A Fast Galerkin Scheme for Linear Integro-differential Equations

We describe an expansion method for the solution of first orderand second order ordinary integro-differential equations, whichis a generalization of the Fast Galerkin scheme for second kindintegral equations (Delves, 1977a; Delves, Abd-Elal & Hendry,1979). The method retains the O(N² In N) operation count ofthat scheme, and pays particular attention to the way in whichthe boundary conditions are incorporated, with the aim of retainingalso the stable structure of the Fast Galerkin equations, andits very rapid convergence. An error analysis, and numericalexamples, indicate that these aims are met.     

An Augmented Galerkin Method for First Kind Fredholm Equations

We describe an augmented Galerkin technique for the numericalsolution of first kind Fredholm equations, which is simple touse and which has the considerable advantage of providing acheaply computed numerical criterion for the existence of asolution of the equations under study. The method has guaranteedstability, and leads to a standard linear programming problem(when posed in the l₁ or l norms). It is much faster than themethod recommended in a recent review by Lewis (1975); numericalcomparisons indicate that it achieves comparable accuracy. Theexistence criterion also appears effective in practice.