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 and Continuous Dynamical Systems - B, 2015, 20 (5) : 1355-1375. doi: 10.3934/dcdsb.2015.20.1355
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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), 1-15. doi: 10.1023/A:1022072215887.

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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), 97-104. doi: 10.1007/s10598-006-0001-9.

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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), 187-205. doi: 10.1007/BF01176745.

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Q. K. Du, Evaluations of certain hypersingular integrals on interval, Internat. J. Numer. Methods Engrg., 51 (2001), 1195-1210. doi: 10.1002/nme.218.

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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), Article ID 428596, 18 pages.

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J. Huang, Z. Wang and R. Zhu, Asymptotic error expansion for hypersingular integrals, Adv. Comput. Math., 38 (2013), 257-279. doi: 10.1007/s10444-011-9236-x.

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[21]

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[22]

N. I. Ioakimidis, The Gauss-Laguerre quadrature rule for finite-part integrals, Comm. Numer. Methods Engrg., 9 (1993), 439-450. doi: 10.1002/cnm.1640090509.

[23]

M. A. Kelmanson, Hypersingular boundary integrals in cusped two-dimensional free-surface Stokes flow, J. Fluid Mech., 514 (2004), 313-325. doi: 10.1017/S0022112004000515.

[24]

P. Kolm and V. Rokhlin, Numerical quadratures for singular and hypersingular integrals, Comput. Math. Appl., 41 (2001), 327-352. doi: 10.1016/S0898-1221(00)00277-7.

[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), 2721-2733. doi: 10.1098/rspa.2002.1001.

[26]

A. M. Korsunsky, Gauss-Chebyshev quadrature formulae for strongly singular integrals, Quart. Appl. Math., 56 (1998), 461-472.

[27]

L. A. de Lacerda and L. C. Wrobel, Hypersingular boundary integral equation for axisymmetric elasticity, Internat. J. Numer. Methods Engrg., 52 (2001), 1337-1354. doi: 10.1002/nme.259.

[28]

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[29]

I. K. Lifanov, L. N. Poltavskii and G. M. Vainikko, Hypersingular Integral Equations and Their Applications, Chapman & Hall/CRC, Boca Raton, FL, 2004.

[30]

A. M. Lin'kov and S. G. Mogilevskaya, Complex hypersingular integrals and integral equations in plane elasticity, Acta Mech., 105 (1994), 189-205. doi: 10.1007/BF01183951.

[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), 375-386. doi: 10.1016/S0045-7825(98)00292-8.

[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), 271-287. doi: 10.1016/0045-7825(92)90136-8.

[33]

G. Monegato, Numerical evaluation of hypersingular integrals, J. Comput. Appl. Math., 50 (1994), 9-31. doi: 10.1016/0377-0427(94)90287-9.

[34]

G. Monegato and J. N. Lyness, The Euler-Maclaurin expansion and finite-part integrals, Numer. Math., 81 (1998), 273-291. doi: 10.1007/s002110050392.

[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), 272-297. doi: 10.1002/nme.1633.

[36]

L. M. Romero and F. G. Benitez, Traffic flow continuum modeling by hypersingular boundary integral equations, Internat. J. Numer. Methods Engrg., 82 (2010), 47-63. doi: 10.1002/nme.2754.

[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), 2596-2618. doi: 10.1016/j.cma.2006.12.004.

[38]

A. Salvadori, Hypersingular boundary integral equations and the approximation of the stress tensor, Internat. J. Numer. Methods Engrg., 72 (2007), 722-743. doi: 10.1002/nme.2041.

[39]

S. G. Samko, Hypersingular Integrals and Their Applications, Analytical Methods and Special Functions, 5. Taylor & Francis, Ltd., London, 2002.

[40]

A. Sidi, Euler-Maclaurin expansions for integrals with arbitrary algebraic endpoint singularities, Math. Comp., 81 (2012), 2159-2173. doi: 10.1090/S0025-5718-2012-02597-X.

[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), 1609-1628. doi: 10.1002/nme.1620361002.

[42]

W. W. Sun and J. M. Wu, Newton-Cotes formulae for the numerical evaluation of certain hypersingular integrals, Computing, 75 (2005), 297-309. doi: 10.1007/s00607-005-0131-5.

[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), 122-157. doi: 10.1002/nme.1195.

[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), 3593-3601. doi: 10.1109/TAP.2010.2071370.

[45]

J. M. Wu and W. W. Sun, The superconvergence of the comosite trapezoidal rule for Hadamard finite-part integrals, Numer. Math., 102 (2005), 343-363. doi: 10.1007/s00211-005-0647-9.

