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Efficient quadrature rules for the singularly oscillatory Bessel transforms and their error analysis

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Abstract

In this paper, we focus on the computation and analysis of the highly oscillatory Bessel transforms with endpoint singularities of algebraic and logarithmic type. Based on the modification of the numerical steepest descent method, we present a new and efficient quadrature rule. Firstly, we divide the considered integrals into two parts by \(J_{m}(z)=\frac {1}{2}\left [H_{m}^{(1)}(z)+H_{m}^{(2)}(z)\right ]\), where each part can be transformed into the Fourier-type integrals. Then, we use the Cauchy’s residue theorem to convert these Fourier-type integrals into the infinite integrals on \([0,+\infty )\). Next, the resulting infinite integrals can be efficiently calculated by constructing some appropriate Gaussian quadrature rules. In addition, we conduct error analysis in inverse powers of the frequency parameter. Finally, several numerical examples are provided to show the efficiency and accuracy of the proposed method.

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References

  1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. DC National Bureau of Standards, Washington (1964)

    MATH  Google Scholar 

  2. Arfken, G.: Mathematical Methods for Physicists, 3rd edn. Academic Press, Orlando (1985)

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  4. Bao, G., Sun, W.: A fast algorithm for the electromagnetic scattering from a large cavity. SIAM J. Sci. Comput. 27, 553–574 (2005)

    Article  MathSciNet  Google Scholar 

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

  6. Brunner, H.: Open problems in the computational solution of Volterra functional equations with highly oscillatory kernels. HOP: Effective computational methods for highly oscillatory solutions Isaac Newton Institute (2007)

  7. Brunner, H.: On the numerical solution of first-kind Volterra integral equations with highly oscillatory kernels, Isaac newton institute. HOP: Highly oscillatory problems: From theory to applications, 13-17 September (2010)

  8. Chen, R.: Numerical approximations to integrals with a highly oscillatory Bessel kernel. Appl. Numer. Math. 62, 636–648 (2012)

    Article  MathSciNet  Google Scholar 

  9. 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 

  10. Chen, R., An, C.: On the evaluation of infinite integrals involving Bessel functions. Appl. Math. Comput. 235, 212–220 (2014)

    MathSciNet  MATH  Google Scholar 

  11. Chen, R.: Numerical approximations for highly oscillatory Bessel transforms and applications. J. Math. Anal. Appl. 421, 1635–1650 (2015)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  13. Davis, P.J., Rabinowitz, P.: Methods of Numerical Integration, 2nd edn. Academic Press, New York (1984)

    MATH  Google Scholar 

  14. Davis, P.J., Duncan, D.B.: Stability and convergence of collocation schemes for retarded potential integral equations. SIAM J. Numer. Anal. 42, 1167–1188 (2004)

    Article  MathSciNet  Google Scholar 

  15. Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental unctions, vol. I. McGraw-Hill, New York (1955)

    MATH  Google Scholar 

  16. Evans, G.A., Webster, J.R.: A high order progressive method for the evaluation of irregular oscillatory integrals. Appl. Numer. Math. 23, 205–218 (1997)

    Article  MathSciNet  Google Scholar 

  17. Evans, G.A., Chung, K.C.: Some theoretical aspects of generalised quadrature methods. J. Complex. 19, 272–285 (2003)

    Article  MathSciNet  Google Scholar 

  18. Gautschi, W.: Gauss quadrature routines for two classes of logarithmic weight functions. Numer. Algor. 55, 205–218 (2010)

    Article  MathSciNet  Google Scholar 

  19. Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, 7th edn. Academic Press, New York (2007)

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  21. Huybrechs, D., Vandewalle, S.: A sparse discretization for integral equation formulations of high frequency scattering problems. SIAM J. Sci. Comput. 29, 2305–2328 (2007)

    Article  MathSciNet  Google Scholar 

  22. Kang, H., An, C.: Differentiation formulas of some hypergeometric functions with respect to all parameters. Appl. Math. Comput. 258, 454–464 (2015)

    MathSciNet  MATH  Google Scholar 

  23. Kang, H., Ling, C.: Computation of integrals with oscillatory singular factors of algebraic and logarithmic type. J. Comput. Appl. Math. 285, 72–85 (2015)

