一类Volterra积分方程配置法数值解的分析

发布时间：2018-04-12 22:06

  本文选题：第三类非线性VIE + 非紧算子　； 参考：《哈尔滨工业大学》2017年硕士论文

【摘要】：积分方程是数学的一个重要分支,而Volterra积分方程(VIE)在积分方程中占有重要地位。VIE的研究遍及物理、生物、化学等多个领域。常见的热传导模型、Lighthill模型、等时摆问题等都涉及到VIE。但是,对于一般的VIE,其解析解很难得到,所以求解VIE的数值解受到人们的广泛关注。近年来,配置法被许多学者应用到求解VIE中,并取得了一些成果。本文研究了配置法求解非线性第三类VIE,数值解的存在唯一性和收敛性得到系统的研究。首先,我们回顾了一些积分方程和积分算子的背景知识:cordial Volterra积分算子及其紧性、cordial Volterra积分方程(CVIE)解析解的存在唯一性。特别地,我们讨论了非线性第三类VIE相关算子的紧性。此外,我们给出了非线性第三类VIE解析解的存在唯一性。其次,配置法运用到非线性第三类VIE,我们采用与第二类VIE相似的方法讨论了带有紧算子的第三类VIE对应配置方程的可解性。但是对于非紧算子,为了保证可解性,我们在第一个区间运用了隐函数定理,在后面的区间则采用了改进的几何网格方法。最后,我们讨论了配置法的收敛阶。我们定义了误差函数,它满足一个线性离散VIE.收敛性是由这个线性离散Volterra积分算子的逆算子的一致有界性得出的,而其逆算子可以用一种适当的范数进行估计。
[Abstract]:Integral equation is an important branch of mathematics, and Volterra integral equation has played an important role in the integral equation. Vie has been studied in many fields, such as physics, biology, chemistry and so on.Some common heat conduction models, such as Lighthill model and isochronous pendulum, are all related to VIE.However, for the general VIEs, the analytical solutions are difficult to obtain, so the numerical solutions to the VIE are paid more and more attention.In recent years, collocation method has been applied to solve VIE by many scholars, and some achievements have been made.In this paper, we study the solution of nonlinear VIEs by collocation method. The existence, uniqueness and convergence of numerical solutions are studied systematically.First of all, we review the background knowledge of some integral equations and integral operators. The existence and uniqueness of analytic solutions of the compact cordial Volterra integral operators and their compact cordial Volterra integral equations are reviewed.In particular, we discuss the compactness of the nonlinear third class of VIE correlation operators.In addition, we give the existence and uniqueness of the nonlinear third class VIE analytic solution.Secondly, the collocation method is applied to the nonlinear third class of VIEs. The solvability of the third type of VIE corresponding collocation equations with compact operators is discussed by using the method similar to that of the second kind of VIE.But for noncompact operators, in order to ensure solvability, we use implicit function theorem in the first interval and the improved geometric mesh method in the following interval.Finally, we discuss the convergence order of collocation method.We define the error function, which satisfies a linear discrete VIE.The convergence is derived from the uniform boundedness of the inverse operator of the linear discrete Volterra integral operator, which can be estimated with an appropriate norm.
【学位授予单位】：哈尔滨工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.83
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本文编号：1741621