不定积分的代数解法

基于积分运算是微分运算的逆运算,利用线性代数方法,提出了一种求解几类特殊函数不定积分的新方法,并结合实例验证了方法的有效性.     

A Domain Integral Equation for the Bergman Kernel

Certain integral operators involving the Szegö, the Bergman and the Cauchy kernels are known to have the reproducing property. Both the Szegö and the Bergman kernels have series representations in terms of an orthonormal basis. In this paper we derive the Cauchy kernel by means of biorthogonality. The ideas involved are then applied to construct a non-Hermitian kernel admitting a reproducing property for a space associated with the Bergman kernel. The construction leads to a domain integral equation for the Bergman kernel.^{1 2}     

Application of Two-Parameter Stabilizing Functions in Solving a Convolution-Type Integral Equation by Regularization Method

This paper examines the solution of convolution-type integral equations of the first kind by applying the Tikhonov regularization method with two-parameter stabilizing functions. The class of stabilizing functions is expanded in order to improve the accuracy of the resulting solution. The features of the problem formulation for identification and adaptive signal correction are described. A method for choosing regularization parameters in problems of identification and adaptive signal correction is suggested.     

Boundary Integral Operator for the Fractional Laplace Equation in a Bounded Lipschitz Domain

We study the boundary integral operator induced from fractional Laplace equation in a bounded Lipschitz domain. As an application, we study the boundary value problem of a fractional Laplace equation.     

周期手性结构电磁散射问题的积分方程方法

In this paper, we consider the electromagnetic scattering by a periodic chiral structure. The media is homogeneous and the structure is periodic in one direction and invariant in another direction. The electromagnetic fields inside the chiral medium are governed by Maxwell equations together with the Drude-BornFedorov equations. We simplify the problem to a two-dimensional scattering problem and discuss the existence and the uniqueness of solutions by an integral equation approach. We show that for all but possibly a discrete set of wave numbers, the integral equation has a unique solution.     

Partial Decomposition of a Domain Containing Thin Tubes for Solving the Heat Equation

An initial–boundary value problem for the heat equation in a three-dimensional domain containing thin cylindrical tubes is considered. The Neumann condition is set on the lateral boundaries of the tubes. The original three-dimensional problem is reduced to a hybrid-dimensional one in which the heat equation in the tubes is replaced by the one-dimensional heat equation in shorter cylinders (subtubes), and the three- and one-dimensional equations are matched on the bases of the subtubes. The difference between the solutions of the original and hybrid-dimensional problems is estimated using two geometric characteristics: the distance between the bases of the tubes and subtubes and the reciprocals of the minimal positive eigenvalues of the Neumann problem for the Laplace operator in the tube cross sections.     

An Iterative Variant of the Degenerate Kernel Method for Solving Fredholm Integral Equations

An iterative variant of the classical degenerate kernel method for solving Fredholm integral equations of the second kind is presented and its convergence properties are studied.     

Domain Approximation Method for Solving Multifacility Location Problems on a Sphere

This paper considers the problem of optimally locating new facilities on a sphere among existing facilities so that the sum of all weighted geodesic distance pairs is minimized. A method involving an iterative solution is presented. The procedure involves the approximation of the domain of objective function, which in the limit approaches to that of the original objective function. Computational experience with the procedure is described.     

An Approximate Method for Solving an Inverse Coefficient Problem for the Heat Equation

Journal of Applied and Industrial Mathematics - We construct a numerical method for recovering a variable coefficient in the Cauchy problem and also in the initial boundary value problem for the...     

一类求解抛物型方程侧边值问题的最优滤波方法

该文考虑一类特殊的抛物型方程侧边值问题,即一类含有对流项的非标准逆热传导问题. 给定在x=1处的温度测量值来确定区间(0,1)上的未知解u(x, t). 这是一类不适定问题,即问题的解(如果解存在)不连续依赖于数据.为了求解这一问题, 必须采用某些正则化技巧. 该文给出了一种最优滤波方法, 使得问题的真实解和近似解之间的误差估计达到了Hölder型最优. 同时还证明了问题的解在x=0处的收敛性.     

An Integral Geometry Problem in a Nonconvex Domain

We consider the problem of recovering the solenoidal part of a symmetric tensor field f on a compact Riemannian manifold (M,g) with boundary from the integrals of f over all geodesics joining boundary points. All previous results on the problem are obtained under the assumption that the boundary M is convex. This assumption is related to the fact that the family of maximal geodesics has the structure of a smooth manifold if M is convex and there is no geodesic of infinite length in M. This implies that the ray transform of a smooth field is a smooth function and so we may use analytic techniques. Instead of convexity of M we assume that M is a smooth domain in a larger Riemannian manifold with convex boundary and the problem under consideration admits a stability estimate. We then prove uniqueness of a solution to the problem for     

关于矩阵方程求解的另一点注记

对任意给定的矩阵A∈Pm×n,B∈Pm×s(s≤n),探讨了矩阵方程AX=B有列满秩解,同时BY=A有行满秩解的充分必要条件,并且给出了基于矩阵的等价、齐次方程组的同解、向量组的等价及线性空间语言的推广.     

Method for Solving the Sine-Gordon Equation in Laboratory Coordinates

By solving the direct and inverse scattering problems for a rather ‘unconventional’ eigenvalue problem, we can solve the initial value problem for the sine-Gordon equation entirely in laboratory coordinates. This allows us to extend and accurately define the class of initial value problems which can be solved by ‘inverse scattering transforms.’ Simple examples are given to illustrate both the direct and the inverse scattering methods.     

求解对流扩散方程的Haar小波方法

本文用Haar小波求解对流扩散方程,将满足初始和边界条件的常系数偏微分方程简化为较简单的代数方程组进行求解．实例说明了这种方法具有收敛速度快和计算容易的特点,同时又避免了用Daubechies小波求解微分方程需要计算相关系数的麻烦．本文所使用的方法可以求解一般的微（积）分方程．     

求解二次Lagrangian有限元方程的代数两水平方法

基于矩阵图集的粗化算法,构造一种新的插值算子,提出了瀑布型代数两重网格法;然后结合部分几何信息,提出了求解二次Lagrangian有限元方程的代数两水平方法.数值实验表明该算法稳健性强、计算量更少.     

Solvability of a Nonlinear Integral Equation in Dynamical String Theory

We investigate an integral equation of the convolution type with a cubic nonlinearity on the entire real line. This equation has a direct application in open-string field theory and in p-adic string theory and describes nonlocal interactions. We prove that there exists a one-parameter family of bounded monotonic solutions and calculate the limits of solutions constructed at infinity.     

一种求多项式方程根的参数并行加速迭代法

推广了一种在无重根情况下,利用Newton类迭代法对同时求多项式零点的加速的迭代法.讨论了该方法的收敛性和收敛阶;最后给出数值算例表明:计算收敛阶和定理结论是一致的,且本算法具有较大的收敛范围.