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Stable application of Filon–Clenshaw–Curtis rules to singular oscillatory integrals by exponential transformations

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Abstract

Highly oscillatory integrals, having amplitudes with algebraic (or logarithmic) endpoint singularities, are considered. An integral of this kind is first transformed into a regular oscillatory integral over an unbounded interval. After applying the method of finite sections, a composite modified Filon–Clenshaw–Curtis rule, recently developed by the author, is applied on it. By this strategy the original integral can be computed in a more stable manner, while the convergence orders of the composite Filon–Clenshaw–Curtis rule are preserved. By introducing the concept of an oscillation subinterval, we propose algorithms, which employ composite Filon–Clenshaw–Curtis rules on rather small intervals. The integral outside the oscillation subinterval is non-oscillatory, so it can be computed by traditional quadrature rules for regular integrals, e.g. the Gaussian ones. We present several numerical examples, which illustrate the accuracy of the algorithms.

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Authors and Affiliations

  1. Department of Basic Sciences, Iranian Institute for Encyclopedia Research, PO Box 14655-478, Tehran, Iran

    Hassan Majidian

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  1. Hassan Majidian
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Correspondence to Hassan Majidian.

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Communicated by Michael Floater.

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Majidian, H. Stable application of Filon–Clenshaw–Curtis rules to singular oscillatory integrals by exponential transformations. Bit Numer Math 59, 155–181 (2019). https://doi.org/10.1007/s10543-018-0730-0

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  • DOI: https://doi.org/10.1007/s10543-018-0730-0

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