Several Studies Of Numerical Augmentation For Eigen-problems With Non-compact Integral Operators

This paper focus on the optimal convergence of numerical solutions for non-compact integral equations eigenvalue problem with weakly singular kernel. Wepropose the product integration methods for the weakly singular integrals, discreteproduct integration methods and product quasi-interpolation method. And thesemethods are utilized to solve the eigenvalue problem. The paper is organized asfollows:In chapter2, We consider the eigenvalue problem of a class of non-compactlinear operators given as the sum of a multiplication and a kernel operator. Wefocus on addressing critical issues with weakly singular kernel integral operator.It is shown that such operator can be approximated base on product integrationmethod. This leads to an approximation of A by piecewise polynomial interpolation.The convergence of the method is proved, it is proved that the convergence rate isO(hr+1).In chapter3, for numerical integration of the weakly singular kernel integraloperator, we use a discrete product integration method to approximate. The dis-crete product integration scheme is the product integration scheme in which theweights are computed numerically by Gauss-Legendre-type quadrature. It is shownthat entries of the corresponding matrix of this method can be evaluated exactly.The convergence of the method is proved; it is proved that the convergence rate ismax{hr+1, m2d}. Our numerical results confrm that this discrete product inte-gration method is particularly efcient for solving this problem.In chapter4, a product quasi-interpolation method is proposed for non-compactintegral equations eigenvalue problem with weakly singular kernel. We briefy in-troduce the quasi interpolation of order O(hm). With the help of a new form by achange of variables, we set up a theoretical framework which is convenient for analy-sis of product quasi-interpolation. And we apply this framwork to the eigen-problemwith weakly singular kernel to obtain optimal convergence results.