Skip to main content
SpringerLink
Log in

Regularization of divergent integrals: A comparison of the classical and generalized-functions approaches

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

This article considers methods of weakly singular and hypersingular integral regularization based on the theory of distributions. For regularization of divergent integrals, the Gauss–Ostrogradskii theorem and the second Green’s theorem in the sense of the theory of distribution have been used. Equations that allow easy calculation of weakly singular, singular, and hypersingular integrals in one- and two-dimensional cases for any sufficiently smooth function have been obtained. These equations are compared with classical methods of regularization. The results of numerical calculation using classical approaches and those based of the theory of generalized functions, along with a comparison for different functions, are presented in tables and graphs of the values of divergent integrals versus the position of the colocation point.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Volume 55 of Applied Mathematics Series, Department of Commerce, National Bureau of Standards (1972)

  2. Aliabadi, M.H.: The boundary element method. Application in Solids and Structures, vol. 2. John Wiley (2002)

  3. Aliabadi, M.H., Hall, W.S.: The regularizing transformation integration method for BE. Comparison with Gauss quadrature. Engineering Analysis with Boundary Elements 6, 66–71 (1986)

    Google Scholar 

  4. Ashour, S.A., Ahmed, H.M.: A Gauss quadrature rule for hypersingular integrals. Appl. Math. Comput. 186, 1671–1682 (2007)

    MATH  MathSciNet  Google Scholar 

  5. Boehme, T.K.: Operational calculus and the finite part of divergent integrals. Trans. Am. Math. Soc. 106, 346–368 (1963)

    MATH  MathSciNet  Google Scholar 

  6. Bonnet, M., Guiggiani, M.: Galerkin BEM with direct evaluation of hypersingular integrals. Electron. J. Bound. Elem. 1(2), 95–111 (2003)

    MathSciNet  Google Scholar 

  7. Bureau, F.J.: Divergent integrals and partial differential equations. Commun. Pur. Appl. Math. 8(1), 143–202 (1955)

    MATH  MathSciNet  Google Scholar 

  8. Caianiello, E.R.: Generalized integration procedure for divergent integrals. Il Nuovo cimento 15A(2), 145–161 (1973)

    MathSciNet  Google Scholar 

  9. Carley, M.: Numerical quadratures for singular and hypersingular integrals in boundary element methods. SIAM J. Sci. Comput. 29(3), 1207–1216 (2007)

    MATH  MathSciNet  Google Scholar 

  10. Carley, M.: Numerical quadratures for near-singular and near-hypersingular integrals in boundary element methods. Mathematical Proceedings of the Royal Irish Academy 109A(1), 49(60) (2009)

    MathSciNet  Google Scholar 

  11. Chan, Y.-S., Fannjiang, A.C., Paulino, G.H.: Integral equations with hypersingular kernels—-theory and applications to fracture mechanics. Int. J. Eng. Sci. 41, 683–720 (2003)

    MATH  MathSciNet  Google Scholar 

  12. Chen, G., Zhou, J.: Boundary Element Method, p 646. Academic Press, London (1992)

    Google Scholar 

  13. Chen, J.T., Hong, H.-K.: Review of dual boundary element methods with emphasis on hypersingular integrals and divergent series. Appl. Mech. Rev. 52(1), 17–33 (1999)

    MathSciNet  Google Scholar 

  14. Chen, M.-C.: Application of finite-part integrals to three-dimensional fracture problems for piezoelectric media. Part II: Hypersingular integral equation and theoretical analysis. Int. J. Fract. 121, 133–148 (2003)

    Google Scholar 

  15. Chen, M.-C.: Application of finite-part integrals to three-dimensional fracture problems for piezoelectric media. Part II: Numerical analysis. Int. J. Fract. 121, 149–161 (2003)

    Google Scholar 

  16. Chen, Y.Z., Lin, X.Y., Peng, Z.Q.: Some particular solutions for penny-shaped crack problem by using hypersingular integral equation or differential-integral equation. Arch. Appl. Mech. 68, 271–280 (1998)

    MATH  Google Scholar 

  17. Cheng, A.H.-D., Cheng, D.T.: Heritage and early history of the boundary element method. Engineering Analysis with Boundary Elements 29, 268–302 (2005)

    MATH  Google Scholar 

  18. Courant, R., Hilbert, D.: Methods of Mathematical Physics, vol. II. John Wiley, New York (1968)

    Google Scholar 

  19. Davey, K., Hinduja, S.: Analytical integration of linear three-dimensional triangular elements in BEM. Applied Mathematical Modeling 13, 450–461 (1989)

