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Year 2019, Volume: 2 Issue: 2, 61 - 71, 30.12.2019

Abstract

References

  • [1] L.N.G. Filon, On a quadrature formula for trigonometric integrals, in Proceedings of Royal Society, Proc. Roy. Soc. 49 (1928) 38–47. [2] A. L. Hascelik. Suitable Gauss and Filon-type methods for oscillatory integrals with an algebraic singularity. Applied Numerical Mathematics. 2009 Jan 1;59(1):101-18. [3] S. Xiang and H. Wang, Fast integration of highly oscillatory integrals with exotic oscillators, Math. Comp. 79 (2010) 829–844. [4] S. Xiang, Y. Cho, H. Wang and H. Brunner, Clenshaw–Curtis–Filon-type methods for highly oscillatory Bessel transforms and applications, IMA. J. Numer. Anal. 31 (4) (2011) 1281–1314. [5] D. Levin, Procedures for computing one-and two-dimensional integrals of functions with \ rapid irregular oscillations, Math. Comput. 38(158) (1982) 531–538. [6] J. Li, X. Wang and T. Wang, S. Xiao, An improved Levin quadrature method for highly oscillatory integrals, Appl. Numer. Math. 60 (2010) 833–842. [7] S. Olver, Moment-free numerical integration of highly oscillatory functions, IMA. J. Numer. Anal. 26 (2006) 213–227. [8] M. M. Khan and S. Zaman, On computation of highly oscillatory integrals with Bessel kernel, Earthline Journal of Mathematical Sciences, 3(1) (2020) 51-63. [9] S. Xiang and W. Gui, On generalized quadrature rules for fast oscillatory integrals. Applied Mathematics and Computation. 2008 Mar 15;197(1):60-75. [10] G.A. Evans and K.C. Chung, Some theoretical aspects of generalized quadrature methods, J. Complexity 19 (2003) 272–285. [11] M. M. Khan, Variational Iteration Method for the Solution of Differential Equation of Motion of the Mathematical Pendulum and Duffing-Harmonic Oscillator, Earthline Journal of Mathematical Sciences, 2(1) (2019) 101-109. [12] H. Ahmad, “Variational iteration method with an auxiliary parameter for solving differential equations of the fifth order,” Nonlinear Sci. Lett. A, vol. 9, no. 1, pp. 27–35, 2018. [13] M. Rafiq, H. Ahmad, and S. T. Mohyud-Din, “Variational iteration method with an auxiliary parameter for solving Volterra’s population model,” Nonlinear Sci. Lett. A, vol. 8, no. 4, pp. 389–396, 2017. [14] C.W. Clenshaw and A.R. Curtis, A method for numerical integration on an automatic computer, Numer. Math. 2 (1960) 197–205. [15] Sloan IH and Smith WE. Product integration with the Clenshaw-Curtis points: implementation and error estimates. Numerische Mathematik. 1980 Dec 1;34(4):387-401. [16] A. Iserles and S.P. Nørsett, Efficient quadrature of highly oscillatory integrals using derivatives, Proc. Royal Soc. A 461 (2005) 1383–1399. [17] H. Ahmad, “Variational Iteration Algorithm-I with an Auxiliary Parameter for Solving Fokker-Planck Equation,” Earthline J. Math. Sci., pp. 29–37, Apr. 2019. [18] H. Ahmad and T. A. Khan, “Variational iteration algorithm-I with an auxiliary parameter for wave-like vibration equations,” J. Low Freq. Noise Vib. Act. Control, vol. 38, no. 3–4, pp. 1113–1124, Jan. 2019. [19] H. Ahmad and T. A. Khan, “Variational iteration algorithm I with an auxiliary parameter for the solution of differential equations of motion for simple and damped mass–spring systems,” Noise Vib. Worldw., vol. 51, no. 1-2, pp. 12-20, 2020. [20] H. Ahmad, “Variational iteration algorithm-II with an auxiliary parameter and its optimal determination,” Nonlinear Sci. Lett. A, vol. 9, no. 1, pp. 62–72, 2018. [21] H. Ahmad, T. A. Khan, and C. Cesarano, “Numerical Solutions of Coupled Burgers′ Equations,” Axioms, vol. 8, no. 4, p. 119, Oct. 2019. [22] H. Ahmad, A. R. Seadawy, and T. A. Khan, “Numerical solution of Korteweg-de Vries-Burgers equation by the modified variational Iteration algorithm-II arising in shallow water waves,” Phys. Scr., Dec. 2019. DOI: 10.1088/1402-4896/ab6070. [23] H. Wang and S. Xiang, On the evaluation of Cauchy principal value integrals of oscillatory functions, J. Comput. Appl. Math. 234 (2010) 95–100. [24] H. Kang and S. Xiang, G. He, Computation of integrals with oscillatory and singular integrands using Chebyshev expansions. Journal of Computational and Applied Mathematics. 2013 Apr 1;242:141-56. [25] H. Kang and S. Xiang, Efficient quadrature of highly oscillatory integrals with algebraic singularities. Journal of Computational and Applied Mathematics. 2013 Jan 1;237(1):576-88. [26] S. Zaman and Siraj-ul-Islam, Efficient numerical methods for Bessel type of oscillatory integrals. Journal of Computational and Applied Mathematics. 2017 May 1;315:161-74. [27] A. L. Hascelik, Efficient computation of highly oscillatory Fourier-type integrals with monomial phase functions and Jacobi-type singularities. Applied Numerical Mathematics. 2019 Oct 16. [28] A. Al-Fhaid and S. Zaman, Meshless and wavelets based complex quadrature of highly oscillatory integrals and the integrals with stationary points. Engineering Analysis with Boundary Elements. 2013 Sep 1;37(9):1136-44. [29] I Aziz and W. Khan, Quadrature rules for numerical integration based on Haar wavelets and hybrid functions. Computers & Mathematics with Applications. 2011 May 1;61(9):2770- 81. [30] Siraj-ul-Islam and S. Zaman. New quadrature rules for highly oscillatory integrals with stationary points. Journal of Computational and Applied Mathematics. 2015 Apr 15;278:75-89.