[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), 143-165. doi: 10.1007/s00211-007-0125-7.

[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), 208-213. doi: 10.1007/BF01130295.

[48]

P. Zhang and T. W. Wu, A hypersingular integral formulation for acoustic radiation in moving flows, J. Sound Vibration, 206 (1997), 309-326. doi: 10.1006/jsvi.1997.1039.

[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), 855-876. doi: 10.1080/00207160802226517.

[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), 1225-1235. doi: 10.1016/j.enganabound.2011.05.004.

[51]

V. V. Zozulya, Regularization of the hypersingular integrals in 3-D problems of fracture mechanics, Boundary elements and other mesh reduction methods XXX, WIT Trans. Model. Simul., WIT Press, Southampton, 47 (2008), 219-228. doi: 10.2495/BEO80221.

[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), 763-775. doi: 10.1080/02533839.1999.9670512.

show all references

References:
[1]

I. V. Boykov, Numerical methods of computation of singular and hypersingular integrals, Internat. J. Math. Math. Sci., 28 (2001), 127-179. doi: 10.1155/S0161171201010924.

[2]

I. V. Boykov, E. S. Ventsel and A. I. Boykov, Accuracy optimal methods for evaluating hypersingular integrals, Appl. Numer. Math., 59 (2009), 1366-1385. doi: 10.1016/j.apnum.2008.08.004.

[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), 683-720. doi: 10.1016/S0020-7225(02)00134-9.

[4]

Y. Z. Chen, A numerical solution technique of hypersingular integral equation for curved cracks, Comm. Numer. Methods Engrg., 19 (2003), 645-655. doi: 10.1002/cnm.623.

[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), 1-16.

[6]

A. G. Davydov, E. V. Zakharov and Y. V. Pimenov, Hypersingular integral equations in computational electrodynamics, Comput. Math. Model., 14 (2003), 1-15. doi: 10.1023/A:1022072215887.

[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), 97-104. doi: 10.1007/s10598-006-0001-9.

[8]

Y. F. Dong and H. C. Gea, A non-hypersingular boundary integral formulation for displacement gradients in linear elasticity, Acta Mech., 129 (1998), 187-205. doi: 10.1007/BF01176745.

[9]

Q. K. Du, Evaluations of certain hypersingular integrals on interval, Internat. J. Numer. Methods Engrg., 51 (2001), 1195-1210. doi: 10.1002/nme.218.

[10]

M. Fogiel, Handbook of Mathematical, Scientific, and Engineering, Research and Education Association, New Jersey, 1994.

[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), 1845-1863. doi: 10.1002/(SICI)1097-0207(19991220)46:11<1845::AID-NME747>3.0.CO;2-I.

[12]

L. Gori, E. Pellegrino and E. Santi, Numerical evaluation of certain hypersingular integrals using refinable operators, Math. Comput. Simulation, 82 (2011), 132-143. doi: 10.1016/j.matcom.2010.07.006.

[13]

L. S. Gradsbteyn and L. M. Ryzbik, Table of Integrals, Series and Produts, Elsevier Pte Ltd, Singapore, 2004.

[14]

L. J. Gray, J. M. Glaeser and T. Kapla, Direct evaluation of hypersingular Galerkin surface integrals, SIAM J. Sci. Comput., 25 (2004), 1534-1556. doi: 10.1137/S1064827502405999.

[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), 1135-1158. doi: 10.1002/nme.1620290603.

[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), Article ID 428596, 18 pages.

[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., Series B, 5 (2014), 269-288.

[18]

J. Huang, Z. Wang and R. Zhu, Asymptotic error expansion for hypersingular integrals, Adv. Comput. Math., 38 (2013), 257-279. doi: 10.1007/s10444-011-9236-x.

[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), 2555-2573. doi: 10.1002/(SICI)1097-0207(19960815)39:15<2555::AID-NME966>3.0.CO;2-6.

[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), 39-50. doi: 10.1007/BF00370484.

[21]

N. I. Ioakimidis, Two-dimensional principal value hypersingular integrals for crack problems in three-dimensional elasticity, Acta Mech., 82 (1990), 129-134. doi: 10.1007/BF01173742.

[22]

N. I. Ioakimidis, The Gauss-Laguerre quadrature rule for finite-part integrals, Comm. Numer. Methods Engrg., 9 (1993), 439-450. doi: 10.1002/cnm.1640090509.

