Journal of applied mathematics & informatics

한국전산응용수학회 (The Korean Society for Computational and Applied Mathematics)
권호 표지
DOI QR코드

DOI QR Code

AN EFFICIENT ALGORITHM FOR EVALUATION OF OSCILLATORY INTEGRALS HAVING CAUCHY AND JACOBI TYPE SINGULARITY KERNELS

  • KAYIJUKA, IDRISSA (Department of Applied Statistics, University of Rwanda) ;
  • EGE, SERIFE M. (Department of Mathematics, Ege University) ;
  • KONURALP, ALI (Department of Mathematics, Manisa Celal Bayar University) ;
  • TOPAL, FATMA S. (Department of Mathematics, Ege University)
  • 투고 : 2020.08.22
  • 심사 : 2021.11.22
  • 발행 : 2022.01.30
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

Herein, an algorithm for efficient evaluation of oscillatory Fourier-integrals with Jacobi-Cauchy type singularities is suggested. This method is based on the use of the traditional Clenshaw-Curtis (CC) algorithms in which the given function is approximated by the truncated Chebyshev series, term by term, and the oscillatory factor is approximated by using Bessel function of the first kind. Subsequently, the modified moments are computed efficiently using the numerical steepest descent method or special functions. Furthermore, Algorithm and programming code in MATHEMATICA® 9.0 are provided for the implementation of the method for automatic computation on a computer. Finally, selected numerical examples are given in support of our theoretical analysis.

키워드

참고문헌

  1. R. Piessens and M. Branders, The evaluation and application of some modified moments, BIT, 13 (1973) 443-450. https://doi.org/10.1007/BF01933408
  2. H. Kang, S. Xiang, and G. He, Computation of integrals with oscillatory and singular integrands using Chebyshev expansions, J. Comput. Appl. Math. 242 (2013), 141-156. https://doi.org/10.1016/j.cam.2012.10.016
  3. H. Kang and X. Shao, Fast computation of singular oscillatory fourier transforms, Abstr. Appl. Anal. 1 (2014), 1-8. https://doi.org/10.1155/S1085337596000012
  4. R. Chen, Fast computation of a class of highly oscillatory integrals, Appl. Math. Comput. 227 (2014), 494-501. https://doi.org/10.1016/j.amc.2013.11.068
  5. I. Kayijuka, J.P. Ndabakuranye, and A.I. Hascelik, Computational efficiency of singular and oscillatory integrals with algebraic singularities, Asian J. Math. Comput. Res. 25 (2018), 285-302.
  6. M.R. Capobianco and G. Criscuolo, On quadrature for Cauchy principal value integrals of oscillatory functions, 156 (2003), 471-486. https://doi.org/10.1016/S0377-0427(03)00388-1
  7. H. Wang and S. Xiang, Uniform approximations to Cauchy principal value integrals of oscillatory functions, Appl. Math. Comput. 215 (2009), 1886-1894. https://doi.org/10.1016/j.amc.2009.07.041
  8. P.K. Kythe and M.R. Schaferkotter, Handbook of computational method for integration, Chapman & Hall/CRC, ISBN-13: 978-1584884286, New York, 2004.
  9. A. Erdelyi, Asymptotic representations of Fourier integrals and the method of stationary phase, J. Soc. Indust. Appl. Math. 3 (1955), 17-27. https://doi.org/10.1137/0103002
  10. S. Lu, A Class of Oscillatory Singular Integrals Boundedness, Int. J. Appl. Math. Sci. 2 (2005), 47-64.
  11. W. Sun and N.G. Zamani, Adaptive mesh redistribution for the boundary element in elastostatics, Comput. Struct. 36 (1990), 1081-1088. https://doi.org/10.1016/0045-7949(90)90215-N
  12. G.B. Arfken, Mathematical Methods For Physicists, Sixth Edit., Elsevier Inc., 2005.
  13. M.A. Hamed and B. Cummins, A numerical integration formula for the solution of the singular integral equation for classical crack problems in plane and antiplane elasticity, J. King Saud Un. 3 (1991) 217-230.
  14. H. Brunner, Open problems in the computational solution of Volterra functional equations with highly oscillatory kernels, Isaac Newt. Institute, HOP 2007, 1-14.
  15. H. Brunner, On the numerical solution of first-kind Volterra integral equations with highly oscillatory kernels, Isaac Newton Institute, HOP 2010.
  16. L.N.G. Filon, On a Quadrature Formula for Trigonometric Integrals, Proc. R. Soc. Edinburgh, 1930.
  17. V. Dominguez, Filon-Clenshaw-Curtis rules for a class of highly-oscillatory integrals with logarithmic singularities, J. Comput. Appl. Math. 261 (2014), 299-319. https://doi.org/10.1016/j.cam.2013.11.012
  18. S. Olver, Moment-free numerical approximation of highly oscillatory integrals with stationary points, Eur. J. Appl. Math. 18 (2007), 435-447. https://doi.org/10.1017/S0956792507007012
  19. J.C. Mason and E. Venturino, Integration Methods of Clenshaw-Curtis Type, Based on Four Kinds of Chebyshev Polynomials, Multivar. Approx. Splines, Birkhause, Bessel, 1996.
  20. J.W. Cooley and J.W. Tukey, An Algorithm for the Machine Calculation of Complex Fourier Series, Math. Comput. 19 (1965), 297-301. https://doi.org/10.1090/S0025-5718-1965-0178586-1
  21. C.W. Clenshaw and A.R. Curtis, A method for numerical integration on an automatic computer, Numer. Math. 2 (1960), 197-205. https://doi.org/10.1007/BF01386223
  22. L.N. Trefethen, Is Gauss quadrature better than Clenshaw-Curtis, SIAM Rev. 50 (2008), 67-87. https://doi.org/10.1137/060659831
  23. F.G. RICOMI, On the finite Hilbert transformation, Quarterly J. Math. 2 (1951), 199-211. https://doi.org/10.1093/qmath/2.1.199
  24. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Washington DC, 1964.
  25. F.B. Hildebrand, Introduction to Numerical Analysis, 2nd Ed., Dover Publication Inc., New york, 2003.
  26. D. Huybrechs and S. Vandewalle, On the evaluation of highly oscillatory integrals by analytic continuation, SIAM J. Numer. Anal. 44 (2006), 1026-1048. https://doi.org/10.1137/050636814
  27. J.C. Mason and D.C. Handscomb, Chebyshev Polynomials, A CRC Press Company, 2003.
  28. Idrissa Kayijuka, Serife Muge Ege, Ali Konuralp and Fatma Serap Topal, Clenshaw-Curtis algorithms for an efficient numerical approximation of singular and highly oscillatory Fourier transform integrals, J. Comput. App. Math. 385 (2021), no:113201.
  29. Idrissa Kayijuka, Suliman Alfaqeih and Turgut Ozis, Application of the Cauchy integral approach to Singular and Highly Oscillatory Integrals, Int. J. Comp. Math. 98 (2021), 2097-2114. https://doi.org/10.1080/00207160.2021.1876228
  30. P.J. Davis and P. Rabinowitz, Methods of Numerical Integration, second ed., Academic Press, Orland, FL, 1984.
  31. H. Takemitsu and T. Tatsuo, An automatic quadrature for Cauchy Principal Value integrals, Math. Comp. 56 (1994), 741-754. https://doi.org/10.1090/S0025-5718-1991-1068816-1
  32. G. Criscuolo, A new algorithm for Cauchy principal value and Hadmard finit-part integrals, J. Comput. Appl. Math. 78 (1997), 255-275. https://doi.org/10.1016/S0377-0427(96)00142-2