Efficient Filon-type methods for $$\int_a^bf(x)\,{\rm e}^{{\rm i}\omega g(x)}\,{\rm d}x$$

Abstract

Based on the transformation y = g(x), some new efficient Filon-type methods for integration of highly oscillatory function $$\int_a^bf(x)\,{\rm e}^{{\rm i}\omega g(x)}\,{\rm d}x$$ with an irregular oscillator are presented. One is a moment-free Filon-type method for the case that g(x) has no stationary points in [a,b]. The others are based on the Filon-type method or the asymptotic method together with Filon-type method for the case that g(x) has stationary points. The effectiveness and accuracy are tested by numerical examples.