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An adaptive Filon-type method for oscillatory integrals without stationary points

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Abstract

In this paper, we propose an adaptive Filon-type method to approximate oscillatory integrals. By using some S-shaped functions connecting Gauss-Legendre nodes and Filon-type method, the adaptive schemes behave well for small as well as for large frequencies. Adding frequency-dependent nodes can improve the asymptotic order at the same time. Moreover, there exist some complex nodes leading to a higher asymptotic order method approximating to the integral, and the special nodes work in other methods as well. The error can be further decreased by taking extra Chebyshev nodes. The efficiency and accuracy are tested by some experiments.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, China

    Longbin Zhao & Chengming Huang

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  1. Longbin Zhao
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  2. Chengming Huang
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Correspondence to Chengming Huang.

Additional information

This work was supported by NSF of China (Nos. 11371157 and 91130003) and the Graduates’ Innovation Fund of Huazhong University of Science and Technology (No. 2015650011).

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Zhao, L., Huang, C. An adaptive Filon-type method for oscillatory integrals without stationary points. Numer Algor 75, 753–775 (2017). https://doi.org/10.1007/s11075-016-0219-3

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  • DOI: https://doi.org/10.1007/s11075-016-0219-3

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