无限域上一维热传导方程的Gauss-Laguerre数值解

针对无限域上一维热传导方程的解析解为反常积分形式,直接计算往往比较困难.首先采用Fourier变换给出问题解析解,其次结合解析解的形式和无限域上Gauss型数值积分法精度高的优点,将半无限域上的一维热传导方程问题利用Gauss-Laguerre数值积分计算数值解,对无限域上的一维热传导方程的解析解转化为半无限域上的形式后用Gauss-Laguerre数值积分计算.实验结果表明,本文给出的数值解方法具有很高的精度.     

Black-Sholes方程的解析解及收敛性分析

将经典的Black-Sholes方程转化为一类特殊的线性偏微分方程,并通过对其采用Fourier变换方法求得解析解,最后利用优核的相关性质对所得Black-Sholes方程解析解进行收敛性分析,证明了该解析解的收敛性.     

基于变分法对一类非线性扩散方程解析解研究北大核心CSCD

本文分别针对一类扩散系数为非线性指数函数和幂函数的扩散方程,基于变分原理中的泛函极值理论分别提出了求解该方程Dirichlet边界和Neumann边界问题解析解的新方法,并证明了新方法是泛函问题极值解的充要性．以非饱和土壤水分运动问题为背景,给出了积水和恒通量条件下水平吸渗问题的解析解,并通过数值算例将解析解与数值解进行了比较分析,结果表明本文方法得到的解析解能够准确预测非饱和土壤水分水平吸渗问题的土壤含水量分布,是一种有效方法．因此本文方法为求解这一类非线性扩散方程提供了一种有效的新方法．     

分数阶广义扰动热波方程的与泛函映射解

研究了一类分数阶广义非线性扰动热波方程.首先在典型分数阶热波方程情形下得到解,接着用泛函分析映射方法,求出了分数阶广义非线性扰动热波方程初始边值问题的任意次近似解析解.最后简述了它的物理意义.求得的近似解析解,弥补了单纯用数值方法得到的模拟解的不足.     

精确的磁流体动力学汇流解析解

就一个特殊的磁流体动力学(MHD)流动,即速度幂指数为-1时的汇流,得到著名的Falkner-Skan方程精确的解析解.解析解是封闭的,并有多重解分支.分析了磁场参数和壁面伸长参数的影响.发现了有趣的速度分布现象:即使壁面固定,回流区域依然出现.在一个罕见的FalknerSkan MHD流动中,得到了一组解,以精确封闭的解析公式表示,极大地丰富了著名的Falkner-Skan方程的解析解,也加深了对这重要又有趣方程的理解.     

非线性强迫扰动Klein-Gordon方程的孤波渐近解法

研究了一类非线性强迫扰动Klein-Gordon方程.首先利用双曲正切待定系数法求得了典型的方程孤波解.然后利用泛函变分迭代原理得到了强迫扰动Klein-Gordon方程的一个近似解,并论述了解的一致有效性.所得到的近似解是一个解析式,它还可对近似解进行解析运算,而使用简单的模拟方法所得到的近似解是达不到这种效果的.     

RLW-Burgers方程的一类解析解

本文给出了 RLW-Burgers方程及 Kd V-Burgers方程的一类解析解 ,且可得到 RLW-Burgers方程的振荡激波解 .这些解可以表示为 Burgers方程和 Kd V方程解的线性组合 ,文末还对文 [8]作了讨论 .     

Burgers-Huxley方程的同伦分析解

针对Burgers-Huxley方程定解问题,构造了一种零阶同伦方程,采用同伦方法得到Burgers-Huxley方程定解问题的近似解析解.最后,进行了实例验证和结果分析.     

薄壳小挠度屈曲方程的加权解法

基于薄壳小挠度屈曲方程,提出了一种求临界载荷解析表达式的加权解法.在复杂边界条件下,小挠度屈曲方程的解析解仍然是一个难点.以轴对称屈曲问题为例,从方程中找出临界载荷的影响因素,加权平均得到临界载荷,再用特例解确定影响系数.这一方法利用特殊问题的已知解来求一般问题的解析解,简化了求解过程,拓宽了解决问题的范围;其计算结果与利用Algor有限元程序得到的数值解一致.     

基于变分法对一类非线性扩散方程解析解研究

本文分别针对一类扩散系数为非线性指数函数和幂函数的扩散方程,基于变分原理中的泛函极值理论分别提出了求解该方程Dirichlet边界和Neumann边界问题解析解的新方法,并证明了新方法是泛函问题极值解的充要性.以非饱和土壤水分运动问题为背景,给出了积水和恒通量条件下水平吸渗问题的解析解,并通过数值算例将解析解与数值解进行了比较分析,结果表明本文方法得到的解析解能够准确预测非饱和土壤水分水平吸渗问题的土壤含水量分布,是一种有效方法。因此本文方法为求解这一类非线性扩散方程提供了一种有效的新方法.     

An Auxiliary Equation for the Bellman Equation in a One-Dimensional Ergodic Control

In this paper we consider the Bellman equation in a one-dimensional ergodic control. Our aim is to show the existence and the uniqueness of its solution under general assumptions. For this purpose we introduce an auxiliary equation whose solution gives the invariant measure of the diffusion corresponding to an optimal control. Using this solution, we construct a solution to the Bellman equation. Our method of using this auxiliary equation has two advantages in the one-dimensional case. First, we can solve the Bellman equation under general assumptions. Second, this auxiliary equation gives an optimal Markov control explicitly in many examples. \keywords{Bellman equation, Auxiliary equation, Ergodic control.} \amsclass{49L20, 35G20, 93E20.} Accepted 11 September 2000. Online publication 16 January 2001.     

