一维热传导方程移动边界识别的计算方法

构造了一种正则化的积分方程方法来由Cauchy数据确定一维热传导方程的移动边界．在将区域延拓至规则区域后,通过Fourier方法将问题转化为一个第一类Volterra积分方程．然后分别用Lavrentiev正则化方法以及Tikhonov正则化方法将不稳定的第一类Volterra积分方程转化为适定的第二类积分方程,并分别将积分方程转化为常微分方程组,并用Runge—Kutta方法数值求解,以及直接离散来求解．最后通过自由边界上的条件得到数值的移动边界．通过一些数值试验表明此方法是有效可行的,并且给出的方法无需迭代,数值计算较简单．     

第一类弱奇异核Fredholm积分方程的数值求解

第一类弱奇异核Fredholm积分方程由于奇异及本质的不适定性,给求解带来很大难度．本文首先利用克雷斯变换将方程转化,并对转化后的方程进行高斯一勒让德离散,得到一离散不适定的线性方程组,结合正则化方法对该类问题进行数值求解．最后给出了数值模拟,验证了本文方法的可行性及有效性．     

无界区域抛物方程自然边界元方法

本文应用自然边界元方法求解无界区域抛物型初边值问题。首先将控制方程对时间进行离散化,得到关于时间步长离散化的椭圆型问题。通过Fourier展开,导出相应问题的自然积分方程和Poisson积分公式。研究了自然积分算子的性质,并讨论了自然积分方程的数值解法,最后给出数值例子。从而解决了抛物型问题的自然边界归化和自然边界元方法。     

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．     

带常系数的Cauchy型奇异积分方程的快速方法

Petrov-Galerkin 方法是研究Cauchy型奇异积分方程的最基本的数值方法. 用此方法离散积分方程可得一系数矩阵是稠密的线性方程组. 如果方程组的阶比较大, 则求解此方程组所需要的计算复杂度则会变得很大. 因此, 发展此类方程的快速数值算法就变成了必然. 该文将就对带常系数的Cauchy型奇异积分方程给出一种快速数值方法. 首先用一稀疏矩阵来代替稠密系数矩阵, 其次用数值积分公式离散上述方程组得到其完全离散的形式,然后用多层扩充方法求解此完全离散的线性方程组. 证明经过上述过程得到方程组的逼进解仍然保持了最优阶, 并且整个过程所需要的计算复杂度是拟线性的. 最后通过数值实验证明结论.     

Block Pulse函数法求二维非线性Volterra-Fredholm-Hammerstein积分方程的数值解

为了求解二维空间上的非线性Volterra-Fredholm-Hammerstein积分方程的数值解,借助Block Pulse函数,并构造相应的算子矩阵将待求二维VolterraFredholm-Hammerstein积分方程转化为非线性代数方程组,然后对式中的未知变量进行离散,求得原方程的数值解.数值结果表明,该方法可行且有效.     

L~1空间带弱奇异核的第二类Fredholm积分方程解法探究

针对带有弱奇异核的第二类Fredholm积分方程数值解法问题,介绍了两种方法.一种方法是直接用L~1空间中的离散化方法求其数值解;另一种方法是将弱奇异核通过迭代变为连续核,再用L~1空间中的离散化方法求其数值解,且通过对具体算例作图分析,从而得出直接用L~1空间中离散化方法更好.     

解Burgers方程的一种显式稳定性方法

拟线性Burgers方程在空间离散后转化成常微分方程,再用指数积分方法求解．数值结果表明指数积分法有显式稳定性,有相应Runge-Kutta方法相同的精度．     

带弱奇异核的第二类Fredholm积分方程数值解法比较

针对带有弱奇异核的第二类Fredholm积分方程数值解法问题,介绍了两种方法.一种方法是泰勒级数展开法;另一种方法是将弱奇异核通过迭代变为连续核,再用L¹空间中的离散化方法求其数值解,且通过对具体算例作图分析,从而得出L1空间中的离散化方法求其数值解,且通过对具体算例作图分析,从而得出L¹空间中离散化方法更好.     

