Nodal-type collocation methods for hypersingular integral equations and nonlocal diffusion problems

In this paper, we study nodal-type quadrature rules for approximating hypersingular integrals and their applications to numerical solution of finite-part integral equations and nonlocal diffusion problems. We first derive explicit expressions for the quadrature coefficients and establish corresponding error estimates. Some collocation schemes are then constructed based on these rules to numerically solve certain type of finite-part integral equations and nonlocal diffusion problems in one dimension, and condition number and optimal error estimates for the proposed schemes are also rigorously obtained. On uniform grids, these schemes are of Toeplitz structure which results in many advantages in developing fast linear solvers. Various numerical experiments are also performed to illustrate the theoretical results.