Abstract
Integral equations are an important branch of mathematics and V olterra integral equation (VIE) plays an important role in integral equations. The study of VIE relates to many fields, such as physics, biology and chemistry. The common heat conduction model, Lighthill model and isochronous pendulum problem are modeled by VIE. However, for general VIE, the numerical solution is difficult to be obtained, so solving the numerical solutions of VIE had raised much attention. In recent years, smooth transformation and collocation method have been applied to VIE by many scholars, and some results have been achieved.
In this paper, the smooth transformation and collocation method are studied to solve the third kind linear VIE and the cordial V olterra integral equation (CVIE). The regularity, existence and uniqueness of exact solutions and convergence of numerical solutions, etc. are studied.
Firstly, we apply the smoothing transformation to solve the third kind linear VIE whose the exact solution smoothness is low. Some of the conditions satisfied by the smooth transformation function are given. The compactness of the equation integral operator of the smooth transformation is discussed. According to the different conditions of in the equation, the general form of smooth transformation function is given and the regularity of the exact solution of the transformed equation is discussed.
Secondly, we apply collocation method to the transformed equation and discuss the solvability and study the convergence of the numerical solution of the collocation equation. By the inverse transformation of the smooth transform function, we discuss the convergence order of the numerical solution of the pre-transformation equation and the transformed equation.
Finally, on the basis of discussion about the third kind linear VIE, we discuss a kind of multi-solution CVIE. We add an extra condition to the original equation and study the existence and uniqueness of the exact solution of the equation. We apply the collocation method on the modified mesh to the equation before and after the transformation, and discuss the solvability of the collocation equation.
Keywords:third-kind linear VIE, smoothing transformation, compact operator, collocation method, solvability, convergence
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目录
摘要 ............................................................................................................................... I Abstract............................................................................................................................. I I 第1章绪论 ......................................................................................................... . (1)
1.1 课题来源及背景 (1)
1.1.1 课题的来源 ................................................................................................... .1
1.1.2 课题研究的背景和意义 .............................................................................. ..2 1.2 国内外研究现状及分析…................................................................................ ..2
1.2.1 国外研究现状 .............................................................................................. ..2
1.2.2 国内研究现状 ............................................................................................... .3
1.2.3 国内外文献综述的简析 .............................................................................. ..4 1.3 本文研究的主要内容.......................................................................................... ..5第2章第三类线性V olterra积分方程数值解的分析 (8)
2.1 引言........................................................................................................... . (8)
2.2 积分算子的性质 (8)
2.2.1 第三类线性V olterra积分方程的光滑变换 ............................................... ..8
2.2.2 光滑变换后积分算子的紧性 ....................................................................... .9 2.3 准确解的正则性.. (12)
2.4 本章小结 (16)
第3章配置方法及可解性和收敛性分析 ................................................................ ..17
3.1 引言 (17)
3.2 配置方法 (17)
3.3 配置方程可解性分析 (19)
3.4 配置方程收敛性分析 (20)
3.5 数值算例 (20)
3.6 本章小结 ............................................................................................................. .25第4章Cordial V olterra积分方程数值解的分析 . (26)
4.1 引言 .................................................................................................................... ..26 4.2 准确解的存在唯一性 . (26)
4.3 配置方程的可解性 (27)
4.4 光滑变换后配置方程的可解性 (29)
4.5 本章小结 (32)
结论 (33)
参考文献 ..................................................................................................................... (35)
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哈尔滨工业大学学位论文原创性声明和使用权限 (39)
致谢 (40)
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