二维Fredholm型泛函积分方程数值解法及收敛性分析

发布时间：2018-06-06 09:58

本文选题：二维Fredholm型泛函积分方程 + 径向基函数无网格解法　；参考：《五邑大学》2017年硕士论文

【摘要】：本文利用径向基函数无网格解法、最佳平方逼近解法、不动点迭代与加速迭代解法等高效数值解法对二维Fredholm型泛函积分方程进行求解,分别给出了数值算法格式、误差估计和收敛性分析的结果,进而给出数值例子阐明所提方法的可行性与可靠性.第一章主要给出了泛函积分方程解析解的存在唯一性定理及其适定性条件.第二章利用径向基函数无网格解法对二维Fredholm型泛函积分方程进行求解,并给出其数值算法格式、误差估计和收敛性分析,进而给出数值例子阐明了方法的可行性与可靠性.最佳平方逼近方法主要用于函数逼近问题,本文将此方法用于数值求解积分方程问题.第三章利用最佳平方逼近解法对二维Fredholm型泛函积分方程进行求解,并给出其数值算法格式、误差估计和收敛性分析,进而给出数值例子阐明了方法的可行性与可靠性并与第二章所提方法进行比较分析.不动点迭代与加速迭代方法主要用于非线性方程的求根问题,本文将此方法运用于数值求解泛函积分方程问题.第四章利用不动点迭代,Aitken加速迭代及Steffensen加速迭代解法对二维Fredholm型泛函积分方程进行求解,并给出其数值算法格式、误差估计和收敛性分析,进而给出数值例子阐明了方法的可行性与可靠性.
[Abstract]:In this paper, two dimensional Fredholm functional integral equations are solved by using the radial basis function meshless method, the best square approximation method, the fixed point iteration method and the accelerated iterative method. The results of error estimation and convergence analysis are given, and numerical examples are given to illustrate the feasibility and reliability of the proposed method. In chapter 1, we give the existence and uniqueness theorem of analytic solution of functional integral equation and the conditions of its fitness. In chapter 2, the radial basis function meshless method is used to solve the two-dimensional Fredholm functional integral equation, and its numerical algorithm format, error estimation and convergence analysis are given, and numerical examples are given to illustrate the feasibility and reliability of the method. The best square approximation method is mainly used in the function approximation problem. In this paper, the method is used to solve the integral equation numerically. In chapter 3, the optimal square approximation method is used to solve the two-dimensional Fredholm functional integral equation, and its numerical algorithm format, error estimation and convergence analysis are given. A numerical example is given to illustrate the feasibility and reliability of the method. The fixed point iterative and accelerated iterative methods are mainly used to find the root of nonlinear equations. This method is applied to solve the problem of functional integral equations numerically in this paper. In chapter 4, the fixed point accelerated iteration and Steffensen accelerated iterative method are used to solve the two-dimensional Fredholm functional integral equation. The numerical algorithm scheme, error estimation and convergence analysis are given. A numerical example is given to illustrate the feasibility and reliability of the method.
【学位授予单位】：五邑大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.83

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