    MATH  Google Scholar 

  20. Davis, P.J., Rabinowitz, P., 2nd edn: Methods of Numerical Integration. Academic Press, New York (1984)

    MATH  Google Scholar 

  21. de Klerk, J.H.: Hypersingular integral equations—past, present, future. Nonlinear Anal. 63, 533–540 (2005)

    Google Scholar 

  22. Diligenti, M., Monegato, G.: Integral evaluation in the BEM solution of (hyper)singular integral equations. 2D problems on polygonal domains. J. Comput. Appl. Math. 81, 29–57 (1997)

    MATH  MathSciNet  Google Scholar 

  23. Dumont, N.A.: On the efficient numerical evaluation of integrals with complex singularity poles. Engineering Analysis with Boundary Elements 13, 155–168 (1994)

    Google Scholar 

  24. Dumont, N.A.: Cauchy principal values, finite-part integrals and interval normalization: some basic considerations. In: Atluri, S. N., Yagawa, G., Cruse, T. A. (eds.) Computational Mechanics 95, pp 2830–2835. Springer-Verlag, Berlin (1995)

    Google Scholar 

  25. Dumont, N.A, Noronha, M.A.M.: A simple, accurate scheme for the numerical evaluation of integrals with complex singularity poles. Comput. Mech. 22, 42–49 (1998)

    MATH  MathSciNet  Google Scholar 

  26. Erdogan, F.E., Gupta, G.D., Cook, T.S.: The Numerical Solutions of Singular Integral Equations. Methods of Analysis and Solutions of Crack Problems, pp 368–425. Noordhoff Intern. Publ., Leyden (1973)

    Google Scholar 

  27. Fata, S.N.: Explicit expressions for 3D boundary integrals in potential theory. Int. J. Numer. Methods Eng. 78, 32–47 (2009)

    MATH  MathSciNet  Google Scholar 

  28. Frangi, A., Guiggiani, M.: Boundary element analysis of Kirchhoff plates with direct evaluation of hypersingular integrals. Int. J. Numer. Methods Eng. 46(11), 1845–1863 (1999)

    MATH  MathSciNet  Google Scholar 

  29. Frangi, A., Guiggiani, M.: A direct approach for boundary integral equations with high-order singularities. Int. J. Numer. Methods Eng. 49, 871–898 (2000)

    MATH  MathSciNet  Google Scholar 

  30. Gao, X.-W.: An effective method for numerical evaluation of general 2D and 3D high order singular boundary integrals. Comput. Methods Appl. Mech. Eng. 199, 2856–2864 (2010)

    MATH  Google Scholar 

  31. Gautschi, W.: Orthogonal Polynomials.. Computation and Approximation, p 312. Oxford University Press, Oxford (2004)

    MATH  Google Scholar 

  32. Gel’fand, I.M., Shilov, G.E.: Generalized Functions, vol. 1. Academic Press, New York (1964)

    Google Scholar 

  33. Gray, L.J., Glaeser, J., Kaplan, T.: Direct evaluation of hypersingular Galerkin surface integrals. SIAM J. Sci. Comput. 25, 1534–1556 (2004)

    MATH  MathSciNet  Google Scholar 

  34. Gray, L.J., Salvadori, A., Phan, A.-V., Mantic, V.: Direct Evaluation of Hypersingular Galerkin Surface Integrals. II. Electron. J. Bound. Elem. 4(3), 105–130 (2006)

    MathSciNet  Google Scholar 

  35. Guiggiani, M.: Computing principal value integrals in 3D BEM for time-harmonic elastodynamics. A direct approach. Communications in Applied Numerical Methods 8, 141–149 (1992)

    MATH  Google Scholar 

  36. Guiggiani, M.: Formulation and numerical treatment of boundary integral equations with hypersingular kernels. In: Sladek, V., Sladek, J (eds.) Singular Integrals in Boundary Element Methods. WIT Press, Southampton (1998)

    Google Scholar 

  37. Guiggiani, M., Gigante, A.: A general algorithm for multidimensional Cauchy principal value integrals in the boundary element method. ASME J. Appl. Mech. 57, 906–915 (1990)

    MATH  MathSciNet  Google Scholar 

  38. Guiggiani, M., Krishnasamy, G., Rudolphi, T.J., Rizzo, F.J.: A general algorithm for the numerical solution of hypersingular boundary integral equations. ASME J. Appl. Mech. 59, 604–14 (1992)

