An application of approximation theory to the numerical solution of a Fredholm equation with a mild singularity

摘要

We present a numerical scheme for approximating the solution x = x(t) of the Fredholm equation x(t)−∫10|t−s|αa(t,s)x(s)dsf(t), 0 < α <, 0 ⩽ t ⩽ 1. Although the solution may have singularities in its derivatives at the endpoints of the interval [0,1], we achieve fast convergence by using spline approximation results of Rice, and results of deBoor describing approximation by splines with variable knots.