An Automatic quadrature Scheme for Bounded and Unbounded Weighted Hypersingular Integrals on the Variable Interval
DOI:
https://doi.org/10.66901/g953et43Keywords:
Automatic quadrature formula, Chebyshev polynomilas, Smoothness, Accuracy.Abstract
In this note, bounded and unbounded Hadamard type singular integrals or hypersingular integrals (HSIs) are considered. An automatic quadrature schemes (AQS) for each of the HSIs (boinded and unbounded), are constructed by approximating density function via the truncated series of first and second kind of Chebyshev polynomials respectively and implimenting techniques of Hasegava and Torri [24]. For the case of fixed interval, standard Gauss-Chebyshev quadratue formula are also constructed. Numerical results revealed that the proposed methods are highly accurate for the case where are polynomials and rational functions.
References
[1] Bogolubov N.N., Mesheryakov V.A., Tavkhelidze A.N., An application of Muskhelishvili’s methods in the theory of elementary particles. Proceedings of Symposium on Solid Mechanics and Related Problems of Analysis, Tbilisi, Vol. 1: 5–11 (1971) (in Russian).
[2] Brown D.D., Jackson A.D., The Nucleon–Nucleon Interaction, North-Holland, Amsterdam.
[3] Ioakimidis N.I., Two methods for the numerical solution of Bueckher’s SIEs for plane elasticity crack problems, Comput. Methods Appl. Mech. Eng. 2: 169–177, (1982).
[4] Lifanov I.K., Singular Integral Equations and Discrete Vortices, VSP, Utrecht, The Netherlands, (1996).
[5] Lifanov I.K., Poltavskii L.N., Vainikko G.M., Hypersingular Integral Equations and Their Applications, Chapman Hall/CRC, CRC Press Company, London, New York. (2004).
[6] Boykov I.V., Ventseland E.S., Boykova A.I., Accuracy optimal methods for evaluating hypersingular integrals, J. Appl. Numer. Maths., 59, (2009), 1366–1385.
[7] Zhang X., Wu J., Yu D., Super-convergence of the composite Simpson’s rule for a certain finite-part integral and its applications, J. Appl. Numer. Maths., 223, (2009), 598–613.
[8] Hui C.Y., Shia D., Evaluation of hypersingular integrals using Gaussian Quadrature, Int. J. Numer. Meth. Engineering, 44, (1999), 205–214.
[9] Kutt H.R., Quadrature formulae for finite part integrals. Report WISK 178, The National Research Institute for Mathematical Sciences, Pretoria. (1975).
[10] Chen Y.Z.,. A numerical solution technique of hypersingular integral equation for curved cracks, Eng. Frac. Mech., 46(2), (1993), 215–283.
[11] Nik Long N.M.A., Eskhuvatov Z.K., (2009). Hypersingular integral equation for multiple curved cracks problem in plane elasticity, Int. J. Solids Struct. 46: 2611–2617.
[12] Chan Y.Sha., Fannjiang A.C., Paulino G.H., Feng B.F., Finite Part Integrals and Hypersingular Kernels, Advances in Dynamical Systems, 14 (2007), 264–269.
[13] Chan Y.Sha, Fannjiang A.C., Paulino G.H. Integral equations with hypersingular kernels––theory and applications to fracture mechanics, International Journal of Engineering Science, 41 (2003), 683–720.
[14] Kim Ph.. A Chebyshev Quadrature Rule for One Sided Finite Parts Integrals, JAT, 111, (2001), 196–219.
[15] Obays S.J., Eshkuvatov Z.K., Nik Long N.M.A., On Error Estimation of Automatic Quadrature Scheme for the Evaluation of Hadamard Integral of Second Order Singularity, U.P.B. Sci. Bull., Series A, 75(1), (2013), 85-98.
[16] Bichi S.L., Eshkuvatov Z.K., Nik Long N.M.A., (2013). A Note on Evaluating Weighted Hadamard Integral of Second Order Singularity Using Airfoil Polynomials, Uzbek Mat. Journal, 4: 94–104.
[17] Eskhuvatov Z.K., Ahmedov A., Nik Long N.M.A., Amalina N.J., (2011). Approximating Cauchy-type singular integral by an automatic quadrature scheme. Journal of Computational and Applied Mathematics, 235 (2011): 4675–4686.
[18] Shadimetov Kh.M., Akhmedov D.M. An optimal approximate solution of the Ist kind Fredholm singular integral equations. Filomat Vol. 38:30, (2024), 10765–10796.
[19] Akhmedov D.M., Pirnapasov S.O. Optimal approximation techniques for solving Hadamard-type integral equations, Uzbek Mathematical Journal, Vol. 69, Issue 4, (2025), pp.15-21.
[20] Eshkuvatov Z.K., Mamatova H.X., Ismail Sh., Dereman F., Modified Homotopy Perturbation Method for Solving Singular Integral Equations of the First Kind. Journal of Computational and Applied Mathematics, Vol. 471, 1 January 2026, 116753.
[21] Eshkuvatov Z.K., Mamatova H.X., Altynbekov Sh.Y., Kuyliev Sh, Nurillaev M, Highly Accurate Numerical Method for First-Kind Singular Integral Equations, MDPI: Mathematics, 14, (2026), 3405.
[22] Mason J.C. & Handscomb D.C.,. Chebyshev polynomials, CRC Press LLC. (2003)
[23] Israilov M.I., Semi-Analytic Solution of Singular Integral Equations of First Kind by Quadrature Formula, Inst. Maths. under National University of Uzbekistan (in Russian). 21, (1990).
[24] Hasegawa T., Torii T.,. An automatic quadrature for Cauchy principal value integrals, Math. Comp. 56, (1991), 741–754.
