The Study Of Integral Equation With Weakly Singular Kernel In Clifford Analysis

Clifford analysis studies the function that is defined on the real vector space Rn and valued on the Clifford algebra.It is the natural expansion of real analysis and complex analysis.The study of singular integral equation is of great significance in real and complex analysis.It is widely used in the practical problems for the special natural phenomena.In the field of real and complex analysis,the singular integral equation has formed a relatively complete theoretical system.In this paper,we mainly study the second kind of integral equations with H(?)lder continuous kernel and weakly singular kernel.By defining a new operator hA,we can overcome the non commutativity of the Clifford algebra.And the continuity of the iterated kernel of the weakly singular kernel is discussed.Furthermore,we obtain the expression,the existence and uniqueness of the solution about the second kind of integral equations with H(?)lder continuous kernel and weakly singular kernel.This paper is divided into three chapters.In chapter 1,the preliminary knowledge and several important lemmas which we needed are given.In chapter 2,firstly,the definition of the second kind of integral equations with H(?)lder continuous kernel is given in Clifford analysis.Secondly we construct a set of functions as the approximate solution of the equation and discuss its convergence.Finally,we study the expression,the existence and uniqueness of the solution about the integral equation.In chapter 3,the continuity of the iterated kernel of the weakly singular kernel in Clifford analysis is studied.And we also study the expression,the existence and uniqueness of the solution about the integral equation.The properties of the integral equation with H(?)lder continuous kernel K1(x,y)are extended to the integral equation with weakly singular kernel K1(x,y)/|x-y|α(0<α<n/3).