[23]

M. A. Kelmanson, Hypersingular boundary integrals in cusped two-dimensional free-surface Stokes flow, J. Fluid Mech., 514 (2004), 313-325. doi: 10.1017/S0022112004000515.

[24]

P. Kolm and V. Rokhlin, Numerical quadratures for singular and hypersingular integrals, Comput. Math. Appl., 41 (2001), 327-352. doi: 10.1016/S0898-1221(00)00277-7.

[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), 2721-2733. doi: 10.1098/rspa.2002.1001.

[26]

A. M. Korsunsky, Gauss-Chebyshev quadrature formulae for strongly singular integrals, Quart. Appl. Math., 56 (1998), 461-472.

[27]

L. A. de Lacerda and L. C. Wrobel, Hypersingular boundary integral equation for axisymmetric elasticity, Internat. J. Numer. Methods Engrg., 52 (2001), 1337-1354. doi: 10.1002/nme.259.

[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), 189-195. doi: 10.1016/j.enganabound.2009.10.005.

[29]

I. K. Lifanov, L. N. Poltavskii and G. M. Vainikko, Hypersingular Integral Equations and Their Applications, Chapman & Hall/CRC, Boca Raton, FL, 2004.

[30]

A. M. Lin'kov and S. G. Mogilevskaya, Complex hypersingular integrals and integral equations in plane elasticity, Acta Mech., 105 (1994), 189-205. doi: 10.1007/BF01183951.

[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), 375-386. doi: 10.1016/S0045-7825(98)00292-8.

[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), 271-287. doi: 10.1016/0045-7825(92)90136-8.

[33]

G. Monegato, Numerical evaluation of hypersingular integrals, J. Comput. Appl. Math., 50 (1994), 9-31. doi: 10.1016/0377-0427(94)90287-9.

[34]

G. Monegato and J. N. Lyness, The Euler-Maclaurin expansion and finite-part integrals, Numer. Math., 81 (1998), 273-291. doi: 10.1007/s002110050392.

[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), 272-297. doi: 10.1002/nme.1633.

[36]

L. M. Romero and F. G. Benitez, Traffic flow continuum modeling by hypersingular boundary integral equations, Internat. J. Numer. Methods Engrg., 82 (2010), 47-63. doi: 10.1002/nme.2754.

[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), 2596-2618. doi: 10.1016/j.cma.2006.12.004.

[38]

A. Salvadori, Hypersingular boundary integral equations and the approximation of the stress tensor, Internat. J. Numer. Methods Engrg., 72 (2007), 722-743. doi: 10.1002/nme.2041.

[39]

S. G. Samko, Hypersingular Integrals and Their Applications, Analytical Methods and Special Functions, 5. Taylor & Francis, Ltd., London, 2002.

[40]

A. Sidi, Euler-Maclaurin expansions for integrals with arbitrary algebraic endpoint singularities, Math. Comp., 81 (2012), 2159-2173. doi: 10.1090/S0025-5718-2012-02597-X.

[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), 1609-1628. doi: 10.1002/nme.1620361002.

[42]

W. W. Sun and J. M. Wu, Newton-Cotes formulae for the numerical evaluation of certain hypersingular integrals, Computing, 75 (2005), 297-309. doi: 10.1007/s00607-005-0131-5.

[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), 122-157. doi: 10.1002/nme.1195.

[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), 3593-3601. doi: 10.1109/TAP.2010.2071370.

[45]

J. M. Wu and W. W. Sun, The superconvergence of the comosite trapezoidal rule for Hadamard finite-part integrals, Numer. Math., 102 (2005), 343-363. doi: 10.1007/s00211-005-0647-9.

[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), 143-165. doi: 10.1007/s00211-007-0125-7.

[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), 208-213. doi: 10.1007/BF01130295.

[48]

P. Zhang and T. W. Wu, A hypersingular integral formulation for acoustic radiation in moving flows, J. Sound Vibration, 206 (1997), 309-326. doi: 10.1006/jsvi.1997.1039.

[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), 855-876. doi: 10.1080/00207160802226517.

[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), 1225-1235. doi: 10.1016/j.enganabound.2011.05.004.

[51]

V. V. Zozulya, Regularization of the hypersingular integrals in 3-D problems of fracture mechanics, Boundary elements and other mesh reduction methods XXX, WIT Trans. Model. Simul., WIT Press, Southampton, 47 (2008), 219-228. doi: 10.2495/BEO80221.

[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), 763-775. doi: 10.1080/02533839.1999.9670512.

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