July  2015, 20(5): 1355-1375. doi: 10.3934/dcdsb.2015.20.1355

Euler-Maclaurin expansions and approximations of hypersingular integrals

1. 

College of Mathematics, Sichuan University, Chengdu,Sichuan, 610064, China, China

2. 

Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65401, United States

Received  March 2013 Revised  January 2015 Published  May 2015

This article presents the Euler-Maclaurin expansions of the hypersingular integrals $\int_{a}^{b}\frac{g(x)}{|x-t|^{m+1}}dx$ and $\int_{a}^{b}% \frac{g(x)}{(x-t)^{m+1}}dx$ with arbitrary singular point $t$ and arbitrary non-negative integer $m$. These general expansions are applicable to a large range of hypersingular integrals, including both popular hypersingular integrals discussed in the literature and other important ones which have not been addressed yet. The corresponding mid-rectangular formulas and extrapolations, which can be calculated in fairly straightforward ways, are investigated. Numerical examples are provided to illustrate the features of the numerical methods and verify the theoretical conclusions.
Citation: Chaolang Hu, Xiaoming He, Tao Lü. Euler-Maclaurin expansions and approximations of hypersingular integrals. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1355-1375. doi: 10.3934/dcdsb.2015.20.1355
References:
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D. L. Clements, M. Lobo and N. Widana, A hypersingular boundary integral equation for a class of problems concerning infiltration from periodic channels,, Electron. J. Bound. Elem., 5 (2007), 1. Google Scholar

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A. G. Davydov, E. V. Zakharov and Y. V. Pimenov, Hypersingular integral equations in computational electrodynamics,, Comput. Math. Model., 14 (2003), 1. doi: 10.1023/A:1022072215887. Google Scholar

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A. G. Davydov and E. V. Zakharov and Y. V. Pimenov, Hypersingular integral equations for the diffraction of electromagnetic waves on homogeneous magneto-dielectric bodies,, Comput. Math. Model., 17 (2006), 97. doi: 10.1007/s10598-006-0001-9. Google Scholar

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Y. F. Dong and H. C. Gea, A non-hypersingular boundary integral formulation for displacement gradients in linear elasticity,, Acta Mech., 129 (1998), 187. doi: 10.1007/BF01176745. Google Scholar

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Q. K. Du, Evaluations of certain hypersingular integrals on interval,, Internat. J. Numer. Methods Engrg., 51 (2001), 1195. doi: 10.1002/nme.218. Google Scholar

[10]

M. Fogiel, Handbook of Mathematical, Scientific, and Engineering,, Research and Education Association, (1994). Google Scholar

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[22]

N. I. Ioakimidis, The Gauss-Laguerre quadrature rule for finite-part integrals,, Comm. Numer. Methods Engrg., 9 (1993), 439. doi: 10.1002/cnm.1640090509. Google Scholar

[23]

M. A. Kelmanson, Hypersingular boundary integrals in cusped two-dimensional free-surface Stokes flow,, J. Fluid Mech., 514 (2004), 313. doi: 10.1017/S0022112004000515. Google Scholar

[24]

P. Kolm and V. Rokhlin, Numerical quadratures for singular and hypersingular integrals,, Comput. Math. Appl., 41 (2001), 327. doi: 10.1016/S0898-1221(00)00277-7. Google Scholar

[25]

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[26]

A. M. Korsunsky, Gauss-Chebyshev quadrature formulae for strongly singular integrals,, Quart. Appl. Math., 56 (1998), 461. Google Scholar

[27]

L. A. de Lacerda and L. C. Wrobel, Hypersingular boundary integral equation for axisymmetric elasticity,, Internat. J. Numer. Methods Engrg., 52 (2001), 1337. doi: 10.1002/nme.259. Google Scholar

[28]

