Numerical method for hypersingular integrals of highly oscillatory functions on the positive semiaxis. (English) Zbl 1540.65089
Summary: This paper deals with a quadrature rule for the numerical evaluation of hypersingular integrals of highly oscillatory functions on the positive semiaxis. The rule is of product type and consists in approximating the density function \(f\) by a truncated interpolation process based on the zeros of generalized Laguerre polynomials and an additional point. We prove the stability and the convergence of the rule, giving error estimates for functions belonging to weighted Sobolev spaces equipped with uniform norm. We also show how the proposed rule can be used for the numerical solution of hypersingular integral equations. Numerical tests which confirm the theoretical estimates and comparisons with other existing quadrature rules are presented.
MSC:
| 65D32 | Numerical quadrature and cubature formulas |
| 41A55 | Approximate quadratures |
| 65D30 | Numerical integration |
| 65R20 | Numerical methods for integral equations |
References:
| [1] | G. Arfken. Mathematical methods for physicists. edn. Academic Press, Orlando, 3rd, 1985. · Zbl 0135.42304 |
| [2] | G. Bao and W. Sun. A fast algorithm for the electromagnetic scattering form a large cavity. SIAM J. Sci. Comput. 27:553-574, 2005. · Zbl 1089.78024 |
| [3] | M.R. Capobianco and G. Criscuolo. On quadrature for Cauchy principal value integrals of oscillatory functions. J. Comput. Appl. Math., 156: 471-486, 2003. · Zbl 1043.65048 |
| [4] | M.R. Capobianco and G. Mastroianni and M.G. Russo, Maria Grazia. Pointwise and uniform approximation of the finite Hilbert transform. Approximation and optimization (Cluj-Napoca 1996)., vol. I: 45-66, Transilvania, Cluj-Napoca, 1997. · Zbl 0883.65106 |
| [5] | R. Chen and D. Yu and J. Chen. Numerical approximations of highly oscillatory Hilbert transforms. Comp. Appl. Math., 39, 2020. · Zbl 1449.65040 |
| [6] | K. Chung and G.A. Evans and J.R. Webster. A method to generate generalized quadrature rules for oscillatory integrals. Appl. Numer. Math., 34: 85-93 2000. · Zbl 0954.65019 |
| [7] | G. Criscuolo. Numerical evaluation of certain strongly singular integrals. IMA J. Numer. Anal., 34,2: 651-674, 2014. · Zbl 1292.65025 |
| [8] | G. Criscuolo and G. Mastroianni. Convergenza di formule Gaussiane per il calcolo delle derivate di integrali a valor principale secondo Cauchy. Calcolo, 24, 2: 179-192, 1987. · Zbl 0643.41018 |
| [9] | P.J. Davis and P. Rabinowitz. Methods of numerical integration, 2nd edn. Academic Press, London, 1984. · Zbl 0537.65020 |
| [10] | M.C. De Bonis and B. Della Vecchia and G. Mastroianni. Approximation of the Hilbert Transform on the real semiaxis using Laguerre zeros. J. Comput. Appl. Math., 140, 1-2: 209-229, 2002. · Zbl 0998.65129 |
| [11] | M.C. De Bonis and M.C. G. Mastroianni. Numerical Treatment of a class of systems of Fredholm integral equations on the real line. Math. Comp., 83, 286: 771-788, 2014. · Zbl 1285.65087 |
| [12] | M.C. De Bonis and D. Occorsio. On the simultaneous approximation of a Hilbert transform and its derivatives on the real semiaxis. Applied Numerical Mathematics, 114: 132-153, 2017. · Zbl 1357.65309 |
| [13] | M.C. De Bonis and D. Occorsio. Approximation of Hilbert and Hadamard transforms on (0, +∞). Applied Numerical Mathematics, 116: 184-194, 2017. · Zbl 1372.65333 |
| [14] | M.C. De Bonis and D. Occorsio. Numerical methods for hypersingular integrals on the real line, Dolomit. Research Notes, 10: 97-157, 2017. · Zbl 1372.65067 |
| [15] | M.C. De Bonis and D. Occorsio. Numerical computation of hypersingular integrals on the real semiaxis. Applied Mathematics and Computation, 313:367-383, 2017. · Zbl 1426.65033 |
| [16] | M.C. De Bonis and D. Occorsio. Error bounds for a Gauss-type quadrature rule to evaluate hypersingular integrals. Filomat, 32(7): 2525-2543, 2018. · Zbl 1499.65085 |
| [17] | M.C. De Bonis and D. Occorsio. A product integration rule for hypersingular integrals on (0, +∞). Electronic Transactions on Numerical Analysis, 50: 129-143, 2018. · Zbl 1415.65063 |
| [18] | M.C. De Bonis and P. Pastore. A quadrature rule for integrals of highly oscillatory functions. Rendiconti del circolo matematico di Palermo, II. 82: 1-25, 2010. |
| [19] | V. Dominguéz and I.G. Graham and V.P. Smyshlyaev. Stability and error estimates for Filon-Clenshaw-Curtis rules for highly oscillatory integrals. IMA J. Numer. Anal., 31: 1253-1289, 2011. · Zbl 1231.65060 |
| [20] | M. Diligenti and G. Monegato. Finite-part integrals: their occurrence and computation. Rend. Circolo Mat. di Palermo, 33: 39-61, 1993. · Zbl 0813.65051 |