    MATH  MathSciNet  Google Scholar 

  39. Gunter, N.M.: Potential Theory And Its Application to Basic Problems of Mathematical Physics. Fredric Ungar, New York (1967)

    Google Scholar 

  40. Gurtin, M.E.: The linear theory of elasticity, In: Encyclopedia of Physics. In: Fluegge, S. (ed.) Vol. IVa-2. Springer, New York–Berlin (1972)

    Google Scholar 

  41. Guz, A.N., Zozulya, V.V.: Brittle Fracture of Constructive Materials Under Dynamic Loading. Kiev, Naukova Dumka (1993). (in Russian)

    Google Scholar 

  42. Guz, A.N., Zozulya, V.V.: Fracture dynamics with allowance for a crack edges contact interaction. International Journal of Nonlinear Sciences and Numerical Simulation 2(3), 173–233 (2001)

    MATH  MathSciNet  Google Scholar 

  43. Guz, A.N., Zozulya, V.V.: Elastodynamic unilateral contact problems with friction for bodies with cracks. Int. Appl. Mech. 38(8), 3–45 (2002)

    MathSciNet  Google Scholar 

  44. Hadamard, J.: Lectures on Cauchy’s Problem in Linear Hyperbolic Differential Equations. Dover, New York (1953)

    Google Scholar 

  45. Hardy, G.H.: Divergent Series. Oxford At The Clarendon Press (1949)

  46. Hörmander, L.: On the theory of general partial differential operators. Acta Mathematica 94, 161–248 (1955)

    MATH  MathSciNet  Google Scholar 

  47. Hong, H.-K., Chen, J.T.: Derivations of Integral Equations of Elasticity. ASCE J. Eng. Mech. 114(6), 1028–1044 (1988)

    Google Scholar 

  48. Hsiao, G.C., Wendland, W.L.: Boundary Integral Equations. Springer-Verlag, Berlin Heidelberg (2008)

    MATH  Google Scholar 

  49. Hussain, F., Karim, M.S., Ahamad, R.: Appropriate Gaussian quadrature formulae for triangles. International Journal of Applied Mathematics and Computation 4(1), 24–38 (2012)

    Google Scholar 

  50. Ioakimidis, N.I.: Application of finite-part integrals to the singular integral equations of crack problems in plane and three-dimensional elasticity. Acta Mech. 45, 31–47 (1982)

    MATH  MathSciNet  Google Scholar 

  51. Iovane, G.: Hypersingular integral equations, Kähler manifolds and Thurston mirroring effect in \(\epsilon ^{(\infty )}\) Cantorian spacetime. Chaos, Solitons and Fractals 31, 1041–1053 (2007)

    Google Scholar 

  52. Iovane, G, Lifanov, I.K., Sumbatyan, M.A.: On direct numerical treatment of hypersingular integral equations arising in mechanics and acoustics. Acta Mech. 162, 99–110 (2003)

    MATH  Google Scholar 

  53. Johnston, B.M., Johnston, P.R., Elliott, D.: A new method for the numerical evaluation of nearly singular integrals on triangular elements in the 3D boundary element method. J. Comput. Appl. Math. 245, 148–161 (2013)

    MATH  MathSciNet  Google Scholar 

  54. Kantor, B.J, Strel’nikova, E.A.: Hypersingular integral equations in the problems of solid mechanics. Kharkiv, Novoe slovo (2005). (in Russian)

    Google Scholar 

  55. Kantorovich, L.V.: On approximate calculation of some type of definite integrals and other applications of the method of singularities extraction. Matematicheskii Sbornik 41(2), 235–245 (1934). (in Russian)

    MathSciNet  Google Scholar 

  56. Karami, G., Derakhshan, D.: An efficient method to evaluate hypersingular and supersingular integrals in boundary integral equations analysis. Engineering Analysis with Boundary Elements 23, 317–326 (1999)

    MATH  Google Scholar 

  57. Karasalo, I.: On evaluation of hypersingular integrals over smooth surfaces. Comput. Mech. 40, 617–625 (2007)

    MATH  Google Scholar 

  58. Kaya, A.C., Erdogan, F.: On the solution of integral equations with strongly singular kernels. Q. Appl. Math. 45, 105–122 (1987)