S. Li and Q. Huang, An improved form of the hypersingular boundary integral equation for exterior acoustic problems,, Eng. Anal. Bound. Elem., 34 (2010), 189. doi: 10.1016/j.enganabound.2009.10.005. Google Scholar

[29]

I. K. Lifanov, L. N. Poltavskii and G. M. Vainikko, Hypersingular Integral Equations and Their Applications,, Chapman & Hall/CRC, (2004). Google Scholar

[30]

A. M. Lin'kov and S. G. Mogilevskaya, Complex hypersingular integrals and integral equations in plane elasticity,, Acta Mech., 105 (1994), 189. doi: 10.1007/BF01183951. Google Scholar

[31]

Y. Liu and S. Chen, A new form of the hypersingular boundary integral equation for 3-D acoustics and its implementation with C0 boundary elements,, Comput. Methods Appl. Mech. Engrg., 173 (1999), 375. doi: 10.1016/S0045-7825(98)00292-8. Google Scholar

[32]

Y. Liu and F. J. Rizzo, A weakly singular form of the hypersingular boundary integral equation applied to 3-D acoustic wave problems,, Comput. Methods Appl. Mech. Engrg., 96 (1992), 271. doi: 10.1016/0045-7825(92)90136-8. Google Scholar

[33]

G. Monegato, Numerical evaluation of hypersingular integrals,, J. Comput. Appl. Math., 50 (1994), 9. doi: 10.1016/0377-0427(94)90287-9. Google Scholar

[34]

G. Monegato and J. N. Lyness, The Euler-Maclaurin expansion and finite-part integrals,, Numer. Math., 81 (1998), 273. doi: 10.1007/s002110050392. Google Scholar

[35]

G. Monegato, R. Orta and R. Tascone, A fast method for the solution of a hypersingular integral equation arising in a waveguide scattering problem,, Internat. J. Numer. Methods Engrg., 67 (2006), 272. doi: 10.1002/nme.1633. Google Scholar

[36]

L. M. Romero and F. G. Benitez, Traffic flow continuum modeling by hypersingular boundary integral equations,, Internat. J. Numer. Methods Engrg., 82 (2010), 47. doi: 10.1002/nme.2754. Google Scholar

[37]

G. Rus and R. Gallego, Hypersingular shape sensitivity boundary integral equation for crack identification under harmonic elastodynamic excitation,, Comput. Methods Appl. Mech. Engrg., 196 (2007), 2596. doi: 10.1016/j.cma.2006.12.004. Google Scholar

[38]

A. Salvadori, Hypersingular boundary integral equations and the approximation of the stress tensor,, Internat. J. Numer. Methods Engrg., 72 (2007), 722. doi: 10.1002/nme.2041. Google Scholar

[39]

S. G. Samko, Hypersingular Integrals and Their Applications,, Analytical Methods and Special Functions, (2002). Google Scholar

[40]

A. Sidi, Euler-Maclaurin expansions for integrals with arbitrary algebraic endpoint singularities,, Math. Comp., 81 (2012), 2159. doi: 10.1090/S0025-5718-2012-02597-X. Google Scholar

[41]

V. Sládek, J. Sládek and M. Tanaka, Regularization of hypersingular and nearly singular integrals in the potential theory and elasticity,, Internat. J. Numer. Methods Engrg., 36 (1993), 1609. doi: 10.1002/nme.1620361002. Google Scholar

[42]

W. W. Sun and J. M. Wu, Newton-Cotes formulae for the numerical evaluation of certain hypersingular integrals,, Computing, 75 (2005), 297. doi: 10.1007/s00607-005-0131-5. Google Scholar

[43]

A. Sutradhar, G. H. Paulino and L. J. Gray, On hypersingular surface integrals in the symmetric Galerkin boundary element method: Application to heat conduction in exponentially graded materials,, Internat. J. Numer. Methods Engrg., 62 (2005), 122. doi: 10.1002/nme.1195. Google Scholar

[44]

