再生核空间中的若干算子方程数值求解方法及其应用

发布时间：2018-04-04 21:31

本文选题：再生核空间　切入点：算子方程　出处：《哈尔滨工业大学》2016年博士论文

【摘要】：在实际问题中,很多复杂方程一般很难求出解析解,寻找有效的数值方法便成了解决该问题的主要途径和研究手段。算子方程能把复杂方程放在特定空间里表示成算子形式,不同的问题对应着不同的算子方程,对算子方程形式与结构的研究就显得尤为重要。本文将综合运用再生核空间和算子方程理论,对再生核空间中的算子方程的性质进行系统的研究,在此基础上用算子方程作为媒介,对几类不同的具体问题进行求解。本文通过引入算子方程将附有各种定解条件的复杂方程放在某一特定空间中,构造出空间上的映射。利用再生核空间优良的数值表现力,也就是将空间内的再生核函数对于固定的变量和空间中的任意函数通过内积运算表现出再生性,即把内积运算转化为求一点的函数值,这样对最基本的数值运算就有了一个连续性的表示,将离散的数值问题连续表现出来,简化了计算,避免了计算误差的积累,保证了计算精度且提高了计算速度,使问题的求解得到了最佳的处理。本文共分为五章,主要讨论再生核空间W_2~m(D)中算子方程的性质及一些微分方程求解方法。本文方法通过计算软件Mathematica进行模拟实现,并把此方法和其它已出现的方法的数值结果进行比较,用具体的数值算例验证本文方法的可行性。本文的主要研究内容概括如下:首先,在再生核空间W_2~m(D)中讨论算子方程的基本概念、性质及其求解算子方程的再生核投影方法,用条件数分析近似解的情况,应用谱分析理论得到一些有用结论。其次,基于再生核理论,根据几类不同的二阶非线性边值问题构造相应的再生核空间,求解出相应的再生核函数,发挥再生核函数在内积运算中的独特作用,用算子方程作为媒介提出新的数值方法。该方法的优点在于能够方便地处理边界条件且保证了数值解的连续性和一致收敛性。再次,提出了求解非共振条件下的弹性梁方程——一类非线性四阶微分方程的三种数值方法。算子的谱方法优点在于通过讨论相应的参数特征值问题的谱结构得到问题的解,再生核-Adomian分解方法结合再生核方法和Adomian分解方法的优势,再生核-搜极小值方法能够将问题的解以级数形式精确表达,保证较快的收敛速度。最后,针对一类变系数微分方程组,应用再生核空间良好的性质,从再生核空间族出发构造新的再生核直和空间W33和W11,引入内积和范数的运算,将该问题利用投影算子逼近做内积运算,得到精确解的级数表达式,通过有限项截断得到近似解,并且近似解的误差随着节点的增加而单调减少。
[Abstract]:In practical problems, it is difficult for many complex equations to find analytical solutions, so finding effective numerical methods becomes the main way and research means to solve this problem.Operator equations can be expressed as operator forms in a particular space, and different problems correspond to different operator equations. Therefore, it is very important to study the form and structure of operator equations.In this paper, the properties of the operator equation in the reproducing kernel space are systematically studied by using the theory of the reproducing kernel space and the operator equation theory. On this basis, the operator equation is used as the medium to solve several different kinds of specific problems.In this paper, by introducing operator equations, the complex equations with various definite solution conditions are placed in a certain space, and the mapping on the space is constructed.The reproducing kernel function in the space is reborn by the inner product operation for the fixed variable and the arbitrary function in the space, that is, the inner product operation is transformed into the function value of the point, by using the good numerical expression force of the reproducing kernel space, the reproducing kernel function in the space is reborn by the inner product operation.In this way, there is a continuous representation of the most basic numerical operation. The discrete numerical problem is shown continuously, the calculation is simplified, the accumulation of calculation errors is avoided, the calculation accuracy is guaranteed and the calculation speed is improved.So that the solution of the problem to get the best treatment.This paper is divided into five chapters. This paper mainly discusses the properties of operator equations in reproducing kernel space W _ 2C _ 2 and some methods for solving differential equations.The method of this paper is simulated by Mathematica and compared with the numerical results of other existing methods. The feasibility of this method is verified by a numerical example.The main research contents of this paper are summarized as follows: firstly, the basic concept of operator equation, its properties and the method of reproducing kernel projection for solving operator equation are discussed in the reproducing kernel space W _ T _ 2mg _ D), and the condition number is used to analyze the approximate solution.Some useful conclusions are obtained by applying spectral analysis theory.Secondly, based on the theory of reproducing kernel, the corresponding reproducing kernel space is constructed according to several kinds of second order nonlinear boundary value problems, and the corresponding reproducing kernel function is solved, which plays a unique role in inner product operation.A new numerical method is proposed by using operator equation as a medium.The advantage of this method is that it can deal with boundary conditions conveniently and ensure the continuity and uniform convergence of numerical solutions.Thirdly, three numerical methods for solving the elastic beam equation, a class of nonlinear fourth-order differential equations, are proposed.The advantage of the spectral method of the operator is that the solution of the problem is obtained by discussing the spectral structure of the corresponding parameter eigenvalue problem, and the reproducing kernel-Adomian decomposition method combines the advantages of the reproducing kernel method and the Adomian decomposition method.The reproducing kernel-search minimization method can accurately express the solution of the problem in the form of series, thus ensuring a faster convergence rate.Finally, for a class of differential equations with variable coefficients, by applying the good properties of reproducing kernel spaces, we construct new reproducing kernel direct sum spaces W33 and W11 from the family of reproducing kernel spaces, and introduce the operations of inner product and norm.By using the projection operator approximation to do the inner product operation, the series expression of the exact solution is obtained, and the approximate solution is obtained by truncating the finite term, and the error of the approximate solution decreases monotonously with the increase of nodes.
【学位授予单位】：哈尔滨工业大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O241.8

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