维尔定理求解二阶变系数线性齐次微分方程的一个理论依据.

四阶线性微分方程的算子解法

通过算子代换引入特征方程的概念,将微分方程化为代数方程,得到了四阶线性微分方程可降阶的充要条件,并给出了求解对应方程通解的方法.