解非线性二阶双曲型方程的一种新数值方法

本文建立了解二阶双曲型方程的一种新数值方法一再生核函数法．利用再生核函数，直接给出每个离散时间层上近似解的显式表达式．此方法的优点是：计算格式绝对稳定，且可显式求解；利用显式表达式，可实现完全并行计算等文中对近似解的收敛性和稳定性进行了理论分析，并给出数值算例．     

一类积分边值问题的再生核数值算法

研究了一类具有积分条件的边值问题.基于再生核理论,巧妙地构造了一个具有积分条件的再生核空间,并且给出了再生核函数表达式.应用泛函分析中的算子理论及逼近论思想,给出了方程的近似解,即再生核数值算法.通过实例验证了算法的可行性和有效性.     

关于边值问题的一种新的再生核数值算法

利用一种新的再生核算法讨论了二阶微分方程边值问题的数值解,并证明了解的收敛性.算法中,定义了再生核空间,避开了施密特正交化过程,借助于逆矩阵给出了近似解的表达式.通过数值算例,分析了数值解的逼近效果,并与泰勒级数法进行了比较,说明了方法的实用性和有效性.     

非线性方程组新的再生核方法

给出了二阶非线性方程组边值问题新的数值方法.该方法基于再生核和最小二乘法.首先建立再生核直积空间,接着证明了近似解的一致收敛性,避免了耗时的施密特正交化过程.数值算例展现出该算法简单、有效.     

广义例外华罗庚域的Bergman核函数

计算出两类广义例外华罗庚域的Bergman核函数的显式表达式. 同时利用Appell的多变量的超几何函数给出了例外华罗庚域的Bergman核函数在边界点附近的渐近性质.     

关于W_2~1[a,b]能否扩大的一些问题

讨论了W12[a,b]能否扩大为含有有间断点函数的再生核空间的问题.结论是:若再生核空间W W12[a,b]含有有间断点的函数,则间断点必固定、间断点个数必有限且非端点a,b.进一步,我们构造了函数含有n个间断点的再生核空间并给出其再生核表达式.     

W_2~m空间中样条插值算子与最佳逼近算子的一致性

由线性微分算子确定的样条是连接多项式样条与希氏空间中抽象算子样条的重要环节,对微分算子样条的研究,既可从更高的观点揭示和概括多项式样条,又可启示我们去发现抽象算子样条的一些新的理论和应用． Ｇｒｅｅｎ函数是研究微分算子样条的重要工具 ［１］,但在微分算子插值样条的计算及将样条用于数值分析中,再生核方法起着更重要的作用．文献［２］［３］给出了与二阶线性微分算子插值样条有关的再生核解析表达式;由此得到了二阶微分算子插值样条与空间Ｗ_２～１［ａ,ｂ］中最佳插值逼近算子的一致性;而且还利用再生核给出了Ｈｉ…     

华罗庚域的Bergman核函数

给出了4类华罗庚域的Bergman核函数的显表达式.其关键之处有两点：一是给出了它们的全纯自同构群;二是引进了semi-Reinhardt域,并给出了它们的完备标准正交函数系.在此基础上,给出了它们的Bergman核函数的显表达式,这也是方法上的创新之处.     

再生核空间偏微分方程解析数值解

１．引言 偏微分方程的近似解法一直是数值计算的重要内容之一。随着计算机的发展,各种实用的新方法也不断涌现．本文在再生核空间Ｈ　（Ｄ）中给出二阶偏微分方程边值问题解析形式的级数解,该级数解具有如下特点：１．级数截断就可直接得到解析数值解;２．解析数值解的误差在空间范数意义下单调下降． 设 Ｄ＝［ａ, ｂ］ ｘ ［ｃ, ｄ］是 Ｒ２中的任一矩形域, Г为边界,０,ｕ（ｘ,ｙ）∈Ｌ２（Ｄ）且是实的绝对连续函数,中规定内积如下： 范数定义为： 山中已证明码（利是一个再生核函数空间,其再生校函数研Ｘ,认（,…表达式…     

计算再生核的Green函数方法

本文讨论了利用Green函数计算再生核的方法,在Wm2空间中利用再生核的和性质以及Green函数理论给出再生核构造的一般方法,并利用此方法计算出W32空间的再生核.     

