利用随机模拟方法求解第二类积分方程

给出一种求解第二类Fredholm和Volterra积分方程的数值算法,算法在数值积分技术的基础上使用Monte Carlo随机模拟方法求积分方程的近似解.通过数值例子证明了该算法是有效的.     

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．     

解带有扰动数据的第一类Volterra积分方程的谱正则化方法

该文的主要目的是通过使用Legendre配置方法和正则化策略来求解带有噪声数据的第一类Volterra积分方程,并给出该方法收敛性分析的严格数学证明.数值实验表明了该方法的有效性.     

基于哈尔小波函数的非线性随机It?-Volterra积分方程的数值解

正1 引言随机积分方程在工程、生物学、海洋学、物理学等领域有着广泛的应用,用其对随机现象进行建模变得越来越重要.由于系统的多样性和不同噪声影响,多种形式的随机方程都用于系统建模.其中,随机Volterral积分方程广受关注.然而,许多随机Volterra积分方程不存在精确解,因此对随机Volterra积分方程寻求较高精度的数值解是有意义的.目前,随机Volterra积分方程有不同的数值解法,如正交基法[4,6-11,13,14,16-18],Wash级数法[2,3],多项式法[1,5,12,15,19,20]等.     

Volterra方程数值解中带自适应步长控制的Runge-Kutta方法(英文)

在本文中我们构造了解第二类Volterra方程的一般Runge-Kutta方法,并且研究了第二类Volterra方程数值解法的自适应步长控制。     

倒向随机Volterra积分方程适应解的表示

倒向随机Volterra积分方程可以看作(确定性)Volterra积分方程和倒向随机微分方程的推广,在随机最优控制理论和数学金融学中有诸多应用.本文利用正倒向随机微分方程适应解表示的思想,得到所研究的一类倒向随机Volterra积分方程适应解的表示.这样的结果对研究适应解的正则性以及数值计算有重要的意义.     

配置方法求解高振荡傅里叶核的Volterra积分方程

本文用配置方法[1],基于moment-free Filon-type方法[2]求解高震荡傅里叶核的第一类Volterra积分方程[3]～[5],数值例子显示此方法的有效性。     

压电弹性动力学中第二类Volterra积分方程的数值解法

对于径向变形的压电空心圆柱和空心球弹性动力学问题,丁皓江等最近的研究表明,可以将它转变为关于一个时间函数的第二类Volterra积分方程,使求解工作得到极大的简化,又使探索第二类Volterra积分方程的快速而又高精度的数值解法成为一个关键．采用插值逼近方法,成功地导出了两个新型的递推公式,不仅计算速度快,且在较大时间步长时仍具有足够的精度,有着广泛的应用价值．     

滞时Volterra积分方程数值方法的数值稳定性分析

本文给出数值方法解Volterra积分方程的稳定性分析,我们判定可约积分方法的数值稳定性基于如下试验方程其中τ是正常数,p和q是复值的。在上述试验方程的情况下,我们研究θ-方法及可约积分方法的稳定性。     

随机Volterra积分方程相容解的稳定性

本文考虑了随机Volterra积分方程相容解的稳定性.应用Lyapunov第二方法,并以推广的Ito公式为工具,给出了随机Volterra积分方程相容解的几乎确定指数稳定和矩指数稳定的充分性原则.     

Approximate solution of a class of singular integral equations of second kind

A simple method based on polynomial approximation of a function is employed to obtain approximate solution of a class of singular integral equations of the second kind. For a hypersingular integral equation of the second kind, this method avoids the complex function-theoretic method and produces the known exact solution to Prandtl's integral equation as a special case. For a particular singular integro-differential equation of the second kind, this also produces an approximate solution which compares favourably with numerical results obtained by various Galerkin methods. The convergence of the method for both the equations is also established.     

On the numerical solution of Cauchy type singular integral equations by the collocation method

The collocation method for the numerical solution of Fredholm integral equations of the second kind is applied, properly modified, to the numerical solution of Cauchy type singular integral equations of the first or the second kind but with constant coefficients. This direct method of numerical solution of Cauchy type singular integral equations is compared afterwards with the corresponding method resulting from applying the collocation method to the Fredholm integral equation of the second kind equivalent to the Cauchy type singular integral equation, as well as with another method, based also on the regularization procedure, for the numerical solution of the same class of equations. Finally, the convergence of the method is discussed.     

