Numerical quadrature methods for singular and nearly singular integrals

Abstract

This thesis is concerned with the development, design, and analysis of simple and efficient numerical quadrature methods for integrals on finite intervals with endpoint singularities, for integrals on the real line of steepest descent type, for integrals on finite intervals with branch point singularities near the interval of integration, and for integrals on the real line of Laplace type with branch point singularities near the path of integration. In Chapter 1 we develop and analyse a numerical quadrature method, known as the variable transformation method, for integrals on finite intervals with endpoint singularities. The idea of this variable transformation method is based on the Euler-Maclaurin formula, and seems to have been suggested first by Korobov in 1963. From the Euler-Maclaurin formula, it is obvious that the trapezium rule is an excellent numerical quadrature method for integrands that are periodic, and for integrands whose derivatives near the endpoints of the interval of integration decay rapidly. To make the integrands always satisfy these properties, the notion is to introduce a mapping function and substitute it into the integrals. This variable transformation method is also sometimes called a periodizing transformation. For integrals on the real line of steepest descent type, integrals on finite intervals with branch point singularities near the interval of integration, and integrals on the real line of Laplace type with branch point singularities near the path of integration, we design numerical quadrature methods and analyses based on the numerical quadrature method for integrals on finite intervals with endpoint singularities via suitable substitutions. These new numerical quadrature rules and analyses are illustrated and supported through numerical experiments. As larger applications we consider in Chapters 3 and 5 the problems of efficient evaluation of the impedance Green's function for the Helmholtz equation in a half-plane and half-space, important problems of acoustic propagation.