Krall-type orthogonal polynomials in several variables

For a bilinear form obtained by adding a Dirac mass to a positive definite moment functional defined in the linear space of polynomials in several variables, explicit formulas of orthogonal polynomials are derived from the orthogonal polynomials associated with the moment functional. Explicit formula for the reproducing kernel is also derived and used to establish certain inequalities for classical orthogonal polynomials.     

W_2~m空间中样条插值算子与最佳逼近算子的一致性

This paper discusses generalized interpolating splines which determined by n order linear differential operators, and the best operators of interpolating approximation in W_2~m spaces, The explicit constructive method for the reproducing kernel in W_2~m space is presented, and proves the uniformity of spline interpolating operators and the best operators of interpolating approximation W_2~m space by reproducing kernel. The explicit expression of approximation error on a bounded ball in W_2~m space, and error estimation of spline operator of approximation are obtained.     

一个二进小波变换像空间的再生核函数

1 引言 小波分析是结合泛函分析、应用数学、逼近论、调和分析、广义函数论等数学知识的结晶,具有深刻的理论意义和广泛的应用范围,被称为”数学显微镜”.基于其多分辨分析的特点以及在时、频两域都具有表征信号局部特征的功能,应用它可以解决许多Fourier变换不能解决的难题,为工程应用提供了一种新的、更有效的分析工具[1],由...     

Construction and calculation of reproducing kernel determined by various linear differential operators

We presented a method to construct and calculate the reproducing kernel for the linear differential operator with constant coefficients and a single latent root; further, we gave the formula for calculation. Additionally, by studying the recurrence relation of the reproducing kernel with arithmetic latent roots, we found that the reproducing kernel with a multi-knots interpolation constraint can be concisely represented by one with an initial-value constraint.     

Construction of cubature formulae with preassigned nodes§

In this article, a technique for developing cubature rules with preassigned nodes is presented to avoid wasting of information in scientific computation. The corresponding constructive method of the cubature rule is also given. As an application of the rules, a cubature formula on disk, which was derived via the method of reproducing kernel in (Xu, Y., 2000, Constructing cubature formulae by the method of reproducing kernel. Numerische Mathematik, 85, 155–173), is reconstructed by using our technique. When the preassigned nodes are selected as the nodes of a cubature formula of lower degree, an embedded cubature formula can be easily obtained by the presented method. Furthermore, some examples are included in the article.     

Solving integro-differential equations with Cauchy kernel

A simple method for solving the Fredholm singular integro-differential equations with Cauchy kernel is proposed based on a new reproducing kernel space. Using a transformation and modifying the traditional reproducing kernel method, the singular term is removed and the analytical representation of the exact solution is obtained in the form of series in the new reproducing kernel space. The advantage of the approach lies in the fact that, on the one hand, by improving the definition of traditional inner product, the representation of new reproducing kernel function becomes simple and requirement for image space of operator is weakened comparing with traditional reproducing kernel method; on the other hand, the approximate solution and its derivatives converge uniformly to the exact solution and its derivatives. Some examples are displayed to demonstrate the validity and applicability of the proposed method.     

Numerical solution of one-dimensional Burgers’ equation using reproducing kernel function

This paper presents a numerical method for one-dimensional Burgers’ equation by the Hopf–Cole transformation and a reproducing kernel function, abbreviated as RKF. The numerical solution is given as explicit integral expressions with the RKF at each time step, so that the computation is fully parallel. The stability and error estimates are derived. Numerical results for some test problems are presented and compared with the exact solutions. Some numerical results are also compared with the results obtained by other methods. The present method is easily implemented and effective.