第二类Fredholm积分方程投影算法的若干研究
发布时间：2018-08-15 19:48
【摘要】：本论文主要研究了第二类Fredholm积分方程全离散多投影外推算法,New Projection方法与迭代Kantorovich方法.首先分析了逼近解的误差渐进展开,随后在渐进展开之下执行Richardson外推,收敛阶能以h的二次幂增长.然后比较了New Projection方法,Kantorovich方法与迭代Kantorovich方法的计算复杂度与误差分析,通过数值结果证明了两个新方法有着更好的优越性.全文总共分为四章:第一章,首先概述了投影法的历史与国内外最新研究进展,然后介绍了本文的研究背景和涉及到的一些常用方法和结论,最后给出了本文的一些预备知识.第二章,针对第二类Fredholm积分方程求解问题,我们先采用了多投影方法给出逼近方程,随后利用Galerkin方法得到迭代解的渐进展开,并在渐进展开基础上执行Richardson外推,从而提高逼近解的精度与收敛阶.第三章,首先对第二类Fredholm积分方程采用多投影方法,紧接着对逼近方程采用Collocation方法得到迭代解的渐进展开,并对其执行Richardson外推,每一次外推都能提高2次收敛阶.第四章,对于弱奇异积分方程的求解,我们首先分别介绍Kantorovich方法,迭代Kantorovich方法与New Projection方法,然后分析了三种方法之间的精度与计算复杂度,最后通过数值算例可以看出俩新方法要优于Kantorovich方法.
[Abstract]:In this paper, the new Projection method and the iterative Kantorovich method for the second kind of Fredholm integral equation are studied. First, the error asymptotic expansion of the approximate solution is analyzed, then the Richardson extrapolation is performed under the asymptotic expansion, and the convergence order can be increased by the quadratic power of h. Then the computational complexity and error analysis of the New Projection method and the iterative Kantorovich method are compared. The numerical results show that the two new methods have better advantages. The paper is divided into four chapters: chapter 1, the history of projective method and the latest research progress at home and abroad are summarized, then the research background and some common methods and conclusions are introduced. Finally, some preliminary knowledge of this paper is given. In the second chapter, for solving the second kind of Fredholm integral equation, we first give the approximation equation by multi-projection method, then we obtain the asymptotic expansion of the iterative solution by using the Galerkin method, and then we carry out the Richardson extrapolation based on the asymptotic expansion. In order to improve the accuracy and convergence order of the approximate solution. In chapter 3, the second kind of Fredholm integral equation is solved by multi-projection method, then the asymptotic expansion of the iterative solution is obtained by using Collocation method for the approximation equation, and the Richardson extrapolation is performed on it. Each extrapolation can improve the order of convergence of the second order. In chapter 4, for the solution of weakly singular integral equations, we first introduce the Kantorovich method, iterative Kantorovich method and New Projection method, and then analyze the accuracy and computational complexity of the three methods. Finally, numerical examples show that the two new methods are better than the Kantorovich method.
【学位授予单位】：广西师范学院
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.5
本文编号：2185227
[Abstract]:In this paper, the new Projection method and the iterative Kantorovich method for the second kind of Fredholm integral equation are studied. First, the error asymptotic expansion of the approximate solution is analyzed, then the Richardson extrapolation is performed under the asymptotic expansion, and the convergence order can be increased by the quadratic power of h. Then the computational complexity and error analysis of the New Projection method and the iterative Kantorovich method are compared. The numerical results show that the two new methods have better advantages. The paper is divided into four chapters: chapter 1, the history of projective method and the latest research progress at home and abroad are summarized, then the research background and some common methods and conclusions are introduced. Finally, some preliminary knowledge of this paper is given. In the second chapter, for solving the second kind of Fredholm integral equation, we first give the approximation equation by multi-projection method, then we obtain the asymptotic expansion of the iterative solution by using the Galerkin method, and then we carry out the Richardson extrapolation based on the asymptotic expansion. In order to improve the accuracy and convergence order of the approximate solution. In chapter 3, the second kind of Fredholm integral equation is solved by multi-projection method, then the asymptotic expansion of the iterative solution is obtained by using Collocation method for the approximation equation, and the Richardson extrapolation is performed on it. Each extrapolation can improve the order of convergence of the second order. In chapter 4, for the solution of weakly singular integral equations, we first introduce the Kantorovich method, iterative Kantorovich method and New Projection method, and then analyze the accuracy and computational complexity of the three methods. Finally, numerical examples show that the two new methods are better than the Kantorovich method.
【学位授予单位】：广西师范学院
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.5
【参考文献】
相关期刊论文 前1条
1 戈承超;隆广庆;;第二类Fredholm积分方程全离散M-Galerkin的外推算法（英文）[J];广西师范学院学报(自然科学版);2016年03期
，本文编号：2185227
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