A Personal Perspective on the History of the Numerical Analysis of Fredholm Integral Equations of the Second Kind Kendall Atkinson The University of Iowa July 25, 2008 Abstract This is a personal perspective on the development of numerical meth- ods for solving Fredholm integral equations of the second kind, discussing work being done principally during the 1950s and 1960s. The princi- pal types of numerical methods being studied were projection methods (Galerkin, collocation) and Nystrm methods. During the 1950s and 1960s, functional analysis became the framework for the analysis of nu- merical methods for solving integral equations, and this inuenced the questions being asked. This paper looks at the history of the analyses being done at that time. 1 Introduction This memoir is about the history of the numerical analysis associated with solving the integral equation x(s)  Z b a K(s; t)x(t) dt = y(s); a  s  b;  6=0 (1) At the time I was in graduate school in the early 1960s, researchers were in- terested principally in this one-dimensional case. It was for a kernel function K that was at least continuous; and generally it was assumed that K(s; t) was several times continuously di/erentiable. This was the type of equation stud- ied by Ivar Fredholm [26], and in his honor such equations are called Fredholm integral equations of the second kind. Today we work with multi-dimensional Fredholm integral equations of the second kind in which the integral operator is completely continuous and the integration region is commonly a surface in R 3 ; in addition, the kernel function K is often discontinuous. The Fredholm theory is still valid for such equations, and this theory is critical for the convergence and stability analysis of associated 1