M. S. Tong and W. C. Chew, A Novel Approach for evaluating hypersingular and strongly singular surface integrals in electromagnetics,, IEEE Trans. Antennas and Propagation, 58 (2010), 3593. doi: 10.1109/TAP.2010.2071370. Google Scholar

[45]

J. M. Wu and W. W. Sun, The superconvergence of the comosite trapezoidal rule for Hadamard finite-part integrals,, Numer. Math., 102 (2005), 343. doi: 10.1007/s00211-005-0647-9. Google Scholar

[46]

J. M. Wu and W. W. Sun, The superconvergence of Newton-Cotes rules for the Hadamard finite-part integrals on an interval,, Numer. Math., 109 (2008), 143. doi: 10.1007/s00211-007-0125-7. Google Scholar

[47]

E. V. Zakharov and I. V. Khaleeva, Hypersingular integral operators in diffraction problems of electromagnetic waves on open surfaces,, Comput. Math. Model., 5 (1994), 208. doi: 10.1007/BF01130295. Google Scholar

[48]

P. Zhang and T. W. Wu, A hypersingular integral formulation for acoustic radiation in moving flows,, J. Sound Vibration, 206 (1997), 309. doi: 10.1006/jsvi.1997.1039. Google Scholar

[49]

X. Zhang, J. Wu and D. H. Yu, The superconvergence of composite trapezoidal rule for Hadamard finite-part integral on a circle and its application,, Int. J. Comput. Math., 87 (2010), 855. doi: 10.1080/00207160802226517. Google Scholar

[50]

C. Zheng, T. Matsumoto, T. Matsumoto and H. Chen, Explicit evaluation of hypersingular boundary integral equations for acoustic sensitivity analysis based on direct differentiation method,, Eng. Anal. Bound. Elem., 35 (2011), 1225. doi: 10.1016/j.enganabound.2011.05.004. Google Scholar

[51]

V. V. Zozulya, Regularization of the hypersingular integrals in 3-D problems of fracture mechanics,, Boundary elements and other mesh reduction methods XXX, 47 (2008), 219. doi: 10.2495/BEO80221. Google Scholar

[52]

V. V. Zozulya and P. I. Gonzalez-Chi, Weakly singular, singular and hypersingular integrals in 3-D elasticity and fracture mechanics,, J. Chinese Inst. Engrs., 22 (1999), 763. doi: 10.1080/02533839.1999.9670512. Google Scholar

show all references

References:
[1]

I. V. Boykov, Numerical methods of computation of singular and hypersingular integrals,, Internat. J. Math. Math. Sci., 28 (2001), 127. doi: 10.1155/S0161171201010924. Google Scholar

[2]

I. V. Boykov, E. S. Ventsel and A. I. Boykov, Accuracy optimal methods for evaluating hypersingular integrals,, Appl. Numer. Math., 59 (2009), 1366. doi: 10.1016/j.apnum.2008.08.004. Google Scholar

[3]

Y. S. Chan, A. C. Fannjiang and G. H. Paulino, Integral equations with hypersingular kernels-theory and applications to fracture mechanics,, Int. J. Eng. Sci., 41 (2003), 683. doi: 10.1016/S0020-7225(02)00134-9. Google Scholar

[4]

Y. Z. Chen, A numerical solution technique of hypersingular integral equation for curved cracks,, Comm. Numer. Methods Engrg., 19 (2003), 645. doi: 10.1002/cnm.623. Google Scholar

[5]

D. L. Clements, M. Lobo and N. Widana, A hypersingular boundary integral equation for a class of problems concerning infiltration from periodic channels,, Electron. J. Bound. Elem., 5 (2007), 1. Google Scholar

[6]

A. G. Davydov, E. V. Zakharov and Y. V. Pimenov, Hypersingular integral equations in computational electrodynamics,, Comput. Math. Model., 14 (2003), 1. doi: 10.1023/A:1022072215887. Google Scholar

