Collocation Methods For Solving Singular Integral Equations And Their Application In Inverse Boundary Value Problems

Singular integral equations have been widely used in physical and engineer-ing. Numerous works have been devoted in developing efficient numerical meth-ods for singular integral equations, in which the collocation method becomes an important one for solving such equations because of simplicity and easy imple-mentation. The effectiveness of collocation method is usually dependent on the efficiency of the numerical integration. In various numerical integration methods, the Newton-Cotes rule draw a lot of attention due to the lower requirements of regularity for density function and free selection of grid.The main work of this paper can be divided into three parts. In the first part, we study the superconvergence phenomenon of any order Newton-Cotes rules for Cauchy singular integral on a circle. By making appropriate asymp-totic error estimate for Newton-Cotes rule, we find that the superconvergence phenomenon will occur when the local coordinate of singular point is zero of a particular function. Based on some properties of these special functions, we can prove the existence of superconvergence points in each subinterval. According to superconvergence results, we propose a new numerical quadrature that can be use to calculate the singular integral when the singular point is node of subinterval, which is very important for design of effective collocation algorithm. Finally, we give some examples to verify the theoretical analysis.The second part of the article is mainly the investigation of the collocation methods of singular integral equations. We apply the superconvergence result-s of singular integral and hypersingular integral to the collocation methods, by selecting the superconvergence points as collocation points,we get some special collocation schemes for solving singular integral equations. Using spectral anal- ysis, the eigenvalue of considered schemes can be expressed as a series, then the optimal error estimates are established. Meanwhile, we give some numeri-cal experiments, the corresponding numerical results agree with the theoretical analysis.In the third part, we focus on the application of singular integral equations in inverse boundary value problem. Based on natural integral equation and its inversion formula on circle, the inverse boundary value problem for Laplace e-quation can be converted to a pair of supersingular integral equation and a weak singular integral equation. Using the interpolation of trigonometric polynomial to approximate singular integrals and construct the relative collocation schemes, we solve the result linear system through Tikhonov regularization method. Nu-merical experiments are gave to show the effectiveness of this method.