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

  • Hassan Majidian
Article

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.

Keywords

Filon–Clenshaw–Curtis rule Highly oscillatory integral Exponential transformation Double exponential transformation Algebraic singularity 

Mathematics Subject Classification

65D30 65T40 

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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of Basic SciencesIranian Institute for Encyclopedia ResearchTehranIran

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