A "Sinc-Galerkin" Method of Solution of Boundary Value Problems

Abstract. This paper illustrates the application of a "Sinc-Galerkin" method to the approximate solution of linear and nonlinear second order ordinary differential equations, and to the approximate solution of some linear elliptic and parabolic partial differential equations in the plane. The method is based on approximating functions and their derivatives by use of the Whittaker cardinal function. The DE is reduced to a system of algebraic equations via new accurate explicit approximations of the inner products, the evaluation of which does not require any numerical integration. Using n function evaluations, the error in the final approximation to the solution of the DE is 0(e~cn ), where c is independent of n, and d denotes the dimension of the region on which the DE is defined. This rate of convergence is optimal in the class of «-point methods which assume that the solution is analytic in the interior of the interval, and which ignore possible singularities of the solution at the endpoints of the interval.