用于解析函数复分析的共轭边界元法

由2个共轭的实调和函数构建1个复解析函数,其复分析在应用数学和力学领域具有重要的作用.提出了一个加权残数方程组,证明了该方程组为2个共轭函数的域内控制方程、边界条件和边界上Cauchy Riemann(柯西-黎曼)条件的近似解,等效为复解析函数的逼近方程.在离散空间中,由该加权残数方程分别推导出2个位势问题的直接边界积分方程和1个表示Cauchy-Riemann条件的有限差分方程,随后解决了弱奇异线性方程组的求解难题,并提出用Cauchy积分公式求内点值的方法,从而建立了一种用于复分析的常单元共轭边界元法.最后,用3个算例证明了所提出方法适用于域内或域外的幂函数、指数函数或对数函数形式的解析函数,而且其误差与2维位势问题是同等量级的.     

非齐次弹性力学问题双互易边界元方法研究北大核心CSCD

基于弹性力学边界元方法理论,将边界元法与双互易法结合,采用指数型基函数对非齐次项进行插值得到双互易边界积分方程.将边界积分方程离散为代数方程组,利用已知边界条件和方程特解求解方程组,得出域内位移和边界面力.指数型基函数的形状参数是由插值点最近距离的最小值决定,采用这种形状参数变化方案,分析径向基函数(RBF)插值精度以及插值稳定性.再次将指数型基函数应用到双互易边界元法中,分析双互易边界元方法下计算精度及稳定性,验证了指数型插值函数作为双互易边界元方法的径向基函数解决弹性力学域内体力项问题的有效性.     

简支梯形底扁球壳自由振动问题的准Green函数方法

以简支梯形底扁球壳的自由振动问题为例,详细阐明了准Green函数方法的思想.即利用问题的基本解和边界方程构造一个准Green函数,此函数满足了问题的齐次边界条件,采用Green公式,将简支梯形底扁球壳自由振动问题的振形控制微分方程化为两个耦合的第二类Fredholm积分方程.边界方程有多种选择,在选定一种边界方程的基础上,可以通过建立一个新的边界方程来表示问题的边界,以克服积分核的奇异性.最后由积分方程的离散化方程组有非平凡解的条件,求得固有频率.数值结果表明,该方法具有较高的精度.     

带两个Carleman位移的奇异积分方程Noether可解的充分必要条件

带Carleman位移的奇异积分方程理论,近年来得到了很大发展。在[1]中建立了这种奇异积分方程的Noether理论,所用的基本方法是建立所谓的对应方程组(是不带位移的奇异积分方程组,它的理论是已知的,参看[2],[3])。在[4]中讨论了带两个Carleman位移的奇异积分方程Noether可解的充分条件,并给出了计算指数的公式。本文目的是在文章[4]的基础上,利用不同的方法解决带两个Carleman位移的奇异积分方程Noether可解的充分必要条件问题,并把所得结果对带两个Carleman位移及未知函数复共轭值的奇异积分方程进行推广。     

大气运动基本方程组的稳定性分析

以分层理论提供的基本方法分析大气运动基本方程组的拓扑学特征;证明局地直角坐标系中的大气运动基本方程组在无穷可微函数类中是稳定方程;给出局部解意义下使方程组典型定解问题适定的充要条件;讨论大气动力学中有关“以过去推测未来”以及当涉及应用问题时如何修改定解条件和下垫面的选择等问题;指出在通常假设下,基本方程组中的3个运动方程和连续方程完全决定了这个方程组的性质．     

利用微分方程求解某些积分方程举例

我们通常称未知函数含在积分形式中的方程为积分方程.积分方程理论是数学的一个专门分支。某些最简单的积分方程，有时可以通过求导，转化为一个微分方程的定解问题。     

Pasternak地基上简支板振动问题的准格林函数方法

提出一种新的数值方法——准格林函数方法．以Pasternak地基上简支多边形薄板的振动问题为例,详细阐明了准格林函数方法的思想．即利用问题的基本解和边界方程构造一个准格林函数,这个函数满足了问题的齐次边界条件,采用格林公式将Pasternak地基上薄板自由振动问题的振型控制微分方程化为两个耦合的第二类Fredholm积分方程．边界方程有多种选择,在选定一种边界方程的基础上,可以通过建立一个新的边界方程来表示问题的边界,以克服积分核的奇异性,最后由积分方程的离散化方程组有非平凡解的条件,求得固有频率．数值方法表明,该方法具有较高的精度．     

