Procedures for Computing One- and Two-Dimensional Integrals of Functions With Rapid Irregular Oscillations

A collocation procedure for efficient integration of rapidly oscillatory functions is presented. The integration problem is transformed into a certain O.D.E. problem, and this is solved by a collocation technique. The method is also extended to two-dimensional integration, and some numerical results are appended showing the efficiency of the method in handling difficult cases of rapid irregular oscillations. 1. The Procedure for One-Dimensional Integrals. We consider integrals of the form (1.1) I=[hf(x)e«^dx, Ja where / is smooth and "nonoscillatory" and | q'(x) |» (b — a)~x. Two practical methods for evaluating rapidly oscillatory integrals are described in [1], the use of approximation as in Filon's method [3], [4] and the speedup method of Longman [5]. Formally both methods are applicable to any integral of the form (1.1), but their best performance is for the case of a constant frequency q' = W. In this note we present an efficient method which is applicable for cases of varying frequency q' using only a small number of values of/ and q' in [a, b] and the values q(a) and q(b). The proposed method follows the spirit of Filon's method. It is based upon the fact that if/were of the form (1.2) f(x) = iq'(x)p(x) +p'(x) = Lmp(x), aMdx = (" -f(p(x)e,'<^) dx (13) = p(b)e'"(h) -p(a)e^a\ Equation (1.2) can be considered as a differential equation for p(x), and any solution of this equation can be used in (1.3) for evaluating /. The general solution of this equation is (1.4) p(x) = e-¡"(x) Received March 24, 1981; revised July 29, 1981. 1980 Mathematics Subject Classification. Primary 65D30; Secondary 65D32. ff(t)eim dt + c 531 ©1982 American Mathematical Society 0025-5718/81/0000-0123/$03.25 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use