某类系数与Fejér缺项级数有关的齐次和非齐次高阶线性微分方程亚纯解的增长性

Nevanlinna理论在复微分方程领域中具有广泛的应用,其中运用该理论研究复线性微分方程亚纯解的增长性和值分布与系数的增长性之间的关系是复微分方程领域中的重要论题．由于缺项级数具有一些特殊性质,当缺项级数作为方程系数时,这些性质即可发挥作用．因此,我们可结合缺项级数的定义和性质研究复线性微分方程亚纯解的性质．在本文中,我们运用Nevanlinna理论并结合Fejér缺项级数的定义和性质对一类齐次和非齐次高阶复线性微分方程进行了研究．当方程的某个系数与Fejér缺项级数有关而其余系数为整函数或亚纯函数时,得到了方程亚纯解的增长级的估计,推广并改进了前人已有结果．     

一类Hammerstein型积分方程及其应用

在不假定核函数非负的条件下，利用锥与半序方法及不动点指数理论，结合线性算子的谱半径，讨论了非线性Hammerstein型积分方程非零解的存在性。并将所得抽象结果具体应用于常微分方程两点边值问题，得到了上列积分方程存在非零解的新结论。     

基于随机市场系数模型的最优投资组合策略

本文将Merton的投资模型拓展到随机市场系数模型。在典型投资问题对应的动态规划中,值函数一般用Bellman方程的粘性解表示。本文通过指数变换把偏微分方程转变成一个半线性的抛物线方程,证明了其值函数连续解的存在性,并在此基础上给出了企业的最优投资组合策略。     

一类具指数函数系数的非线性复差分方程

众所周知,在复微分方程和复差分方程领域中,Malmquist型方程是比Painlev′e方程和Riccati方程形式更一般的非线性方程.在本文中,我们运用Nevanlinna理论的差分模拟结果和微分域理论对一类具指数函数系数的Malmquist型复差分方程进行了研究.当上述Malmquist型复差分方程的有限级超越亚纯解具有较少的零点和极点时,我们得到其增长性和指数函数ez的增长性一致.该结果是对复微分Malmquist定理和复差分Malmquist定理的推广和补充.     

分数阶微分方程耦合系统边值问题解的存在性

应用Green函数将微分方程边值问题可转化为等价的积分方程.近来此方法被应用于讨论分数阶微分方程边值问题正解的存在性.本文讨论非线性分数阶微分方程耦合系统的两点边值问题,应用Green函数,将其转化为等价的积分方程耦合系统,并设非线性项在无穷远处有增长条件,应用Schauder不动点定理证明解而非限于正解的存在性.     

一类高阶微分方程的亚纯解与其小函数的复振荡

利用亚纯函数的Nevanlinna基本理论和方法,本文研究了一类高阶齐次线性微分方程亚纯解的复振荡、亚纯系数的高阶齐次线性微分方程亚纯解与小函数的关系、以及亚纯系数的高阶齐次线性微分方程亚纯解的一阶导数与小函数的关系,得到了亚纯系数高阶齐次线性微分方程的亚纯解和其导数取小函数的精确估计,推广了一些已有文献的结论,得到了更一般,更精确的结果,且文中有例子表明结果是精确的.     

利用函数变换构造非线性发展方程新的复合型精确解

为了获得非线性发展方程新的复合型精确解,本文引入了一种函数变换,把常系数的非线性发展方程转化为二阶非齐次线性常微分方程。在此基础上利用常微分方程的理论和符号计算系统Mathematica及用(2+1)维修改的色散水波方程,构造了新的复合型精确解。这些解中包括指数函数、三角函数和有理函数,通过这几种形式组合而成的复合型单孤子解和双孤子解。     

高阶线性微分方程解与其某些导数的不动点

本文研究了以整函数为系数的高阶线性微分方程的解及其某些导数的不动点问题,指出它们的不动点与解的增长性密切相关,并给出不动点密度的精确估计。     

双参数弹性地基上正交各向异性矩形薄板自由振动的精确解

根据分离变量法得到了双参数弹性地基上正交各向异性矩形薄板自由振动封闭形式的精确解,其中边界条件为CCCC、SCCC和SSCC情况的精确解过去被认为是难以得到的。在分离变量方法中,利用控制微分方程本征值给出振型函数的解析形式和两个空间本征值和时间本征值的关系,再利用边界条件得到振型函数系数和本征值方程或频率方程的精确形式。数值结果与有限元结果及文献结果吻合较好,验证了本文方法和结果的正确性。     

用改进平均法求解自由衰减振动

为了提高平均法求解非线性振动的精度,必须考虑阻尼对振动周期影响,提出可将待解微分方程的圆频率与派生方程圆频率的差异函数表示为阻尼系数的多项式．通常只需取方程中一阶导数项的系数作为差异函数,就可在较大范围内提高平均法解的精度．利用改进平均法具体求解了线性阻尼衰减振动和杜芬型衰减振动,并与精确解或数值解进行了比较,证明改进平均法确实是有效的．     

Fundamental solutions for second order linear elliptic partial differential equations

This paper is concerned with outlining some fundamental solutions and Green's functions for a system of second order linear elliptic partial differential equations in two independent variables. The fundamental solution and a number of Green's functions are given in relatively elementary closed form for some cases when the coefficients in the equations are constant. When the coefficients are variable the fundamental solution is obtained for some particular classes of equations.     

