×

Numerical study on integrals involving the product of Bessel functions and a trigonometric function arising in hydrodynamic problems. (English) Zbl 1458.65022

Summary: This paper explores numerical techniques for evaluating a class of oscillatory infinite integrals of the form \[ \int_0^\infty f(k)J_m(k)J_n(hk)\sin(t\sqrt{k})\operatorname{d}k. \] These oscillatory integrals arise from the transient super Green function associated with cylindrical surfaces in a multi-domain method for three-dimensional hydrodynamic problems in the time domain. The original integral is decomposed into two sub-integrals which are studied in the real \(k\)-axis and the complex plane, respectively. The second sub-integral is reformulated by contour integrals whose integrands are exponentially decreasing and well suited for numerical evaluation. The techniques proposed in this paper are shown to be very efficient and accurate by comparing with results from Mathematica.

MSC:

65D30 Numerical integration
65D20 Computation of special functions and constants, construction of tables
65D32 Numerical quadrature and cubature formulas

Software:

Algorithm 644
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Huybrechs, D.; Vandewalle, S., On the evaluation of highly oscillatory integrals by analytic continuation, SIAM J. Numer. Anal., 44, 3, 1026-1048 (2006) · Zbl 1123.65017
[2] Xiang, S.; Ma, J., Efficient methods for the computation of Pollaczek integrals in the magnetic field, Int. J. Appl. Electromagn. Mech., 41, 227-236 (2013)
[3] Puoskari, M., A method for computing Bessel function integrals, J. Comput. Phys., 75, 334-344 (1988) · Zbl 0637.65014
[4] Sidi, A., A user-friendly extrapolation method for oscillatory infinite integrals, Math. Comp., 51, 183, 249-266 (1988) · Zbl 0694.40004
[5] Lucas, S. K.; Stone, H. A., Evaluating infinite integrals involving Bessel functions of arbitrary order, J. Comput. Appl. Math., 64, 3, 217-231 (1995) · Zbl 0857.65025
[6] Lucas, S. K., Evaluating infinite integrals involving products of Bessel functions of arbitrary order, J. Comput. Appl. Math., 64, 3, 269-282 (1995) · Zbl 0853.65028
[7] Zaman, S.; Siraj-ul-Islam, H. A., Efficient numerical methods for Bessel type of oscillatory integrals, J. Comput. Appl. Math., 315, 161-174 (2017) · Zbl 1421.65009
[8] Ceballos, M. A., Numerical evaluation of integrals involving the product of two Bessel functions and a rational fraction arising in some elastodynamic problems, J. Comput. Appl. Math., 313, 355-382 (2017) · Zbl 1388.74117
[9] Newman, J. N., The approximation of free-surface Green functions, (Martin, P. A.; Wickham, G. R., Wave Asymptotics (1992), Cambridge Univ. Press: Cambridge Univ. Press Cambridge, UK), 107-135, Ch. 4 · Zbl 0803.76015
[10] Bingham, H. B., A note on the relative efficiency of methods for computing the transient free-surface Green function, Ocean Eng., 120, 15-20 (2016)
[11] Chen, X. B.; Liang, H., Wavy properties and analytical modeling of free-surface flows in the development of the multi-domain method, J. Hydrodyn., 28, 6, 971-976 (2016)
[12] X.B. Chen, H. Liang, R.P. Li, X.Y. Feng, Ship seakeeping hydrodynamics by multi-domain method, in: Proceedings of 32nd Symposium on Naval Hydrodynamics, Hamburg, Germany, 2018.
[13] X.B. Chen, F. Noblesse, Super Green functions, in: Proceedings of the 22nd Symposium on Naval Hydrodynamimcs, Washington D.C., 1998, pp. 860-874.
[14] Li, R. P.; Chen, X. B.; Duan, W. Y., Transient wave diffraction around cylinders by a novel boundary element method based on Fourier-Laguerre expansions, Ships Offshore Struct. (2020)
[15] Amos, D. E., A remark on algorithm 644: “A portable package for Bessel functions of a complex argument and nonnegative order”, ACM Trans. Math. Software, 21, 4, 388-393 (1995) · Zbl 0888.65016
[16] Chen, X. B.; Li, R. P., Reformulation of wavenumber integrals describing transient waves, J. Eng. Math., 115, 121-140 (2019) · Zbl 1441.76025
[17] Kahaner, D.; Tietjen, G.; Beckman, R., Gaussian-quadrature formulas for \(\int_0^\infty \operatorname{e}^{- x^2} g ( x ) \operatorname{d} x\), J. Stat. Comput. Simul., 15, 155-160 (1982) · Zbl 0489.65006
[18] Levin, D., Procedures for computing one- and two-dimensional integrals of functions with rapid irregular oscillations, Math. Comp., 38, 158, 531-538 (1982) · Zbl 0482.65013
[19] Levin, D., Fast integration of rapidly oscillatory functions, J. Comput. Appl. Math., 67, 95-101 (1996) · Zbl 0858.65017
[20] Levin, D., Analysis of a collocation method for integrating rapidly oscillatory functions, J. Comput. Appl. Math., 78, 131-138 (1997) · Zbl 0870.65019
[21] Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables, no. 55 (1964), National Bureau of Standards · Zbl 0171.38503
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.