    Article  MathSciNet  Google Scholar 

  24. Kang, H., Ma, J.: Quadrature rules and asymptotic expansions for two classes of oscillatory Bessel integrals with singularities of algebraic or logarithmic type. Appl. Numer. Math. 118, 277–291 (2017)

    Article  MathSciNet  Google Scholar 

  25. Kang, H., Wang, H.: Asymptotic analysis and numerical methods for oscillatory infinite generalized bessel transforms with an irregular oscillator. J. Sci. Comp., vol. 82, (29) (2020)

  26. Kurtoǧlu, D.K., Hasçelik, A.I., Milovanović, G.V.: A method for efficient computation of integrals with oscillatory and singular integrand. Numer. Algor. https://doi.org/10.1007/s11075-019-00859-8 (2019)

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

    Article  MathSciNet  Google Scholar 

  28. Levin, D.: Analysis of a collocation method for integrating rapidly oscillatory functions. J. Comput. Appl. Math. 78, 131–138 (1997)

    Article  MathSciNet  Google Scholar 

  29. Luke, Y.L.: The Special Functions and Their Approximations, vol. 2. Academic Press, New York (1969)

    MATH  Google Scholar 

  30. Mastroianni, G., Milovanović, G.V.: Interpolation Processes: Basic Theory and Applications. Springer-Verlag, Berlin-Heidelberg-New York (2008)

    Book  Google Scholar 

  31. Milovanović, G.V.: Numerical calculation of integrals involving oscillatory and singular kernels and some applications of quadratures. Comput. Math. Appl. 36, 19–39 (1998)

    Article  MathSciNet  Google Scholar 

  32. 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 

  33. Olver, S.: Numerical approximation of vector-valued highly oscillatory integrals. BIT Numer. Math. 47, 637–655 (2007)

    Article  MathSciNet  Google Scholar 

  34. Piessens, R., Branders, M.: Modified Clenshaw-Curtis method for the computation of Bessel function integrals. BIT 23, 370–381 (1983)

    Article  MathSciNet  Google Scholar 

  35. Slater, L.J.: Generalized hypergeometric functions. University of Cambridge, Cambridge(UK) (1966)

    MATH  Google Scholar 

  36. Wang, H.: A unified framework for asymptotic analysis and computation of finite Hankel transform. J. Math. Anal. Appl., vol. 483 (123640) (2019)

  37. Xiang, S., Wang, H.: Fast integration of highly oscillatory integrals with exotic oscillators. Math. Comp. 79, 829–844 (2010)

    Article  MathSciNet  Google Scholar 

  38. Xiang, S., Cho, Y., Wang, H., Brunner, H.: Clenshaw-Curtis-Filon-type methods for highly oscillatory bessel transforms and applications. IMA J. Numer. Anal. 31(4), 1281–1314 (2011)

    Article  MathSciNet  Google Scholar 

  39. Xu, Z., Xiang, S.: Computation of highly oscillatory Bessel transforms with algebraic singularities. arXiv:1605.08568 (2016)

  40. Xu, Z., Xiang, S.: Numerical evaluation of a class of highly oscillatory integrals involving Airy functions. Appl. Math. Comput. 246, 54–63 (2014)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  42. Xu, Z., Xiang, S.: On the evaluation of highly oscillatory finite Hankel transform using special functions. Numer. Algor. 72, 37–56 (2016)

    Article  MathSciNet  Google Scholar 

  43. 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 

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Acknowledgments

The authors would like to express their most sincere thanks to the referees and editors for their very helpful comments and suggestions, which greatly improved the quality of this paper.

Funding

This research was supported by Zhejiang Provincial Natural Science Foundation of China under Grant No. LY18A010009, National Natural Science Foundation of China (Grant Nos. 11301125, 11971138), Research Foundation of Hangzhou Dianzi University (Grant No. KYS075613017)

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Authors and Affiliations

  1. Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou, Zhejiang, 310018, People’s Republic of China

    Hongchao Kang, Chunzhi Xiang & Hong Wang

  2. College of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou, Henan, 450002, People’s Republic of China

    Zhenhua Xu

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  1. Hongchao Kang
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  2. Chunzhi Xiang
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  4. Hong Wang
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Correspondence to Hongchao Kang.

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Kang, H., Xiang, C., Xu, Z. et al. Efficient quadrature rules for the singularly oscillatory Bessel transforms and their error analysis. Numer Algor 88, 1493–1521 (2021). https://doi.org/10.1007/s11075-021-01083-z

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