A Collection Method for the Numerical Solution of Integral Equations

Collocation methods for the solution of equations $(I + K)u = z$ in the space $X = C[a,b]$ when K is a compact linear operator in $[X]$ are studied as projections onto subspaces of spline functions with distinct knots. Existence of solutions of the collocation equations as well as rate of convergence of these approximate solutions to the actual solution are given via Jackson’s theorems. In the event $(I + K)u = z$ is an integral equation, quadratures are introduced which preserve the rate of convergence. In this case collocation with quadrature is shown to be equivalent to full discretization of the Nystrom method as studied by Kantorovich and Anselone.