Numerical Approximation of Highly Oscillatory Integrals with Weak Singularity

Year 2019, Volume: 2 Issue: 2, 61 - 71, 30.12.2019

Abstract

In this paper, we design an accurate scheme for the approximation of highly oscillatory integrals having singularity . The interval of integration [a,b] is divided into two subintervals and then approximate the integral over first interval by hybrid function quadrature (Q_hf [g]), while for the approximation of integrals over the second interval we use Levin meshless method (Q_L^m [g]). For a result, we find the sum of both the integrals. To check the effectiveness of method results of some test problems are calculated by hybrid function quadrature and compared with the results produces by the proposed method.

References

  • [1] L.N.G. Filon, On a quadrature formula for trigonometric integrals, in Proceedings of Royal Society, Proc. Roy. Soc. 49 (1928) 38–47. [2] A. L. Hascelik. Suitable Gauss and Filon-type methods for oscillatory integrals with an algebraic singularity. Applied Numerical Mathematics. 2009 Jan 1;59(1):101-18. [3] S. Xiang and H. Wang, Fast integration of highly oscillatory integrals with exotic oscillators, Math. Comp. 79 (2010) 829–844. [4] S. Xiang, Y. Cho, H. Wang and H. Brunner, Clenshaw–Curtis–Filon-type methods for highly oscillatory Bessel transforms and applications, IMA. J. Numer. Anal. 31 (4) (2011) 1281–1314. [5] D. Levin, Procedures for computing one-and two-dimensional integrals of functions with \ rapid irregular oscillations, Math. Comput. 38(158) (1982) 531–538. [6] J. Li, X. Wang and T. Wang, S. Xiao, An improved Levin quadrature method for highly oscillatory integrals, Appl. Numer. Math. 60 (2010) 833–842. [7] S. Olver, Moment-free numerical integration of highly oscillatory functions, IMA. J. Numer. Anal. 26 (2006) 213–227. [8] M. M. Khan and S. Zaman, On computation of highly oscillatory integrals with Bessel kernel, Earthline Journal of Mathematical Sciences, 3(1) (2020) 51-63. [9] S. Xiang and W. Gui, On generalized quadrature rules for fast oscillatory integrals. Applied Mathematics and Computation. 2008 Mar 15;197(1):60-75. [10] G.A. Evans and K.C. Chung, Some theoretical aspects of generalized quadrature methods, J. Complexity 19 (2003) 272–285. [11] M. M. Khan, Variational Iteration Method for the Solution of Differential Equation of Motion of the Mathematical Pendulum and Duffing-Harmonic Oscillator, Earthline Journal of Mathematical Sciences, 2(1) (2019) 101-109. [12] H. Ahmad, “Variational iteration method with an auxiliary parameter for solving differential equations of the fifth order,” Nonlinear Sci. Lett. A, vol. 9, no. 1, pp. 27–35, 2018. [13] M. Rafiq, H. Ahmad, and S. T. Mohyud-Din, “Variational iteration method with an auxiliary parameter for solving Volterra’s population model,” Nonlinear Sci. Lett. A, vol. 8, no. 4, pp. 389–396, 2017. [14] C.W. Clenshaw and A.R. Curtis, A method for numerical integration on an automatic computer, Numer. Math. 2 (1960) 197–205. [15] Sloan IH and Smith WE. Product integration with the Clenshaw-Curtis points: implementation and error estimates. Numerische Mathematik. 1980 Dec 1;34(4):387-401. [16] A. Iserles and S.P. Nørsett, Efficient quadrature of highly oscillatory integrals using derivatives, Proc. Royal Soc. A 461 (2005) 1383–1399. [17] H. Ahmad, “Variational Iteration Algorithm-I with an Auxiliary Parameter for Solving Fokker-Planck Equation,” Earthline J. Math. Sci., pp. 29–37, Apr. 2019. [18] H. Ahmad and T. A. Khan, “Variational iteration algorithm-I with an auxiliary parameter for wave-like vibration equations,” J. Low Freq. Noise Vib. Act. Control, vol. 38, no. 3–4, pp. 1113–1124, Jan. 2019. [19] H. Ahmad and T. A. Khan, “Variational iteration algorithm I with an auxiliary parameter for the solution of differential equations of motion for simple and damped mass–spring systems,” Noise Vib. Worldw., vol. 51, no. 1-2, pp. 12-20, 2020. [20] H. Ahmad, “Variational iteration algorithm-II with an auxiliary parameter and its optimal determination,” Nonlinear Sci. Lett. A, vol. 9, no. 1, pp. 62–72, 2018. [21] H. Ahmad, T. A. Khan, and C. Cesarano, “Numerical Solutions of Coupled Burgers′ Equations,” Axioms, vol. 8, no. 4, p. 119, Oct. 2019. [22] H. Ahmad, A. R. Seadawy, and T. A. Khan, “Numerical solution of Korteweg-de Vries-Burgers equation by the modified variational Iteration algorithm-II arising in shallow water waves,” Phys. Scr., Dec. 2019. DOI: 10.1088/1402-4896/ab6070. [23] H. Wang and S. Xiang, On the evaluation of Cauchy principal value integrals of oscillatory functions, J. Comput. Appl. Math. 234 (2010) 95–100. [24] H. Kang and S. Xiang, G. He, Computation of integrals with oscillatory and singular integrands using Chebyshev expansions. Journal of Computational and Applied Mathematics. 2013 Apr 1;242:141-56. [25] H. Kang and S. Xiang, Efficient quadrature of highly oscillatory integrals with algebraic singularities. Journal of Computational and Applied Mathematics. 2013 Jan 1;237(1):576-88. [26] S. Zaman and Siraj-ul-Islam, Efficient numerical methods for Bessel type of oscillatory integrals. Journal of Computational and Applied Mathematics. 2017 May 1;315:161-74. [27] A. L. Hascelik, Efficient computation of highly oscillatory Fourier-type integrals with monomial phase functions and Jacobi-type singularities. Applied Numerical Mathematics. 2019 Oct 16. [28] A. Al-Fhaid and S. Zaman, Meshless and wavelets based complex quadrature of highly oscillatory integrals and the integrals with stationary points. Engineering Analysis with Boundary Elements. 2013 Sep 1;37(9):1136-44. [29] I Aziz and W. Khan, Quadrature rules for numerical integration based on Haar wavelets and hybrid functions. Computers & Mathematics with Applications. 2011 May 1;61(9):2770- 81. [30] Siraj-ul-Islam and S. Zaman. New quadrature rules for highly oscillatory integrals with stationary points. Journal of Computational and Applied Mathematics. 2015 Apr 15;278:75-89.