On a Solution of the Quaternion Matrix Equation X - A X B = C and Its Application

This paper first studies the solution of a complex matrix equation X - AXB = C, obtains an explicit solution of the equation by means of characteristic polynomial, and then studies the quaternion matrix equation X - A X B = C, characterizes the existence of a solution to the matrix equation, and derives closed-form solutions of the matrix equation in explicit forms by means of real representations of quaternion matrices. This paper also gives an application to the complex matrix equation X - AXB =C.     

Integrability properties of a generalized reduction of the KdV6 equation

We consider the integrability properties of a generalized version of a similarity reduction of the so-called KdV6 equation, an equation that has recently generated much interest. We give a linear problem for this generalized reduction and show that it satisfies the requirements of the Ablowitz-Ramani-Segur algorithm. In addition we give a Bäcklund transformation to a related equation, giving also an auto-Bäcklund transformation for this last. Our results mirror those for the Korteweg-de Vries equation itself, which has a similarity reduction to an ordinary differential equation which is related by a Bäcklund transformation to the second Painlevé equation, this last having an auto-Bäcklund transformation.     

Solutions of nonlinear differential equations

We consider an ordinary nonlinear differential equation with generalized coefficients as an equation in differentials in the algebra of new generalized functions. The solution of such an equation is a new generalized function. In this article we formulate necessary and sufficient conditions for when the solution of the given equation in the algebra of new generalized functions is associated with an ordinary function. Moreover, a class of all possible associated functions is described.     

Exact WKB analysis of a Schrödinger equation with a merging triplet of two simple poles and one simple turning point,I — Its WKB-theoretic transformation to the Mathieu equation

We develop the exact WKB analysis of an M2P1T (merging two simple poles and one simple turning point) Schrödinger equation. Our emphasis is put on the analysis of the singularity structure of its Borel transformed WKB solutions near fixed singular points relevant to the two simple poles contained in the potential of the equation. In Part I, we focus our attention on the construction and analytic properties of a WKB-theoretic transformation that transforms an M2P1T equation to an algebraic Mathieu equation. That transformation plays an important role in Part II ([7]) when we discuss the singularity structure of Borel transformed WKB solutions of an M2P1T equation.     

Solution of the equivalence problem for the Painlevé IV equation

We solve the equivalence problem for the Painlevé IV equation, formulating the necessary and sufficient conditions in terms of the invariants of point transformations for an arbitrary second-order differential equation to be equivalent to the Painlevé IV equation. We separately consider three pairwise nonequivalent cases: both equation parameters are zero, a = b = 0; only one parameter is zero, b = 0; and the parameter b ?? 0. In all cases, we give an explicit point substitution transforming an equation satisfying the described test into the Painlevé IV equation and also give expressions for the equation parameters in terms of invariants.     

Mixing “In the sense of Ibragimov.” Estimate for the rate of approach of a family of integral functionals of a solution of a differential equation with periodic coefficients to a family of wiener processes. Some applications. I

We prove that a bounded 1-periodic function of a solution of a time-homogeneous diffusion equation with 1-periodic coefficients forms a process that satisfies the condition of uniform strong mixing. We obtain an estimate for the rate of approach of a certain normalized integral functional of a solution of an ordinary time-homogeneous stochastic differential equation with 1-periodic coefficients to a family of Wiener processes in probability in the metric of space C [0, T]. As an example, we consider an ordinary differential equation perturbed by a rapidly oscillating centered process that is a 1-periodic function of a solution of a time-homogeneous stochastic differential equation with 1-periodic coefficients. We obtain an estimate for the rate of approach of a solution of this equation to a solution of the corresponding It? stochastic equation.     

Equations with Random Gaussian Operators with Unbounded Mean Value

We consider an equation in a Hilbert space with a random operator represented as a sum of a deterministic, closed, densely defined operator and a Gaussian strong random operator. We represent a solution of an equation with random right-hand side in terms of stochastic derivatives of solutions of an equation with deterministic right-hand side. We consider applications of this representation to the anticipating Cauchy problem for a stochastic partial differential equation.     

Coefficient characterization of linear differential equations with maximal symmetries

A characterization of the general linear equation in standard form admitting a maximal symmetry algebra is obtained in terms of a simple set of conditions relating the coefficients of the equation. As a consequence, it is shown that in its general form such an equation can be expressed in terms of only two arbitrary functions, and its connection with the Laguerre–Forsyth form is clarified. The characterizing conditions are also used to derive an infinite family of semi-invariants, each corresponding to an arbitrary order of the linear equation. Finally a simplifying ansatz is established, which allows an easier determination of the infinitesimal generators of the induced pseudo group of equivalence transformations, for all the three most common canonical forms of the equation.     

On existence and asymptotic stability of solutions of a nonlinear integral equation

We prove an existence theorem for a nonlinear integral equation being a Volterra counterpart of an integral equation arising in the traffic theory. The method used in the proof allows us to obtain additional characterization in terms of asymptotic stability of solutions of an equation in question.