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A method for efficient computation of integrals with oscillatory and singular integrand

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Abstract

A method based on modification of numerical steepest descent method to efficiently compute highly oscillatory integrals having endpoint singularities of algebraic and logarithmic type is proposed in this paper. The three-term recursion coefficients for orthogonal polynomials with respect to Gautschi’s weight function \(w^{G}(t;s)=t^{s}(t-1-\log t){\mathrm {e}}^{-t}\) (s > − 1) on \((0,\infty )\), as well as the corresponding quadrature formulas of Gaussian type, are used in this method. Finally, in order to illustrate the efficiency of the presented method a few numerical examples are included. The obtained results show that the proposed method is very efficient and economical in terms of computation time.

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References

  1. Ball, J. S., Beebe, N. H. F.: Efficient Gauss-related quadrature for two classes of logarithmic weight functions. ACM Trans. Math. Softw. 33(3), 21 (2007). Article 19

    Article  MathSciNet  Google Scholar 

  2. Brown, J. W., Churchill, R. V.: Complex variables and applications. The University of Michigan-Dearborn (2009)

  3. Chen, R.: On the evaluation of Bessel transformations with the oscillators via asymptotic series of Whittaker functions. J. Comput. Appl. Math. 250, 107–121 (2013)

    Article  MathSciNet  Google Scholar 

  4. Chen, R.: Fast Computation of a class of highly oscillatory integrals. Appl. Math. Comput. 227, 494–501 (2014)

    MathSciNet  MATH  Google Scholar 

  5. Clenshaw, C. W., Curtis, A. R.: A method for numerical integration on an automatic computer. Numer. Math. 2, 197–205 (1960)

    Article  MathSciNet  Google Scholar 

  6. Cvetković, A. S., Milovanović, G. V.: The Mathematica package “OrthogonalPolynomials”. Facta Univ. Ser. Math. Inform. 19, 17–36 (2004)

    MathSciNet  MATH  Google Scholar 

  7. Evans, G. A., Webster, J. R.: The numerical evaluation of a class of singular, oscillatory integrals. Intern. J. Comput. Math. 68, 309–322 (1996)

    Article  MathSciNet  Google Scholar 

  8. Filon, L. N. G.: On a quadrature formula for trigonometric integrals. Proc. Roy. Soc. Berlin 49, 38–47 (1928)

    MATH  Google Scholar 

  9. Flinn, E. A.: A modification of Filon’s method of numerical integration. J. Assoc. Comput. Mach. 7, 181–184 (1960)

    Article  MathSciNet  Google Scholar 

  10. Gautschi, W.: On generating orthogonal polynomials. SIAM J. Sci. Statist. Comput. 3, 289–317 (1982)

    Article  MathSciNet  Google Scholar 

  11. Gautschi, W.: Orthogonal polynomials: Computation and approximation. Oxford University Press, New York (2004)

    Book  Google Scholar 

  12. Gautschi, W.: Gauss quadrature routines for two classes of logarithmic weight functions. Numer. Algor. 55, 265–277 (2010)

    Article  MathSciNet  Google Scholar 

  13. Gautschi, W.: Orthogonal polynomials in matlab: Exercises and solutions, Software-Environments-Tools. SIAM, Philadelphia (2016)

    Book  Google Scholar 

  14. Gautschi, W.: A software repository for orthogonal polynomials. Software, Environments, and Tools. SIAM, Philadelphia (2018)

    Book  Google Scholar 

  15. Golub, G. H., Welsch, J. H.: Calculation of Gauss quadrature rules. Math. Comp. 23, 221–230 (1969)

    Article  MathSciNet  Google Scholar 

  16. Hasçelik, A. I.: Suitable Gauss and Filon-type methods for oscillatory integrals with an algebraic singularity. Appl. Numer. Math. 59, 101–118 (2009)

    Article  MathSciNet  Google Scholar 

  17. Hasçelik, A. I.: On numerical computation of integrals with integrands of the form \(f(x)\sin \limits ({\omega }/x^r)\) on [0, 1]. J. Comput. Appl. Math. 223, 399–408 (2009)

    Article  MathSciNet  Google Scholar 

  18. Hasçelik, A. I., Kılıç, D.: Efficient computation of a class of singular oscillatory integrals by steepest descent method. Appl. Math. Sci. 8, 1535–1542 (2014)

    MathSciNet  Google Scholar 

  19. Huybrechs, D., Vandewalle, S.: On the evaluation of highly oscillatory integrals by analytic continuation. SIAM J. Numer. Math. 44, 1026–1048 (2006)

