【摘要】：本論文主要研究了第二類Fredholm積分方程全離散多投影外推算法,New Projection方法與迭代Kantorovich方法.首先分析了逼近解的誤差漸進(jìn)展開,隨后在漸進(jìn)展開之下執(zhí)行Richardson外推,收斂階能以h的二次冪增長(zhǎng).然后比較了New Projection方法,Kantorovich方法與迭代Kantorovich方法的計(jì)算復(fù)雜度與誤差分析,通過(guò)數(shù)值結(jié)果證明了兩個(gè)新方法有著更好的優(yōu)越性.全文總共分為四章:第一章,首先概述了投影法的歷史與國(guó)內(nèi)外最新研究進(jìn)展,然后介紹了本文的研究背景和涉及到的一些常用方法和結(jié)論,最后給出了本文的一些預(yù)備知識(shí).第二章,針對(duì)第二類Fredholm積分方程求解問(wèn)題,我們先采用了多投影方法給出逼近方程,隨后利用Galerkin方法得到迭代解的漸進(jìn)展開,并在漸進(jìn)展開基礎(chǔ)上執(zhí)行Richardson外推,從而提高逼近解的精度與收斂階.第三章,首先對(duì)第二類Fredholm積分方程采用多投影方法,緊接著對(duì)逼近方程采用Collocation方法得到迭代解的漸進(jìn)展開,并對(duì)其執(zhí)行Richardson外推,每一次外推都能提高2次收斂階.第四章,對(duì)于弱奇異積分方程的求解,我們首先分別介紹Kantorovich方法,迭代Kantorovich方法與New Projection方法,然后分析了三種方法之間的精度與計(jì)算復(fù)雜度,最后通過(guò)數(shù)值算例可以看出倆新方法要優(yōu)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.
【學(xué)位授予單位】：廣西師范學(xué)院
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O175.5

【參考文獻(xiàn)】

相關(guān)期刊論文 前1條

1 戈承超;隆廣慶;;第二類Fredholm積分方程全離散M-Galerkin的外推算法（英文）[J];廣西師范學(xué)院學(xué)報(bào)(自然科學(xué)版);2016年03期

，

本文編號(hào)：2185227