    MATH  MathSciNet  Google Scholar 

  59. Kieser, R., Schwab, C., Wendland, W.L.: Numerical Evaluation of Singular and Finite-Part Integrals on Curved Surfaces Using Symbolic Manipulation. Computing 49, 279–301 (1992)

    MATH  MathSciNet  Google Scholar 

  60. Kim, P., Choi, U.J.: Two trigonometric quadrature formulae for evaluating hypersingular integrals. Int. J. Numer. Methods Eng. 56, 469–486 (2003)

    MATH  Google Scholar 

  61. Kolm, P., Rokhlin, V.: Numerical quadratures for singular and hypersingular integrals. Computers and Mathematics with Applications 41, 327–352 (2001)

    MATH  MathSciNet  Google Scholar 

  62. Krylov, V.I.: Approximate Calculation of Integrals, p 386. Dover Publications, Inc. (2005)

  63. Kupradze, V.D.: Potential Methods in the Theory of Elasticity, p 339. Daniel Davey & Co., Inc. (1965a)

  64. Kupradze, V.D., Gegelia, T.G., Basheleishvili, M.O., Burchuladze, T.V.: Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity, p 929. North-Holland Publ. Comp., Amsterdam (1979)

    MATH  Google Scholar 

  65. Kythe, P.K., Schaferkotter, M.R.: Handbook of Computational Methods for Integration, p 604. Chapman & Hall/CRC Press (2005)

  66. Lebedev, L.P., Vorovich, I.I., Gladwell, G.K.: Functional Analysis. Applications in Mechanics and Inverse Problems. Springer, Netherlands (1996)

    MATH  Google Scholar 

  67. Lighthill, M.: Introduction to Fourier Analysis and Generalized Functions. Cambridge Univ. Press, Cambridge (1959)

    Google Scholar 

  68. Lifanov, I.K., Lifanov, P.I.: On some exact solutions of singular integral equations in the class of generalized functions and their numerical computation. Diff. Equat. 40(12), 1770–1780 (2004)

    MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  70. Linkov, A.M.: Complex hypersingular integrals and integral equations in plane elasticity. Acta Mech. 105, 18–205 (1994)

    MathSciNet  Google Scholar 

  71. Martin, P.A., Rizzo, F.J.: Hypersingular integrals: How smooth must the density be? Int. J. Numer. Methods Eng. 39, 687–704 (1996)

    MATH  MathSciNet  Google Scholar 

  72. Michlin, S.G.: Singular integral equations. Uspehi Maematiceskih Nauk 3(25), 29–112 (1948). (in Russian)

    Google Scholar 

  73. Michlin, S.G.: Multidimensional Singular Integrals and Integral Equations. Pergamon Press, Oxford (1965)

    Google Scholar 

  74. Monegato, G.: Numerical evaluation of hypersingular integrals. J. Comput. Appl. Math. 50, 9–31 (1994)

    MATH  MathSciNet  Google Scholar 

  75. Monegato, G.: Definitions, properties and applications of finite-part integrals. J. Comput. Appl. Math. 229, 425–439 (2009)

    MATH  MathSciNet  Google Scholar 

  76. Moore, M.N.J., Gray, L.J., Kaplan, T.: Evaluation of supersingular integrals: Second-order boundary derivatives. Int. J. Numer. Methods Eng. 69, 1930–1947 (2007)

    MATH  MathSciNet  Google Scholar 

  77. Mousavi, S.E., Xiao, H., Sukumar, N.: Generalized Gaussian quadrature rules on arbitrary polygons. Int. J. Numer. Methods Eng. 82(1), 99–113 (2010)

    MATH  MathSciNet  Google Scholar 

  78. Mukherjee, S.: CPV and HFP integrals and their applications in the boundary element method. Int. J. Solids Struct. 37, 6623–6634 (2000)

    MATH  Google Scholar 

  79. Mukherjee, S.: Finite parts of singular and hypersingular integrals with irregular boundary source points. Engineering Analysis with Boundary Elements 24, 767–776 (2000)

    MATH  Google Scholar 

  80. Mukherjeea, S., Mukherjee, Y.X., Wenjing, Y.: Cauchy principal values and finite parts of boundary integrals—revisited. Engineering Analysis with Boundary Elements 29, 844–849 (2005)

    Google Scholar 

  81. Mukherjee, Y.X., Mukherjee, S.: Error analysis and adaptivity in three-dimensional linear elasticity by the usual and hypersingular boundary contour method. Int. J. Solids Struct. 38, 161–178 (2001)