[7]

A. G. Davydov and E. V. Zakharov and Y. V. Pimenov, Hypersingular integral equations for the diffraction of electromagnetic waves on homogeneous magneto-dielectric bodies,, Comput. Math. Model., 17 (2006), 97. doi: 10.1007/s10598-006-0001-9. Google Scholar

[8]

Y. F. Dong and H. C. Gea, A non-hypersingular boundary integral formulation for displacement gradients in linear elasticity,, Acta Mech., 129 (1998), 187. doi: 10.1007/BF01176745. Google Scholar

[9]

Q. K. Du, Evaluations of certain hypersingular integrals on interval,, Internat. J. Numer. Methods Engrg., 51 (2001), 1195. doi: 10.1002/nme.218. Google Scholar

[10]

M. Fogiel, Handbook of Mathematical, Scientific, and Engineering,, Research and Education Association, (1994). Google Scholar

[11]

A. Frangi and M. Guiggiani, Boundary element analysis of kirchhoff plates with direct evaluation of hypersingular integrals,, Int. J. Numer. Meth. Engng., 46 (1999), 1845. doi: 10.1002/(SICI)1097-0207(19991220)46:11<1845::AID-NME747>3.0.CO;2-I. Google Scholar

[12]

L. Gori, E. Pellegrino and E. Santi, Numerical evaluation of certain hypersingular integrals using refinable operators,, Math. Comput. Simulation, 82 (2011), 132. doi: 10.1016/j.matcom.2010.07.006. Google Scholar

[13]

L. S. Gradsbteyn and L. M. Ryzbik, Table of Integrals, Series and Produts,, Elsevier Pte Ltd, (2004). Google Scholar

[14]

L. J. Gray, J. M. Glaeser and T. Kapla, Direct evaluation of hypersingular Galerkin surface integrals,, SIAM J. Sci. Comput., 25 (2004), 1534. doi: 10.1137/S1064827502405999. Google Scholar

[15]

L. J. Gray, L. F. Martha and A. R. Ingraffea, Hypersingular integrals in boundary element fracture analysis,, Internat. J. Numer. Methods Engrg., 29 (1990), 1135. doi: 10.1002/nme.1620290603. Google Scholar

[16]

C. L. Hu, J. Lu and X. M. He, Productivity formulae of an infinite-conductivity hydraulically fractured well producing at constant wellbore pressure based on numerical solutions of a weakly singular integral equation of the first kind,, Math. Probl. Eng., (2012). Google Scholar

[17]

C. L. Hu, J. Lu and X. M. He, Numerical solutions of hypersingular integral equation with application to productivity formulae of horizontal wells producing at constant wellbore pressure,, Int. J. Numer. Anal. Mod., 5 (2014), 269. Google Scholar

[18]

J. Huang, Z. Wang and R. Zhu, Asymptotic error expansion for hypersingular integrals,, Adv. Comput. Math., 38 (2013), 257. doi: 10.1007/s10444-011-9236-x. Google Scholar

[19]

O. Huber, R. Dallner, P. Partheymüller and G. Kuhn, Evaluation of the stress tensor in 3-D elastoplasticity by direct solving of hypersingular integrals,, Internat. J. Numer. Methods Engrg., 39 (1996), 2555. doi: 10.1002/(SICI)1097-0207(19960815)39:15<2555::AID-NME966>3.0.CO;2-6. Google Scholar

[20]

O. Huber, A. Lang and G. Kuhn, Evaluation of the stress tensor in 3D elastostatics by direct solving of hypersingular integrals,, Comput. Mech., 12 (1993), 39. doi: 10.1007/BF00370484. Google Scholar

[21]

N. I. Ioakimidis, Two-dimensional principal value hypersingular integrals for crack problems in three-dimensional elasticity,, Acta Mech., 82 (1990), 129. doi: 10.1007/BF01173742. Google Scholar

