Clenshaw-Curtis algorithms for an efficient numerical approximation of singular and highly oscillatory Fourier transform integrals


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Kayıjuka I., Ege Ş. M. , Konuralp A., Topal F. S.

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, no.113201, ss.1-22, 2020 (SCI İndekslerine Giren Dergi)

  • Basım Tarihi: 2020
  • Doi Numarası: 10.1016/j.cam.2020.113201
  • Dergi Adı: JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
  • Sayfa Sayıları: ss.1-22

Özet

This paper investigates the implementation of Clenshaw-Curtis algorithms on singular and highlyoscillatory integrals for efficient evaluation of the finite Fourier-type transform of integrands withendpoint singularities. In these methods, integrands are truncated by orthogonal polynomials andspecial function series term by term.  Then their singularity types are computed using third andfourth-order homogeneous recurrence relations.  The first approach reveals its efficiency for low,moderate and very high frequencies, whereas the second one, is more efficient for small values offrequencies. Moreover, all the results were found quite satisfactory. Algorithms and programming codein MATHEMATICAR©9.0 are provided for the implementation of methods for automatic computationon a computer. Lastly, illustrative numerical experiments and comparison of the proposed Clenshaw-Curtis algorithms to the steepest descent method are mentioned in support of our theoretical analysisin the examples section.