Clifford分析中具有弱奇异核的积分方程的研究

发布时间：2018-05-07 03:44

本文选题：Clifford分析 + 第二类积分方程　；参考：《河北师范大学》2017年硕士论文

【摘要】：Clifford分析研究的是定义在实向量空间Rn上,取值于Clifford代数An(R)的函数.它是实分析和复分析的自然推广.在实分析和复分析中奇异积分方程的研究具有非常重要的意义,在许多实际问题中针对一些特殊的自然现象有广泛的应用.在实分析和复分析范围内,奇异积分方程已经形成了比较完善的理论体系.本文在此基础上，主要研究了 Clifford分析中具有H(?)lder连续核和弱奇异核的第二类积分方程，通过定义新的算子hA克服了 Clifford代数的不可交换性,并讨论了弱奇异核的迭核的连续性,进一步得到了Clifford分析中具有H(?)lder·连续核和弱奇异核的第二类积分方程的解的表达式,解的存在性和解的唯一性.本文分为以下三章。第一章给出了本文所需的一些预备知识和几个重要引理.第二章首先给出了 Clifford分析中具有H(?)lder连续核的第二类积分方程的定义,然后构造了一列函数作为此方程的近似解,讨论了函数列的收敛性.其次研究了此积分方程的解的表达式,解的存在性和解的唯一性.第三章研究了 Clifford分析中弱奇异核的迭核的连续性以及具有弱奇异核的第二类积分方程的解的表达式,解的存在性和解的唯一性.即把具有H(?)lder连续核K1(x,y)的积分方程的性质推广到具有弱奇异核K1(x-y|α(0αn/3)的积分方程上.
[Abstract]:Clifford analysis is a function defined on the real vector space R _ n and valued on the Clifford algebra Ani _ R). It is the natural generalization of real analysis and complex analysis. The study of singular integral equations in real analysis and complex analysis is of great significance and has been widely used in many practical problems for some special natural phenomena. In the range of real analysis and complex analysis, the singular integral equation has formed a relatively perfect theoretical system. In this paper, the second kind of integral equation with H(?)lder continuous kernel and weakly singular kernel in Clifford analysis is studied. By defining a new operator ha, the noncommutativity of Clifford algebra is overcome, and the continuity of overlapped kernel of weakly singular kernel is discussed. Furthermore, the expression of the solution of the second kind of integral equation with H(?)lder continuous kernel and weakly singular kernel in Clifford analysis is obtained, and the existence and uniqueness of the solution are obtained. This article is divided into the following three chapters. In the first chapter, some preparatory knowledge and some important Lemma are given. In chapter 2, we first give the definition of the second kind of integral equation with H(?)lder continuous kernel in Clifford analysis, then construct a series of functions as the approximate solution of the equation, and discuss the convergence of the sequence of functions. Secondly, the expression of the solution of the integral equation and the existence and uniqueness of the solution are studied. In chapter 3, we study the continuity of overlapped nuclei of weakly singular nuclei in Clifford analysis and the expressions of solutions of the second kind of integral equations with weak singular kernels, the existence and uniqueness of the solutions. That is, the property of integral equation with H(?)lder continuous kernel K1X XY) is extended to the integral equation with weakly singular kernel K1(x-y 伪 0 伪 n / 3).
【学位授予单位】：河北师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.83

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