[22]

N. I. Ioakimidis, The Gauss-Laguerre quadrature rule for finite-part integrals,, Comm. Numer. Methods Engrg., 9 (1993), 439. doi: 10.1002/cnm.1640090509. Google Scholar

[23]

M. A. Kelmanson, Hypersingular boundary integrals in cusped two-dimensional free-surface Stokes flow,, J. Fluid Mech., 514 (2004), 313. doi: 10.1017/S0022112004000515. Google Scholar

[24]

P. Kolm and V. Rokhlin, Numerical quadratures for singular and hypersingular integrals,, Comput. Math. Appl., 41 (2001), 327. doi: 10.1016/S0898-1221(00)00277-7. Google Scholar

[25]

A. M. Korsunsky, On the use of interpolative quadratures for hypersingular integrals in fracture mechanics,, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 458 (2002), 2721. doi: 10.1098/rspa.2002.1001. Google Scholar

[26]

A. M. Korsunsky, Gauss-Chebyshev quadrature formulae for strongly singular integrals,, Quart. Appl. Math., 56 (1998), 461. Google Scholar

[27]

L. A. de Lacerda and L. C. Wrobel, Hypersingular boundary integral equation for axisymmetric elasticity,, Internat. J. Numer. Methods Engrg., 52 (2001), 1337. doi: 10.1002/nme.259. Google Scholar

[28]

S. Li and Q. Huang, An improved form of the hypersingular boundary integral equation for exterior acoustic problems,, Eng. Anal. Bound. Elem., 34 (2010), 189. doi: 10.1016/j.enganabound.2009.10.005. Google Scholar

[29]

I. K. Lifanov, L. N. Poltavskii and G. M. Vainikko, Hypersingular Integral Equations and Their Applications,, Chapman & Hall/CRC, (2004). Google Scholar

[30]

A. M. Lin'kov and S. G. Mogilevskaya, Complex hypersingular integrals and integral equations in plane elasticity,, Acta Mech., 105 (1994), 189. doi: 10.1007/BF01183951. Google Scholar

[31]

Y. Liu and S. Chen, A new form of the hypersingular boundary integral equation for 3-D acoustics and its implementation with C0 boundary elements,, Comput. Methods Appl. Mech. Engrg., 173 (1999), 375. doi: 10.1016/S0045-7825(98)00292-8. Google Scholar

[32]

Y. Liu and F. J. Rizzo, A weakly singular form of the hypersingular boundary integral equation applied to 3-D acoustic wave problems,, Comput. Methods Appl. Mech. Engrg., 96 (1992), 271. doi: 10.1016/0045-7825(92)90136-8. Google Scholar

[33]

G. Monegato, Numerical evaluation of hypersingular integrals,, J. Comput. Appl. Math., 50 (1994), 9. doi: 10.1016/0377-0427(94)90287-9. Google Scholar

[34]

G. Monegato and J. N. Lyness, The Euler-Maclaurin expansion and finite-part integrals,, Numer. Math., 81 (1998), 273. doi: 10.1007/s002110050392. Google Scholar

[35]

G. Monegato, R. Orta and R. Tascone, A fast method for the solution of a hypersingular integral equation arising in a waveguide scattering problem,, Internat. J. Numer. Methods Engrg., 67 (2006), 272. doi: 10.1002/nme.1633. Google Scholar

[36]

L. M. Romero and F. G. Benitez, Traffic flow continuum modeling by hypersingular boundary integral equations,, Internat. J. Numer. Methods Engrg., 82 (2010), 47. doi: 10.1002/nme.2754. Google Scholar

[37]

G. Rus and R. Gallego, Hypersingular shape sensitivity boundary integral equation for crack identification under harmonic elastodynamic excitation,, Comput. Methods Appl. Mech. Engrg., 196 (2007), 2596. doi: 10.1016/j.cma.2006.12.004. Google Scholar

