Abstract
A mixed-type formulation composed of Gauss-Laguerre quadrature and meshless collocation is presented for approximation of oscillatory integrals containing Hankel function of the first kind. An adaptive splitting procedure is implemented to resolve singularity of the transformed integral at x = 0. The singularity ridden integral is computed by the multi-resolution quadrature. Bessel function of the second kind is transformed into a Bessel function of the first kind before being approximated by the meshless collocation method (Zaman and Siraj-ul-Islam, J. Comput. Appl. Math. 315, 161–174, 2017). Theoretical error bounds of the proposed methods are established. Numerical results obtained from the benchmark problems are presented in the tabular and graphical forms for verification of theoretical error bounds and accuracy of the methods.
Similar content being viewed by others
References
Arfken, G.B., Weber, H.J., Harris, F.E.: Mathematical methods for physicists: a comprehensive guide. Academic press (2011)
Aziz, I., Siraj-ul-Islam, Khan, W.: Quadrature rules for numerical integration based on Haar wavelets and hybrid functions. Comput. Math. Appl. 61, 2770–2781 (2011)
Bao, G., Sun, W.: A fast algorithm for the electromagnetic scattering from a large cavity. SIAM J. Sci. Comput. 27(2), 553–574 (2005)
Brunet, M., Morestin, F., Walter-Leberre, H.: Failure analysis of anisotropic sheet-metals using a non-local plastic damage model. J. Mater. Proc. Tech. 170(1), 457–470 (2005)
Chandler-Wilde, S.N., Langdon, S.: A Galerkin boundary element method for high frequency scattering by convex polygons. SIAM J. Numer. Anal. 45(2), 610–640 (2007)
Chen, R.: Numerical approximations to integrals with a highly oscillatory Bessel kernel. Appl. Numer. Math. 62(5), 636–648 (2012)
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)
Chen, R.: Numerical approximations for highly oscillatory Bessel transforms and applications. J. Math. Anal. Appl. 421(2), 1635–1650 (2015)
Elliott, D., Johnston, P.R.: Gauss-Legendre quadrature for the evaluation of integrals involving the Hankel function. J. Comput. Appl. Math. 211(1), 23–35 (2008)
Evans, G.A., Chung, K.C.: Some theoretical aspects of generalized quadrature methods. J. Complexity 19(3), 272–285 (2003)
Ferreira, A.J.M., Fasshauer, G.E., Batra, R.C., Rodrigues, J.D.: Static deformations and vibration analysis of composite and sandwich plates using a layerwise theory and RBF-PS discretizations with optimal shape parameter. Compos. Struct. 86, 328–343 (2008)
Gradshteyn, I.S., Ryzhik, I.M.: : Table of integrals, series, and products, 7th edn. Academic Press, New York (2007)
Huybrechs, D., Vandewalle, S.: A sparse discretization for integral equation formulations of high frequency scattering problems. SIAM J. Sci. Comput. 29(6), 2305–2328 (2007)
Ioakimidis, N.I.: Application of the gauss-Laguerre and radau-Laguerre quadrature rules to the numerical solution of Cauchy type singular integral equations. Comput. Struct. 14(1), 63–70 (1981)
Jentschura, U.D., Lötstedt, E.: Numerical calculation of Bessel, Hankel and Airy functions. Comput. Phys. Commun. 183(3), 506–519 (2012)
Kzaz, M.: Convergence acceleration of the gauss-Laguerre quadrature formula. Appl. Num. Math. 29, 201–220 (1999)
Levin, D.: Fast integration of rapidly oscillatory functions. J. Comput. Appl. Math. 67(1), 95–101 (1996)
Levin, D.: Analysis of a collocation method for integrating rapidly oscillatory functions. J. Comput. Appl. Math. 78(1), 131–138 (1997)
Lu, T., Yevick, D.: A vectorial boundary element method analysis of integrated optical waveguides. J. Lightw. Tech. 21(8), 1793 (2003)
Piessens, R., Branders, M.: Modified Clenshaw-Curtis method for the computation of Bessel function integrals. BIT Numer. Math. 23(3), 370–381 (1983)
Ramesh, P.S., Lean, M.H.: A boundary integral formulation for natural convection flows. I. J. Numer. Method Biomed. Eng. 8(6), 407–415 (1992)
Rippa, S.: An algorithm for selecting a good value for the parameter c in radial basis function interpolation. Adv. Comp. Math. 11, 193–210 (1999)
Schippers, H.: Boundary-integral-equation methods for screen problems in acoustic and electromagnetic aerospace research. J. Eng. Math. 34(1-2), 143–162 (1998)
Siraj-ul-Islam, Al-Fhaid, A.S., Zaman, S.: Meshless and wavelets based complex quadrature of highly oscillatory integrals and the integrals with stationary points. Eng. Anal. Bound. Elem. 37, 1136–1144 (2013)
Siraj-ul-Islam, Aziz, I., Haq, F.: A comparative study of numerical integration based on Haar wavelets and hybrid functions. Comput. Math. Appl. 59(6), 2026–2036 (2010)
Siraj-ul-Islam, Zaman, S.: New quadrature rules for highly oscillatory integrals with stationary points. J. Comput. Appl. Math. 278, 75–89 (2015)
Watson, G.N.: A treatise on the theory of Bessel functions. Cambridge University Press (1995)
Xiang, S.: Numerical analysis of a fast integration method for highly oscillatory functions. BIT Numer. Math. 47(2), 469–482 (2007)
Xiang, S., Cho, Y.J., Wang, H., Brunner, H.: Clenshaw–Curtis-Filon-type methods for highly oscillatory Bessel transforms and applications. IMA J. Numer. Analy. 31(4), 1281–1314 (2011)
Xiang, S., Gui, W.: On generalized quadrature rules for fast oscillatory integrals. Appl. Math. Comput. 197(1), 60–75 (2008)
Xiang, S., Wang, H.: Fast integration of highly oscillatory integrals with exotic oscillators. Math Comput. 79(270), 829–844 (2010)
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)
Xu, Z., Milovanović, G.V., Xiang, S.: Efficient computation of highly oscillatory integrals with Hankel kernel. Appl. Math. Comput. 261, 312–322 (2015)
Xu, Z., Xiang, S.: On the evaluation of highly oscillatory finite Hankel transform using special functions. Numer. Algorithms 72(1), 37–56 (2015)
Zaman, S., Siraj-ul-Islam: Efficient numerical methods for Bessel type of oscillatory integrals. J. Comput. Appl. Math. 315, 161–174 (2017)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Zaman, S., Siraj-ul-Islam On numerical evaluation of integrals involving oscillatory Bessel and Hankel functions. Numer Algor 82, 1325–1343 (2019). https://doi.org/10.1007/s11075-019-00657-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-019-00657-2