利用方程思想求解几何图形的面积

解决某些数学问题的时候,需要通过已知量去求出未知量,这时解决问题的指导思想就是想方设法抓住问题的相等关系,建立数学中的方程或方程组的模型,通过方程或方程组来解决问题,这就是方程思想.利用方程思想可以求一些几何图形的面积,甚至用其他方法无法解决的面积问题,运用方程思想就可     

复变函数、积分变换与数理方程一体化教改方案的实施与体会

复变函数、积分变换与数理方程一体化教改方案的实施与体会方照琴（浙江工业大学）“复变函数、积分变换与数理方程一体化”是我校“工科数学一体化”教改项目中的一个组成部分，主要解决工程数学中该模块的整体优化问题。根据一体化教改的基本原则，我们、ｔｉｒｔ述三门...     

含共圆循环对称裂纹的一维六方准晶平面应变第一基本问题

讨论了一维六方准晶在整个周期平面循环对称基本域中含一个共圆循环对称裂纹的全平面应变第一基本问题.利用复变函数方法,将弹性平衡的问题转化为唯一可解的Fredholm奇异积分方程.引入保角映射并结合裂纹共圆的特点得出了此问题的解析解.此问题的结果对工程断裂问题具有理论意义.     

Use of radial basis functions for solving the second‐order parabolic equation with nonlocal boundary conditions

Nonlocal mathematical models appear in various problems of physics and engineering. In these models the integral term may appear in the boundary conditions. In this paper the problem of solving the one‐dimensional parabolic partial differential equation subject to given initial and nonlocal boundary conditions is considered. These kinds of problems have certainly been one of the fastest growing areas in various application fields. The presence of an integral term in a boundary condition can greatly complicate the application of standard numerical techniques. As a well‐known class of meshless methods, the radial basis functions are used for finding an approximation of the solution of the present problem. Numerical examples are given at the end of the paper to compare the efficiency of the radial basis functions with famous finite‐difference methods. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Cost-efficient numerical algorithm for solving the linear inverse problem of finding a variable magnetization

The paper is devoted to developing an original cost-efficient algorithm for solving the inverse problem of finding a variable magnetization in a rectangular parallelepiped. The problem is ill-posed and is described by the integral Fredholm equation. It is shown that after discretization of the area and approximation of the integral operator, this problem is reduced to solving a system of linear algebraic equations with the Toeplitz-block-Toeplitz matrix. We have constructed the memory efficient variant of the stabilized biconjugate gradient method BiCGSTABmem. This optimized algorithm exploits the special structure of the matrix to reduce the memory requirements and computing time. The efficient implementation is developed for multicore CPU and GPU. A series of the model problems with synthetic and real magnetic data are solved. Investigation of efficiency and speedup of parallel algorithm is performed.     

2010年全国研究生数模竞赛D题的评阅综述

2010年全国研究生数学建模竞赛D题的素材取自一项科研项目.将本题建为何种模型取决于建模人的数学基础和对本题描述运动过程的认识深度.评述介绍了三种建模思路:刚体平面运动模型,最优控制模型和微分方程模型,其中微分方程模型是在阅卷过程中从考生答卷中发现的,从答卷中发现了不少有共性的毛病,反应出国内大学和研究生数学教育中的弊病,值得有识之士们重视.     

带有缺失数据的结构方程模型中的模型选择问题

结构方程模型在社会学、教育学、医学、市场营销学和行为学中有很广泛的应用。在这些领域中,缺失数据比较常见,很多学者提出了带有缺失数据的结构方程模型,并对此模型进行过很多研究。在这一类模型的应用中,模型选择非常重要,本文将一个基于贝叶斯准则的统计量,称为L_v测度,应用到此类模型中进行模型选择。最后,本文通过一个模拟研究及实例分析来说明L_v测度的有效性及应用,并在实例分析中给出了根据贝叶斯因子进行模型选择的结果,以此来进一步说明该测度的有效性。     

非线性四阶方程正解存在问题

本文讨论了一个四阶非线性方程在二类不同边界条件下正解的存在问题，即多点边值问题和积分型的边值问题．采用的方法是锥拉伸和压缩不动点定理，这里的结果推广了这类四阶方程边值问题的结果．     