| [21] | T. Diogo and P. Lima and D. Occorsio. A numerical method for finite-part integrals. Dolomites Research Notes on Approximation, 13: 1-11, 2020. · Zbl 1540.65092 |
| [22] | F. Filbir and D. Occorsio and W. Themistoclakis Approximation of Finite Hilbert and Hadamard transforms by using equally spaced nodes. Mathematics, 8, 4: doi 10.3390/math804054, 542 , 2020. · doi:10.3390/math804054 |
| [23] | I.S. Gradshteyn and I.M. Ryzhik. Table of Integrals, Series, and Products. Seventh edition, 2007. · Zbl 1208.65001 |
| [24] | D. Huybrechs and S. Vandewalle. On the evaluation of highly oscillatory integrals by analytic continuation. SIAM J. Numer. Anal.,44: 1026-1048, 2006. · Zbl 1123.65017 |
| [25] | N.I. Ioakimidis. On the uniform convergence of Gaussian quadrature rules for Cauchy principal value integrals and their derivatives. Math. Comp. 44: 191-198, 1985. · Zbl 0562.65088 |
| [26] | A. Iserles and S.P. Nørsett. Efficient quadrature of highly-oscillatory integrals using derivatives. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461:1383-1399, 2005. · Zbl 1145.65309 |
| [27] | D. Levin. Fast integration of rapidly oscillatory functions. J. Comput. Appl.Math., 67: 95-101, 1996. · Zbl 0858.65017 |
| [28] | G. Mastroianni and G. Monegato Truncated quadrature rules over (0, ∞) and Nyström type methods. SIAM Jour. Num. Anal. 41, 5 : 1870-1892, 2003. · Zbl 1056.65022 |
| [29] | G. Mastroianni and G. Monegato Some new applications of truncated Gauss-Laguerre quadrature formulas. Numer. Algor. 49: 283-297, 2008. · Zbl 1181.65044 |
| [30] | G. Mastroianni and G.V. Milovanović. Some numerical methods for second-kind Fredholm integral equations on the real semiaxis. IMA J. Numer. Anal., 29(4): 1046-1066, 2009. · Zbl 1180.65177 |
| [31] | G. Mastroianni and G.V. Milovanović. Interpolation Processes Basic Theory and Applications. Springer-Verlag, Springer Monographs in Mathematics, Berlin, Heidelberg, 2009. |
| [32] | G. Mastroianni and D. Occorsio. Lagrange interpolation at Laguerre zeros in some weighted uniform spaces. Acta Math. Hung., 91 (1-2): 27-52, 2001. · Zbl 0980.41002 |
| [33] | G. Mastroianni and D. Occorsio. Some quadrature formulae with non standard weights. J. Comput. Appl. Math., 235, 3: 602-614, 2010. · Zbl 1210.41008 |
| [34] | N. Mastronardi and D. Occorsio. Some numerical algorithms to evaluate Hadamard finite-part integrals. J. Comput. Appl. Math., 70, 1: 75-93, 1996. · Zbl 0858.65018 |
| [35] | G. Monegato. Numerical evaluation of hypersingular integrals. J. Comput. Appl. Math., 50: 9-31, 1994. · Zbl 0818.65016 |
| [36] | G. Monegato. Definitions, properties and applications of finite-part integrals. J. Comput. Appl. Math., 229: 425-439, 2009. · Zbl 1166.65061 |
| [37] | I. Notarangelo. Approximation of the Hilbert Transform on the Real Line Using Freud Weights. In book: Approximation and Computation, Edition: Series Optimization and Its Applications. J. SpringerEditors: W. Gautschi, G.Mastroianni, T. Rassias.: 233-252, 2010.- · Zbl 1215.65189 |
| [38] | D. Occorsio. A method to evaluate the Hilbert transform on (0, +∞). Appl. Math. Comput., 217, 12: 5667-5679, 2011. · Zbl 1209.65139 |
| [39] | D. Occorsio and G. Serafini. Cubature formulae for nearly singular and highly oscillating integrals. Calcolo, 55, 1, doi 10.1007/s10092-018-0243-x, 4: 2018. · Zbl 06859129 · doi:10.1007/s10092-018-0243-x |
| [40] | D. Occorsio and M.G. Russo and W. Themistoclakis. Filtered integration rules for finite weighted Hilbert transforms. J. Comput. and Appl. Math., 114166, doi https://doi.org/10.1016/j.cam.2022.114166: 2022. · Zbl 1492.65060 · doi:10.1016/j.cam.2022.114166:2022 |
| [41] | A. Sidi. Unified compact numerical quadrature formulas for Hadamard finite parts of singular integrals of periodic functions, Calcolo, 58,22, 2021. · Zbl 1472.65031 |
| [42] | H. Wang and S. Xiang. Uniform approximations to Cauchy principal value integrals of oscillatory functions. Appl. Math. Comput., 215: 1886-1894, 2009. · Zbl 1179.65165 |
| [43] | H. Wang and L. Zhang and D. Huybrechs. Asymptotic expansions and fast computation of oscillatory Hilbert transforms. Numer. Math., 123: 709-743, 2013. · Zbl 1263.65131 |
| [44] | S. Xiang. Efficient Filon-type methods for b a f (x)e iωg(x) d x. Numer. Math., 105: 633-658, 2007. · Zbl 1158.65020 |
| [45] | S. Xiang and C. Fang and Z. Xu. On uniform approximations to hypersingular finite-part integrals. J. Math. Anal. Appl., 435: 1210-1228, 2016. · Zbl 1329.41036 |
| [46] | J. Wu and W. Sun. The superconvergence of Newton-Cotes rules for the Hadamard finite-part integral on an interval. Numer. Math., 109, 1: 2008. · Zbl 1147.65026 |
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.
