Application of the Cauchy integral approach to singular and highly oscillatory integrals

References

• M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Washington, D.C., 1964.
   Google Scholar
• M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables, 10 Print, Reprinted by Dover Publications, New York, 1965.
   Google Scholar
• P. Agarwal, S. Jain, S. Agarwal, and M. Nagpal, On a new class of integrals involving Bessel functions of the first kind, Commun. Numer. Anal. 2014 (2014), pp. 1–7.
   Google Scholar
• G.B. Arfken and H. Weber, Mathematical Methods for Physicists, 6th ed., Elsevier Inc., California, 2005.
   Google Scholar
• J.W. Brown and R.V. Churchill, Complex Variables and Applications, 8th ed., McGraw-Hill, New York, 2009.
   Google Scholar
• H. Brunner, Open problems in the computational solution of Volterra functional equations with highly oscillatory kernels, Isaac Newt. Institute, HOP 2007, 2007, pp. 1–14.
   Google Scholar
• R. Chen, Fast computation of a class of highly oscillatory integrals, Appl. Math. Comput.227(11071260) (2014), pp. 494–501.
   Google Scholar
• C. Fang, Efficient methods for highly oscillatory integrals with weak and Cauchy singularities, Int. J. Comput. Math. 93(9) (2016), pp. 1597–1610.
   Web of Science ®Google Scholar
• L.N.G. Filon, On a quadrature formula for trigonometric integrals, Proc. R. Soc. Edinb. 49 (1930), pp. 38–47.
   Google Scholar
• W. Gautschi, A package of routines for generating orthogonal polynomials and Gauss-types quadrature rules, ACM Trans. Math. Softw. 20(1) (1994), pp. 21–62.
   Web of Science ®Google Scholar
• W. Gautschi, Orthogonal Polynomials, ‘Computation and Approximation’, Oxford University Press, New York, 2004.
   Google Scholar
• I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, 7th ed., Academic Press, New York, 2007.
   Google Scholar
• 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 Univ. Eng. Sci.3(2) (1991), pp. 217–231.
   Google Scholar
• F.B. Hildebrand, Introduction to Numerical Analysis, 2nd ed., Dover Publication, New York, 1974.
   Google Scholar
• D. Huybrechs and S. Vandewalle, On the evaluation of highly oscillatory integrals by analytic continuation, SIAM J. Numer. Anal., 44(2006), pp. 1026–1048.
   Google Scholar
• H. Kang and X. Shao, Fast computation of singular oscillatory Fourier transforms, Abstr. Appl. Anal.2014(1) (2014), pp. 1–8.
   Google Scholar
• H. Kang, S. Xiang, and G. He, Computation of integrals with oscillatory and singular integrands using Chebyshev expansions, J. Comput. Appl. Math. 242(1) (2013), pp. 141–156.
   Google Scholar
• I. Kayijuka, J.P. Ndabakuranye, and A.I. Hasçelik, Computational efficiency of singular and oscillatory integrals with algebraic singularities, Asian J. Math. Comput. Res. 25(5) (2018), pp. 285–302.
   Google Scholar
• I. Kayijuka, Ş. Müge Ege, A. Konuralp, and F. Serap Topal, Fast approximation of algebraic and logarithmic hypersingular type singular integrals with highly oscillatory kernel, Int. J. Anal. Appl.18(6) (2020), pp. 965–980.
   Google Scholar
• I. Kayijuka, Ş. Müge Ege, A. Konuralp, and F. Serap Topal, Clenshaw–Curtis algorithms for an efficient numerical approximation of singular and highly oscillatory Fourier transform integrals, J. Comput. Appl. Math. 385 (2021), pp. 113201.
   Google Scholar
• V.I. Krylov, Approximate Calculation of Integrals, Macmillan, New York, 2006.
   Google Scholar
• P.K. Kythe, M.R. Schäferkotter, Handbook of Computational Method for Integration, Chapman and Hall/CRC, New York, 2004.
   Google Scholar
• D. Levin, Fast integration of rapidly oscillatory functions, J. Comput. Appl. Math. 67 (1996), pp. 95–101.
   Web of Science ®Google Scholar
• G.V. Milovanović, Computing integrals of highly oscillatory special functions using complex integration methods and Gaussian quadratures, Dolomites Res. Notes Approx. 10 (Special Issue)(2017), pp. 79–96.
   Google Scholar
• M. Mori, The double exponential formula for oscillatory functions over the half infinite interval, J. Comput. Appl. Math. 38(1991), pp.353–360.
   Web of Science ®Google Scholar
• W.F. Moss and M.J. Christense, Scattering and heat transfer by a strip, J. Integr. Equ. 4 (1980), pp. 299–317.
   Google Scholar
• S. Rekhviashvili, A. Pskhu, P. Agarwal, and S. Jain, Application of the fractional oscillator model to describe damped vibrations, Turkish J. Phys. 43(3) (2019), pp. 236–242.
   Google Scholar
• Wolfram Research, The Mathematica BOOK, 5th ed., Cambridge University Press, Cambridge, 2003.
   Google Scholar
• V.L. Rvachev, The pressure on an elastic half space of a beam which has the form of a strip, Prikl. Mat. Meck. 20 (1956), pp. 248–256.
   Google Scholar
• W. Sun and N.G. Zamani, Adaptive mesh redistribution for the boundary element in elastostatics, Comput. Struct. 36(6) (1990), pp. 1081–1088.
   Web of Science ®Google Scholar
• G.T. Symm and M.A. Jaswon, Integral Equation Methods in Potential Theory and Elastostatics, 2nd ed., Academic Press, London, 1977.
   Google Scholar