Numerical solution of the inverse electrocardiography problem with the use of the Tikhonov regularization method

The inverse electrocardiography problem related to medical diagnostics is considered in terms of potentials. Within the framework of the quasi-stationary model of the electric field of the heart, the solution of the problem is reduced to the solution of the Cauchy problem for the Laplace equation in R ³. A numerical algorithm based on the Tikhonov regularization method is proposed for the solution of this problem. The Cauchy problem for the Laplace equation is reduced to an operator equation of the first kind, which is solved via minimization of the Tikhonov functional with the regularization parameter chosen according to the discrepancy principle. In addition, an algorithm based on numerical solution of the corresponding Euler equation is proposed for minimization of the Tikhonov functional. The Euler equation is solved using an iteration method that involves solution of mixed boundary value problems for the Laplace equation. An individual mixed problem is solved by means of the method of boundary integral equations of the potential theory. In the study, the inverse electrocardiography problem is solved in region Ω close to the real geometry of the torso and heart.     

Qualitative difference between solutions of stationary model Boltzmann equations in the linear and nonlinear cases

We consider problems for the nonlinear Boltzmann equation in the framework of two models: a new nonlinear model and the Bhatnagar-Gross-Krook model. The corresponding transformations reduce these problems to nonlinear systems of integral equations. In the framework of the new nonlinear model, we prove the existence of a positive bounded solution of the nonlinear system of integral equations and present examples of functions describing the nonlinearity in this model. The obtained form of the Boltzmann equation in the framework of the Bhatnagar-Gross-Krook model allows analyzing the problem and indicates a method for solving it. We show that there is a qualitative difference between the solutions in the linear and nonlinear cases: the temperature is a bounded function in the nonlinear case, while it increases linearly at infinity in the linear approximation. We establish that in the framework of the new nonlinear model, equations describing the distributions of temperature, concentration, and mean-mass velocity are mutually consistent, which cannot be asserted in the case of the Bhatnagar-Gross-Krook model.     

Numerical Treatment of Homogeneous Semi-Markov Processes in Transient Case–a Straightforward Approach

This paper presents the numerical solution of the process evolution equation of a homogeneous semi-Markov process (HSMP) with a general quadrature method. Furthermore, results that justify this approach proving that the numerical solution tends to the evolution equation of the continuous time HSMP are given. The results obtained generalize classical results on integral equation numerical solutions applying them to particular kinds of integral equation systems. A method for obtaining the discrete time HSMP is shown by applying a very particular quadrature formula for the discretization. Following that, the problem of obtaining the continuous time HSMP from the discrete one is considered. In addition, the discrete time HSMP in matrix form is presented and the fact that the solution of the evolution equation of this process always exists is proved. Afterwards, an algorithm for solving the discrete time HSMP is given. Finally, a simple application of the HSMP is given for a real data social security example.     

Spectral Approach to D‐bar Problems

We present the first numerical approach to D‐bar problems having spectral convergence for real analytic, rapidly decreasing potentials. The proposed method starts from a formulation of the problem in terms of an integral equation that is numerically solved with Fourier techniques. The singular integrand is regularized analytically. The resulting integral equation is approximated via a discrete system that is solved with Krylov methods. As an example, the D‐bar problem for the Davey‐Stewartson II equations is considered. The result is used to test direct numerical solutions of the PDE.© 2017 Wiley Periodicals, Inc.     

Mixed boundary element solution for three-dimensional convection-diffusion problem with a velocity profile

A mixed boundary element formulation is presented for convection-diffusion problems with a velocity profile. In this formulation the convection-diffusion equation is considered as a nonlinear diffusion equation with inhomogeneous terms in which the convective term is involved additionally, because the spatial distribution of the drift velocity cannot be straightforwardly expressed in boundary integral form. Accordingly, a corresponding boundary integral equation may be described usually in the form of a so-called hybrid-type boundary integral equation.

In the present paper, mixed boundary elements are employed in a discrete model of the original convection-diffusion system. In the mixed element, potentials are approximated linearly, and their normal derivatives to boundaries are assumed constant. A simple iterative scheme is adopted in order to solve hybrid-type mixed boundary element equations. Simple three-dimensional models are dealt with in numerical experiments. The proposed approach gives more accurate and stable solutions compared with constant boundary elements which have been reported.