The Direct Method Of Soution For Singular Integral Equations With Soutions Having Singularities Of Order One

In this paper, we study the direct method of solution for a class of singular integral equations with solutions having singularities of order one. This paper mainly contains three parts.In the part of introduction, we give an interpretation about the background, motivation, and the principle results of this paper.In chapter 1, we introduce some notions and some classical results related to this paper, such as Cauchy type integral as well as the limiting value of Cauchy integral and so on.In chapter 2, we consider the direct method of solution for singular integral equations with solutions having singularities of order one. Firstly, we give the definition of singularities of order one and some related lemmas. Secondly, we give an important lemma. This lemma is the basis of this paper and also has an independent sense. Finally, by introducing Hermite interpolation polynomials and combining with the works of [7-10], we discuss the direct method of solution for singular integral equations with solutions having singularities of order one in terms of two cases of normal type and non-normal type. The necessary and sufficient condition for the solvability and the close form of the solution are obtained.There are mainly three innovations in this paper.(1)It is the first time to study the direct method of solution for singular integral equations with solutions having singularities of order one. The necessary and sufficient condition for the solvability and the close form of the solution are obtained.(2)It is different from [7-10] that we introduce Hermite interpolation polynomials to investigate the direct method of solution. This is the key to our strategy. It has some advandges as follows: Taking use of Hermite interpolation polynomials, we can eliminate singularities appeared in the problem such that problem as well as the analysis of the problem and calculation become more simple. Morever, the number of and the form of conditions for the solvability seem more simple and easy to check.(3)We establish an important lemma which is not only applicable to the direct method of singular integral equations but also to some other branches such as interpolation and quadrature theory in an independent sense.