ABSTRACT
Spline collocation methods have evolved as valuable techniques for the solution of a broad class of problems covering ordinary and partial differential equations, functional equations, integral equations and integro-differential equations. The suboptimality of nodal spline collocation led researchers to seek spline collocation methods of optimal order. In their fundamental work, C. de Boor and B. Swartz showed that optimal rates of convergence can be attained by collocating at certain Gauss points in each subinterval of the partition on which the spline space is defined. The advantage of the methods over finite element Galerkin methods is that the calculation of the coefficients in the equations determining the approximate solution is very fast since no integrals need be evaluated or approximated. Basically, the method involves the determination of an approximate solution in a suitable set of functions, sometimes called trial functions, by requiring the approximate solution to satisfy the boundary conditions and the differential equation at certain points, called the collocation points.