Numerical solution of Fredholm integral equations of the second kind by using integral mean value theorem II. High dimensional problems

In this work, we generalize the numerical method discussed in [Z. Avazzadeh, M. Heydari, G.B. Loghmani, Numerical solution of Fredholm integral equations of the second kind by using integral mean value theorem, Appl. math. modelling, 35 (2011) 2374–2383] for solving linear and nonlinear Fredholm integral and integro-differential equations of the second kind. The presented method can be used for solving integral equations in high dimensions. In this work, we describe the integral mean value method (IMVM) as the technical algorithm for solving high dimensional integral equations. The main idea in this method is applying the integral mean value theorem. However the mean value theorem is valid for multiple integrals, we apply one dimensional integral mean value theorem directly to fulfill required linearly independent equations. We solve some examples to investigate the applicability and simplicity of the method. The numerical results confirm that the method is efficient and simple.     

Method of boundary integral equations as applied to the numerical solution of the three-dimensional Dirichlet problem for the laplace equation in a piecewise homogeneous medium

A Dirichlet problem is considered in a three-dimensional domain filled with a piecewise homogeneous medium. The uniqueness of its solution is proved. A system of Fredholm boundary integral equations of the second kind is constructed using the method of surface potentials, and a system of boundary integral equations of the first kind is derived directly from Green’s identity. A technique for the numerical solution of integral equations is proposed, and results of numerical experiments are presented.     

Numerical method for solving linear Fredholm fuzzy integral equations of the second kind

In this paper we use parametric form of fuzzy number and convert a linear fuzzy Fredholm integral equation to two linear system of integral equation of the second kind in crisp case. We can use one of the numerical method such as Nystrom and find the approximation solution of the system and hence obtain an approximation for fuzzy solution of the linear fuzzy Fredholm integral equations of the second kind. The proposed method is illustrated by solving some numerical examples.     

On the asymptotic convergence of a collocation method for some boundary integral equations of the first kind

In this paper we present a certain collocation method for the numerical solution of a class of boundary integral equations of the first kind with logarithmic kernel as principle part. The transformation of the boundary value problem into boundary singular integral equation of the first kind via single-layer potential is discussed. A discretization and error representation for the numerical solution of boundary integral equations has been given. Quadrature formulae have been proposed and the error arising due to the quadrature formulae used has been estimated. The convergence of the solution with respect to the proposed numerical algorithm is shown and finally some numerical results have been presented.     

A meshless local Galerkin method for the numerical solution of Hammerstein integral equations based on the moving least squares technique

In this paper, a computational scheme is proposed to estimate the solution of one- and two-dimensional Fredholm-Hammerstein integral equations of the second kind. The method approximates the solution using the discrete Galerkin method based on the moving least squares (MLS) approach as a locally weighted least squares polynomial fitting. The discrete Galerkin technique for integral equations results from the numerical integration of all integrals in the system corresponding to the Galerkin method. Since the proposed method is constructed on a set of scattered points, it does not require any background meshes and so we can call it as the meshless local discrete Galerkin method. The implication of the scheme for solving two-dimensional integral equations is independent of the geometry of the domain. The new method is simple, efficient and more flexible for most classes of nonlinear integral equations. The error analysis of the method is provided. The convergence accuracy of the new technique is tested over several Hammerstein integral equations and obtained results confirm the theoretical error estimates.     

Legendre Galerkin method for weakly singular Fredholm integral equations and the corresponding eigenvalue problem

In this paper, we consider Galerkin method for weakly singular Fredholm integral equations of the second kind and its corresponding eigenvalue problem using Legendre polynomial basis functions of degree ≤n. We obtain the convergence rates for the approximated solution and iterated solution in weakly singular Fredholm integral equations of the second kind and also obtain the error bounds for the approximated eigenelements in the corresponding eigenvalue problem. We illustrate our results with numerical examples.     

Numerical solution of Fredholm integral equations of first kind by two-dimensional trigonometric wavelets in holder space <Emphasis Type="Italic">C</Emphasis><Superscript>α</Superscript>([<Emphasis Type="Italic">a,b</Emphasis>])

In this article, we employ trigonometric wavelet bases to numerical solution of Fredholm integral equations of first kind in Holder space. Employment of Galerkin method for trigonometric wavelets in Fredholm integral equations of first kind has resulted in occurrence of two-dimensional trigonometric wavelets. Here, we present the convergence of two-dimensional trigonometric wavelets in numerical solution in Holder space C ^α([a, b]).     

积分算子方程的连续型配置方法

1．引言由于对积分算子方程来说，配置法比Galerkin法具计算量小的优点（少算一重积分），故配置法更受人们重视．但已有的文献几乎都是将配置空间取作非连续的分片多项式样条空间，以得到某种超收敛结果（如［1，2］）．这种方法存在下列不足：（a）光滑核Volterra积分方程与光滑核Fredholm积分方程具完全不同的收敛性质［1］，且需用不同的方法获得其加速收敛结果（比较［31与［4］），尽管Volterra积分方程在理论上被看作是Fredholm积分方程的特殊情形；（b）光滑核Volterra积分方程的配置解不具任何超收敛性，其迭代配置解也只在结点…