On The Numerical Solution For Singular Integral Equations With Cosecant Kernel

The numerical solution is important for the theory and applications of singular integral equations(SIE).On one hand ,some problems on engineering can be transformed to SIE,so, the requirment for numerical solution arises naturally; on the other hand ,it is difficult or even impossible to give get the closed-form solution for general SIE. People usually study the solvable conditions and the relations between SIE and its associated SIE,for example the Noether theorem. Hence,it is necessary and important to find the approximate solution for SIE.At the same time , the study of numerical solution can develop the SIE theory and strengthen the relations between SIE with operator theory, compute theory,approximate theory,orthogonal polynomial theory,Fourier analysis and so on.In the known investigations,there are a good few numerical methods on the SIE with Cauchy kernel including the collocation method,spline method,Galerkin method an so on,especially various collocation method have been the topic of a great many of papers in the past twenty years. But the study on numerical solution for SIE with Hilbert kernel is less.Of course, it is more difficult to study numerical method for SIE with Hilbert kernel.The research on the numerical method for SIE with cosecant kernel is almost empty.In this paper,we will construct a kind quadrature rule for the singular integral with cosecant kernel firstly.we use a method similar to the classcial method — seperation singular point which can transform the original problem to the quadrature rule for proper integral of periodic function.The method we use didn't appear in literature up to now to the author's knowledge. On the base of quadrature rule for singular integral with cosecant kernel, we propose the collocation method for the SIE with cosecant kernel.The framework of this dissertation consists of the following four parts:The introduction is an introduction on the background,motivation,and the principle results of this dissertation.In chapter l,we study mainly the quadrature rule for the singular integal with cosecant kernel, we give out the quadrature representation ,the remainder and discuss the convengence.At the same time ,we study the quadrature rule for properintegral of anti-periodic function with weight function , paratrigonometric type and the highest paratrigonometric precision quadrature rules are constructed.In chapter 2,we discuss the properties of singular integral operators associated with the SIE with cosecant kernel.The unisolving operator and the restricting operator can help us to grasp the conception of index of SIE with cosecant kernel.We simplify the classical results andIn chapter 3,we propose the a kind numerical method called collocation method. We construct the singular quadrature operators(SQO) respponding singular integral operators(SIO),the properties of SQO are similar to SlO.We replace the SIO by rsepponding SQO and get the direct function equation (or approximation equation),then by discretizing the function equation,we get the linear algebra equation(or numerical equation).We can slove the numerical equation to get the solution of approximation equation.This way is simple and convenient,but it is very difficult to discuss the existence and convengence of approximate solution. So,we propose the other source of collocation-indirect quadrture method which base on the regularizing equation.Prom the regularizing equation,we can construct respponding (in-direct)approximation equation and (indirect)numerical equation.Then we use the abstract properties of SIO and SQO to prove the equivalence of direct method and indirect method, On this base ,we discuss the existence and convengence of approximation solution.In chapter 4,we study some classes of SIE with periodic kernel which have singularity in solutions.Some classical results are generalized.There are mainly four innovations in this dissertation.(l)We create a new method to seperate singular point in singular integral with coseant kernel by using the idea appearing in the study on paratrigonometric inter-polation.Then we construct the quadrature formula first time.(2)It is the first time to study the properties of SIO associated with SIE with cosevant kernel by use the new result on paratrigonometric.We simplify the classical results and make it easy to grasp the conception of index by introducing the unisolving operator and restricting operators.(3)It is the first time to propose the collocation method for SIE with cosecantkernel.We use abstract properties of SIO and SQO and discretizing operators to prove the coincidence,avoiding the complicated calculation.(4) We study the SIE with periodic kernel systematically and generalize some classcical results about SIE with periodic kernel.