    Article  MathSciNet  Google Scholar 

  20. Iserles, A., Nørsett, S.P.: Efficient quadrature of highly ossillatory integrals using derivatives. Proc. Royal Soc. A. 461, 1383–1399 (2005)

    Article  Google Scholar 

  21. Kang, H., Xiang, S., He, G.: Computation of integrals with oscillatory and singular integrands using Chebyshev expansions. J. Comp. Appl. Math. 227, 141–156 (2013)

    Article  MathSciNet  Google Scholar 

  22. Levin, D.: Procedures for computing one- and two-dimensional integrals of functions with rapid irregular oscillations. Math. Comp. 38, 531–538 (1982)

    Article  MathSciNet  Google Scholar 

  23. Levin, D.: Fast integration of rapidly oscillatory functions. J. Comput. Appl. Math. 67, 95–101 (1996)

    Article  MathSciNet  Google Scholar 

  24. Mastroianni, G., Milovanović, G. V.: Interpolation processes: Basic theory and applications. Springer Monographs in Mathematics. Springer-Verlag, Berlin (2008)

    Book  Google Scholar 

  25. Milovanović, G. V.: Numerical calculation of integrals involving oscillatory and singular kernels and some applications of quadratures. Comput. Math. Appl. 36, 9–39 (1998)

    Article  MathSciNet  Google Scholar 

  26. Milovanović, G. V.: Chapter 23: Computer algorithms and software packages. In: Brezinski, C., Sameh, A. (eds.) Walter Gautschi: Selected works with commentaries, vol. 3, pp 9–10. Basel, Birkhäuser (2014)

  27. Milovanović, G. V.: Computing integrals of highly oscillatory special functions using complex integration methods and Gaussian quadratures. Dolomites Res. Notes Approx. Special Issue 10, 79–96 (2017)

    Google Scholar 

  28. Milovanović, G. V., Cvetković, A. S.: Special classes of orthogonal polynomials and corresponding quadratures of Gaussian type. Math. Balkanica 26, 169–184 (2012)

    MathSciNet  MATH  Google Scholar 

  29. Olver, S.: Moment-free numerical integration of highly oscillatory functions. IMA J. Numer. Anal. 26, 213–227 (2006)

    Article  MathSciNet  Google Scholar 

  30. Ooura, T., Mori, M.: The double exponential formula for oscillatory functions over the half-infinite interval. J. Comput. Appl. Math. 38, 353–360 (1991)

    Article  Google Scholar 

  31. Piessens, R., Branders, M.: The evaluation and application of some modified moments. J. Comp. Appl. Math. 13, 443–450 (1973)

    MathSciNet  MATH  Google Scholar 

  32. Piessens, R., Branders, M.: On the computation of Fourier transforms of singular functions. J. Comp. Appl. Math. 43, 159–169 (1992)

    Article  MathSciNet  Google Scholar 

  33. Xu, Z., Milovanović, G. V.: Efficient method for the computation of oscillatory Bessel transform and Bessel Hilbert transform. J. Comput. Appl. Math. 308, 117–137 (2016)

    Article  MathSciNet  Google Scholar 

  34. Xu, Z., Milovanović, G. V., Xiang, S.: Efficient computation of highly oscillatory integrals with Hankel kernel. App. Math. Comput. 261, 312–322 (2015)

    Article  MathSciNet  Google Scholar 

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Funding

The third author was supported in part by the Serbian Academy of Sciences and Arts, Project No. Φ − 96.

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Authors and Affiliations

  1. Department of Mathematics, University of Gaziantep, 27310, Gaziantep, Turkey

    Dilan Kılıç Kurtoǧlu & A. Ihsan Hasçelik

  2. Serbian Academy of Sciences and Arts, Belgrade, 11000, Serbia

    Gradimir V. Milovanović

  3. Faculty of Sciences and Mathematics, University of Niš, Niš, 18000, Serbia

    Gradimir V. Milovanović

Authors
  1. Dilan Kılıç Kurtoǧlu
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  2. A. Ihsan Hasçelik
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  3. Gradimir V. Milovanović
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Correspondence to Dilan Kılıç Kurtoǧlu.

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Kurtoǧlu, D.K., Hasçelik, A.I. & Milovanović, G.V. A method for efficient computation of integrals with oscillatory and singular integrand. Numer Algor 85, 1155–1173 (2020). https://doi.org/10.1007/s11075-019-00859-8

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