An efficient method for evaluating the integral of a class of highly oscillatory functions

• Published in: Journal of computational and applied mathematics Vol. 230; no. 2; pp. 433 - 442
• Main Authors: Harris, Paul J., Chen, Ke
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier B.V 15.08.2009; Elsevier
• Subjects: Exact sciences and technology; High frequency; Mathematical analysis; Mathematics; Measure and integration; Numerical analysis; Numerical analysis. Scientific computation; Numerical approximation; Numerical integration; Oscillatory integrands; Partial differential equations; Sciences and techniques of general use; Sequences, series, summability; High frequency; Oscillatory integrands; Numerical integration; Numerical approximation; Numerical analysis; Applied mathematics; Fourier analysis; Numerical method; Critical point; Computing method
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/j.cam.2008.12.026

Highly oscillatory integrals require special techniques for their effective evaluation. Various studies have been conducted to find computational methods for evaluating such integrals. In this paper we present an efficient numerical method to evaluate a class of generalised Fourier integrals (on a line or a square) with integrands of the form f ( x ) e i k g ( x ) , under the assumption that in the domain of integration, both f and g are sufficiently smooth and that g does not have any stationary/critical points. Numerical analysis and results are given to illustrate the effectiveness of our method for computing generalised Fourier integrals.