A New Collocation Method for Solving Certain Hadamard Finite-Part Integral Equation

Abstract

In this paper, we study a new nodal-type trapezoidal rule for approximating Hadamard finite-part integrals, and its application to numerical solution of certain finite-part integral equation. We start with a nodal-type trapezoidal rule discussed in [21], and then establish its error expansion analysis, from which a new nodal-type trapezoidal rule with higher order accuracy is proposed and corresponding error analysis is also obtained. Based on the proposed rule, a new collocation scheme is then constructed to solve certain finite-part integral equation, with the optimal error estimate being rigorously derived. Some numerical experiments are also performed to verify the theoretical results.