【作者】 郭国安；

【导师】 杜金元；

【作者基本信息】 武汉大学 ， 基础数学， 2005， 硕士

【摘要】 Riemann边值问题的研究大多限于正则型Riemann边值问题,封闭曲线上非正则型Riemann边值问题的研究最早始于F.D.Gakhov的著作[1],在其问题的解法中为了引入一些Hermite插值多项式一般要求已知函数具有足够高阶H(?)lder连续导数,在国内林玉波教授研究了较多的非正则型Riemann边值问题,这方面已有参考文献[4-9],杜金元教授[11]引入Peano导数并以此为基础构造一种广义Hermite插值多项式给出了封闭曲线上非正则型Riemann边值问题的一种新解法;随后H.Begehr教授[10]在非切向极限导数意义下构造Hermite插值多项式,给出了该问题的一种新解法.作为预备知识,本文第一章对此作了简要介绍。 第二章是本文的主要部分,分别给出了实轴上一类非正则型Riemann边值问题的提法、齐次问题的解法、两种导数的关系及非齐次问题的求解,本文运用杜金元教授[11]的方法获得了实轴上非正则型Riemann边值问题的封闭解及可解性条件,且在问题可解的情况下论证了函数Ψ(z)的非切向极限导数和Peano导数存在且相等,从而获得了统一的Hermite插值多项式,同样关于封闭曲线上非正则型Riemann边值问题,采用本文论证方法证得了函数Ψ(z)的非切向极限导数和Peano导数存在且相等,从而较好地统一了[10]、[11]中的Hermite插值多项式。更多还原

【Abstract】 The most researches on Riemann boundary value problem are confined to normal type. The Riemann boundary value problem of non-normal type on smooth closed contours first arised in the book [1] written by F.D.Gakhov. Its method of solution requires that the input functions have enough high order continuous Holder derivatives. Prof.Lin Yubo systematically explored the Riemann boudanry value problem of non-normal type in [4-9] in home. Prof.Du Jinyuan constructed a generalized interpolatory polynomial by introducing Peano derivatives and got a new method of solution; thereafter, H.Begehr constructed Hermite interpolatory polynomial in the meaning of non-tangential limit derivatives and got another method of solution. As preparation, these will be presented in the first chapter of this paper.The second chapter is the main part of this paper, in which the formulation of the Riemann boundary value problem of non-normal type on the real axis, the solution method of homogeneous problem, the relation between the two kinds of different derivatives and the inhomogeneous problem will be thoroughly given. In this paper, the solution and the solvability of the Riemann boundary value problem of non-normal type on the real axis will be given. Furthermore, it is shown that the twokinds of derivatives of the function Ψ(z) are existing and equivalent in the case ofthe solution about the original problem, therefore, we get uniformly Hermite interpolatory polynomial. The relation between the two kinds of different derivativesof the function Ψ(z) are similar for smooth closed contours by means of the same proof.更多还原