, Volume 105, Issue 4, pp 633-658
Date: 10 Jan 2007

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

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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.