[38]

A. Salvadori, Hypersingular boundary integral equations and the approximation of the stress tensor,, Internat. J. Numer. Methods Engrg., 72 (2007), 722. doi: 10.1002/nme.2041. Google Scholar

[39]

S. G. Samko, Hypersingular Integrals and Their Applications,, Analytical Methods and Special Functions, (2002). Google Scholar

[40]

A. Sidi, Euler-Maclaurin expansions for integrals with arbitrary algebraic endpoint singularities,, Math. Comp., 81 (2012), 2159. doi: 10.1090/S0025-5718-2012-02597-X. Google Scholar

[41]

V. Sládek, J. Sládek and M. Tanaka, Regularization of hypersingular and nearly singular integrals in the potential theory and elasticity,, Internat. J. Numer. Methods Engrg., 36 (1993), 1609. doi: 10.1002/nme.1620361002. Google Scholar

[42]

W. W. Sun and J. M. Wu, Newton-Cotes formulae for the numerical evaluation of certain hypersingular integrals,, Computing, 75 (2005), 297. doi: 10.1007/s00607-005-0131-5. Google Scholar

[43]

A. Sutradhar, G. H. Paulino and L. J. Gray, On hypersingular surface integrals in the symmetric Galerkin boundary element method: Application to heat conduction in exponentially graded materials,, Internat. J. Numer. Methods Engrg., 62 (2005), 122. doi: 10.1002/nme.1195. Google Scholar

[44]

M. S. Tong and W. C. Chew, A Novel Approach for evaluating hypersingular and strongly singular surface integrals in electromagnetics,, IEEE Trans. Antennas and Propagation, 58 (2010), 3593. doi: 10.1109/TAP.2010.2071370. Google Scholar

[45]

J. M. Wu and W. W. Sun, The superconvergence of the comosite trapezoidal rule for Hadamard finite-part integrals,, Numer. Math., 102 (2005), 343. doi: 10.1007/s00211-005-0647-9. Google Scholar

[46]

J. M. Wu and W. W. Sun, The superconvergence of Newton-Cotes rules for the Hadamard finite-part integrals on an interval,, Numer. Math., 109 (2008), 143. doi: 10.1007/s00211-007-0125-7. Google Scholar

[47]

E. V. Zakharov and I. V. Khaleeva, Hypersingular integral operators in diffraction problems of electromagnetic waves on open surfaces,, Comput. Math. Model., 5 (1994), 208. doi: 10.1007/BF01130295. Google Scholar

[48]

P. Zhang and T. W. Wu, A hypersingular integral formulation for acoustic radiation in moving flows,, J. Sound Vibration, 206 (1997), 309. doi: 10.1006/jsvi.1997.1039. Google Scholar

[49]

X. Zhang, J. Wu and D. H. Yu, The superconvergence of composite trapezoidal rule for Hadamard finite-part integral on a circle and its application,, Int. J. Comput. Math., 87 (2010), 855. doi: 10.1080/00207160802226517. Google Scholar

[50]

C. Zheng, T. Matsumoto, T. Matsumoto and H. Chen, Explicit evaluation of hypersingular boundary integral equations for acoustic sensitivity analysis based on direct differentiation method,, Eng. Anal. Bound. Elem., 35 (2011), 1225. doi: 10.1016/j.enganabound.2011.05.004. Google Scholar

[51]

V. V. Zozulya, Regularization of the hypersingular integrals in 3-D problems of fracture mechanics,, Boundary elements and other mesh reduction methods XXX, 47 (2008), 219. doi: 10.2495/BEO80221. Google Scholar

[52]

V. V. Zozulya and P. I. Gonzalez-Chi, Weakly singular, singular and hypersingular integrals in 3-D elasticity and fracture mechanics,, J. Chinese Inst. Engrs., 22 (1999), 763. doi: 10.1080/02533839.1999.9670512. Google Scholar

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