Details

Primary Language English
Subjects Applied Mathematics
Journal Section Articles
Authors

İrfan ULLAH This is me

Muhammad Munib KHAN

Publication Date December 30, 2019
Acceptance Date February 2, 2020
Published in Issue Year 2019 Volume: 2 Issue: 2

Cite

APA ULLAH, İ., & KHAN, M. M. (2019). Numerical Approximation of Highly Oscillatory Integrals with Weak Singularity. International Journal of Informatics and Applied Mathematics, 2(2), 61-71.
AMA ULLAH İ, KHAN MM. Numerical Approximation of Highly Oscillatory Integrals with Weak Singularity. IJIAM. December 2019;2(2):61-71.
Chicago ULLAH, İrfan, and Muhammad Munib KHAN. “Numerical Approximation of Highly Oscillatory Integrals With Weak Singularity”. International Journal of Informatics and Applied Mathematics 2, no. 2 (December 2019): 61-71.
EndNote ULLAH İ, KHAN MM (December 1, 2019) Numerical Approximation of Highly Oscillatory Integrals with Weak Singularity. International Journal of Informatics and Applied Mathematics 2 2 61–71.
IEEE İ. ULLAH and M. M. KHAN, “Numerical Approximation of Highly Oscillatory Integrals with Weak Singularity”, IJIAM, vol. 2, no. 2, pp. 61–71, 2019.
ISNAD ULLAH, İrfan - KHAN, Muhammad Munib. “Numerical Approximation of Highly Oscillatory Integrals With Weak Singularity”. International Journal of Informatics and Applied Mathematics 2/2 (December 2019), 61-71.
JAMA ULLAH İ, KHAN MM. Numerical Approximation of Highly Oscillatory Integrals with Weak Singularity. IJIAM. 2019;2:61–71.
MLA ULLAH, İrfan and Muhammad Munib KHAN. “Numerical Approximation of Highly Oscillatory Integrals With Weak Singularity”. International Journal of Informatics and Applied Mathematics, vol. 2, no. 2, 2019, pp. 61-71.
Vancouver ULLAH İ, KHAN MM. Numerical Approximation of Highly Oscillatory Integrals with Weak Singularity. IJIAM. 2019;2(2):61-7.

International Journal of Informatics and Applied Mathematics