一类高阶中立型微分方程周期解的存在性

本文研究一类高阶中立型泛函微分方程周期解的存在性,利用一些分析技巧和k-集压缩映射理论得到了该类方程至少存在一个周期解的两类充分条件.所得结果将现有关于常微分方程的结论推广到了泛函微分方程情形,同时减少或减弱了已有结果中的一些条件,从方程的形式和周期解的存在性条件两个方面推广和改进了文献中的相应工作.     

Trefftz Methods for Time Dependent Partial Differential Equations

In this paper we present a mesh-free approach to numerically solving a class of second order time dependent partial differential equations which include equations of parabolic, hyperbolic and parabolic-hyperbolic types. For numerical purposes, a variety of transformations is used to convert these equations to standard reaction-diffusion and wave equation forms. To solve initial boundary value problems for these equations, the time dependence is removed by either the Laplace or the Laguerre transform or time differencing, which converts the problem into one of solving a sequence of boundary value problems for inhomogeneous modified Helmholtz equations. These boundary value problems are then solved by a combination of the method of particular solutions and Trefftz methods. To do this, a variety of techniques is proposed for numerically computing a particular solution for the inhomogeneous modified Helmholtz equation. Here, we focus on the Dual Reciprocity Method where the source term is approximated by radial basis functions, polynomial or trigonometric functions. Analytic particular solutions are presented for each of these approximations. The Trefftz method is then used to solve the resulting homogenous equation obtained after the approximate particular solution is subtracted off. Two types of Trefftz bases are considered, F-Trefftz bases based on the fundamental solution of the modified Helmholtz equation, and T-Trefftz bases based on separation of variables solutions. Various techniques for satisfying the boundary conditions are considered, and a discussion is given of techniques for mitigating the ill-conditioning of the resulting linear systems. Finally, some numerical results are presented illustrating the accuracy and efficacy of this methodology.     

Linear or nonlinear hyperbolic wave problems with input sets (Part I)

Summary In this first part, collections of linear hyperbolic initial boundary value problems are treated which are defined via sets of coefficient functions in the differential equations. If the solutions are oscillatory, the nonlinear dependency of the solutions on coefficients becomes more and more ill-conditioned as time progresses, unless there is a sufficiently strong damping term in the differential equation. For the problem of dynamic buckling, the theory of the Neumann series yields a sufficient condition for the uniform boundedness of the oscillatory solutions which are induced by arbitrary continuous transient perturbations whose range is restricted to a suitable interval. In this part I, there is an introductory discussion of the Taylor-representation of sets of solutions in terms of constant coefficients. Via such a Taylor-representation, it is shown that solutions of the distortionless telephone line are insensitive to sufficiently small variations of the constant coefficients in this hyperbolic differential equation.This research has been substantially supported by NATO Senior Scientist Grant SA. 5-2-05B(1761)178(79) MDL and the A. v. Humboldt Foundation.     

Boundary integral equation formulation for the generalized micro-polar thermoviscoelasticity

A formulation of the boundary integral equation method for generalized linear micro-polar thermoviscoelasticity is given. Fundamental solutions, in Laplace transform domain, of the corresponding differential equations are obtained. The initial, mixed boundary value problem is considered as an example illustrating the BIE formulation. The results are applicable to the generalized thermoelasticity theories: Lord-Shulman with one relaxation time, Green-Lindsay with two relaxation times, Green-Naghdi theories, and Chandrasekharaiah and Tzou with dual-phase lag, as well as to the dynamic coupled theory. The cases of generalized linear micro-polar thermoviscoelasticity of Kelvin-Voigt model, generalized linear thermoviscoelasticity and generalized thermoelasticity can be obtained from the given results.     

Least-squares differential quadrature time integration scheme in the dual reciprocity boundary element method solution of diffusive–convective problems

Least-squares differential quadrature method (DQM) is used for solving the ordinary differential equations in time, obtained from the application of dual reciprocity boundary element method (DRBEM) for the spatial partial derivatives in diffusive–convective type problems with variable coefficients. The DRBEM enables us to use the fundamental solution of Laplace equation, which is easy to implement computationally. The terms except the Laplacian are considered as the nonhomogeneity in the equation, which are approximated in terms of radial basis functions. The application of DQM for time derivative discretization when it is combined with the DRBEM gives an overdetermined system of linear equations since both boundary and initial conditions are imposed. The least squares approximation is used for solving the overdetermined system. Thus, the solution is obtained at any time level without using an iterative scheme. Numerical results are in good agreement with the theoretical solutions of the diffusive–convective problems considered.     

传递函数法在非局部弹性梁动力学分析中的应用

采用传递函数方法进行了非局部弹性梁的动力学分析。非局部弹性梁内一点的应力与梁某一区域内任意一点的应变均有关系。本文基于Eringen的非局部弹性积分型本构关系,采用幂指数型核函数,利用Laplace变换导出梁的四阶偏微分形式振动方程,通过定义状态向量,将控制方程化为一阶微分方程组,并采用传递函数方法进行了求解,针对两种边界条件给出了非局部弹性梁的固有频率和固有振型。结果表明,同阶频率下,非局部弹性梁的频率比局部梁的频率低,振型基本一致。