对流扩散方程的混合时间间断时空有限元方法

构造并分析二阶对流扩散方程的混合时间间断时空有限元格式．利用混合有限元方法将二阶方程降阶,利用空间连续而时间允许间断的时空有限元方法离散低阶方程．证明数值解的稳定性、存在唯一性和收敛性．最后通过数值结果验证该算法的有效性和可行性．     

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.     

Richardson extrapolation of iterated discrete Galerkin solution for two-dimensional Fredholm integral equations

In this paper, we study the numerical solution of two-dimensional Fredholm integral equation by discrete Galerkin and iterated discrete Galerkin method. We are able to derive an asymptotic error expansion of the iterated discrete Galerkin solution. This expansion covers arbitrarily high powers of the discretization parameters if the solution of the integral equation is smooth. The expansion gives rise to Richardson-type extrapolation schemes which rapidly improve the original rate of the convergence. Numerical experiments confirm our theoretical results.     

Multigrid method for nonlinear integral equations

In this paper, we introduce a multigrid method for solving the nonliear Urysohn integral equation. The algorithm is derived from a discrete resolvent equation which approximates the continuous resolvent equation of the nonlinear Urysohn integral equation. The algorithm is mathematically equivalent to Atkinson’s adaptive twogrid iteration. But the two are different computationally. We show the convergence of the algorithm and its equivalence to Atkinson’s adaptive twogrid iteration. In our numerical example, we compare our algorithm to other multigrid methods for solving the nonliear Urysohn integral equation including the nonlinear multigrid method introduced by Hackbush.     

Convergence of a numerical scheme of the type of the vortex loop method on a closed surface with an approximation to the surface shape

We consider a linear integral equation with a hypersingular integral treated in the sense of the Hadamard finite value. This equation arises when solving the Neumann boundary value problem for the Laplace equation with the use of the representation of the solution in the form of a double layer potential. We study the case in which an exterior or interior boundary value problem is solved in a domain whose boundary is a smooth closed surface and the integral equation is written out on that surface. For the numerical solution of the integral equation, the surface is approximated by spatial polygons whose vertices lie on the surface. We construct a numerical scheme for solving the integral equation on the basis of such an approximation to the surface with the use of quadrature formulas of the type of the method of discrete singularities with regularization. We prove that the numerical solutions converge to the exact solution of the hypersingular integral equation uniformly on the grid.     

无单元Galerkin法在热传导中的应用

对热传导问题的微分方程采用无单元Galerkin法进行数值求解.首先,将微分方程用Galerkin加权残量法转化为等效的积分形式.然后,先将时间变量看作参数,对空间变量进行离散化,得到方程的半离散形式,接着,对时间采用向后Euler—Galerkin格式进行离散,得到方程的全离散形式最后,编制MATLAB程序,上机计算.列举了两个热传导算例,通过计算说明EFG法适用于热传导问题,且其计算速度快,精确度高、前后处理也十分方便,是一种具有潜力的温度场数值计算的新方法.     

RESEARCH ANNOUNCEMENTS High order Zumerical Solution of Integral Transport Equation in slab Geometry

There are some common numerical methods for solving neutron transport equation, which including the well-known discrete ordinates method, PN approximation and integral transport methods[1]. There exists certain singularities in the solution of transport equation near the boundary and interface[2]. It gives rise to the difficulty in the construction of high order accurate numerical methods. The numerical solution obtained by now can not attain the second order convergent accuracy[3,4].     

A generalization of discrete Gronwall inequality and its application to weakly singular Volterra integral equation of the second kind

This paper presents a new discrete Gronwall inequality. Using the inequality, we prove convergence and error estimate of the numerical solutions of the second weakly singular Volterra integral equation, where discrete equation is derived by Novot's quadrature formula.     

Stability of discrete volterra equations of hammerstein type

Stability conditions for Volterra equations with discrete time are obtained using direct Liapunov method, without usual assumption of the summability of the series of the coeffcients. Using such conditions, the stability of some numerical methods for second kind Volterra integral equation is analyzed.