Key Takeaway: This paper presents numerical methods for integrating periodic or oscillatory solutions of ordinary differential equations using trigonometric polynomials, with potential applications in numerical quadrature.

Abstract

There are many numerical methods available for the step-by-step integration of ordinary differential equations. Only few of them, however, take advantage of special properties of the solution that may be known in advance. Examples of such methods are those developed by BROCK and MURRAY [9], and by DENNIS Eg], for exponential type solutions, and a method by URABE and MISE [b~ designed for solutions in whose Taylor expansion the most significant terms are of relatively high order. The present paper is concerned with the case of periodic or oscillatory solutions where the frequency, or some suitable substitute, can be estimated in advance. Our methods will integrate exactly appropriate trigonometric polynomials of given order, just as classical methods integrate exactly algebraic polynomials of given degree. The resulting methods depend on a parameter, v=h~o, where h is the step length and ~o the frequency in question, and they reduce to classical methods if v-~0. Our results have also obvious applications to numerical quadrature. They will, however, not be considered in this paper. 1. Linear functionals of algebraic and trigonometric order In this section [a, b~ is a finite closed interval and C ~ [a, b~ (s > 0) denotes the linear space of functions x(t) having s continuous derivatives in Fa, b~. We assume C s [a, b~ normed by s (t.tt IIxll = )2 m~x Ix~ (ttt. a=0 a~t~b