    MATH  Google Scholar 

  82. Mukherjee, S., Mukherjee, Y.X.: Boundary Methods: Elements, Contours, and Nodes. CRC/Taylor & Francis, Boca Raton (2005)

    Google Scholar 

  83. Muskhelishvili, N.I.: Singular Integral Equations, p 452. P, Noordhoff Ltd. (1953)

  84. Nagarajan, A., Lutz, E.D., Mukherjee, S.: A novel boundary element method for linear elasticity with no numerical integration for 2D and line integrals for 3D problems. J. Appl. Mech. 61, 264–269 (1994)

    MATH  MathSciNet  Google Scholar 

  85. Ninham, B.W.: Generalised Functions and Divergent Integrals. Numer. Math. 8, 44–57 (1966)

    MathSciNet  Google Scholar 

  86. Niu, Z., Wendland, W.L., Wang, X., Zhou, H.A: semi-analytical algorithm for the evaluation of the nearly singular integrals in three-dimensional boundary element methods. Comput. Methods Appl. Mech. Engrg. 194, 1057–1074 (2005)

    MATH  Google Scholar 

  87. Ortner, V.N.: Regularisierte Faltung von Distributionen. Teil l. Zur Berechnung von Fundamentallösungen. Z. Angew. Math. Phys. (ZAMP) 31, 133–154 (1980)

    MATH  MathSciNet  Google Scholar 

  88. Ortner, V.N.: Regularisierte Faltung von Distributionen. Teil 2. Eine Tabelle von Fundamentallösungen. Z. Angew. Math. Phys. (ZAMP) 31, 155–173 (1980)

    MathSciNet  Google Scholar 

  89. Polyzos, D., Tsepoura, K.G., Tsinopoulos, S.V., Beskos, D.E.: A boundary element method for solving 2-D and 3-D static gradient elastic problems. Part I: Integral formulation. Comput. Methods Appl. Mech. Eng. 192, 2845–2873 (2003)

    MATH  MathSciNet  Google Scholar 

  90. Qin, T.Y., Yu, Y.S., Noda, N.A.: Finite-part integral and boundary element method to solve three-dimensional crack problems in piezoelectric materials. Int. J. Solids Struct. 44, 4770–4783 (2007)

    MATH  Google Scholar 

  91. Rathod, H.T., Karim, M.S.: An explicit integration scheme based on recursion for the curved triangular finite elements. Comput. Struct. 80, 43–76 (2002)

    Google Scholar 

  92. Saez, A., Gallego, R., Dominguez, J.: Hypersingular quarter-point boundary elements for crack problems. Int. J. Numer. Methods Eng. 38, 1681–1701 (1995)

    MATH  Google Scholar 

  93. Salvadori, A.: Analytical integrations in 2D BEM elasticity. Int. J. Numer. Methods Eng. 53, 1695–1719 (2002)

    MATH  MathSciNet  Google Scholar 

  94. Salvadori, A.: Analytical integration of hypersingular kernel in 3D BEM problems. Comput. Methods Appl. Mech. Eng. 190, 3957–3975 (2001)

    MATH  MathSciNet  Google Scholar 

  95. Salvadori, A.: Analytical integrations in 3D BEM for elliptic problems: evaluation and implementation. Int. J. Numer. Methods Eng. 84, 505–542 (2010)

    MATH  MathSciNet  Google Scholar 

  96. Salvadori, A., Temponi, A.: Analytical integrations for the approximation of 3D hyperbolic scalar boundary integral equations. Engineering Analysis with Boundary Elements 34, 977–994 (2010)

    MATH  MathSciNet  Google Scholar 

  97. Samko, S.G.: Hypersingular iIntegrals and Their Applications. Taylor & Francis (2002)

  98. Sanz, J.A., Solis, M., Dominguez, J.: Hypersingular BEM for piezoelectric solids: formulation and applications for fracture mechanics. CMES: Comput. Model. Eng. Sci. 17(3), 215–229 (2007)

    MATH  MathSciNet  Google Scholar 

  99. Schwab, C, Wendland, W.L.: Kernel properties and representations of boundary integral operators. Mathematische Nachrichten 156, 187–218 (1992)

    MATH  MathSciNet  Google Scholar 

  100. Schwartz, L.: Theorie des distributions, Vols. I and II. Hermann, Paris (1957)

    Google Scholar 

  101. Sladek, V., Sladek, J. (eds.): Singular Integrals in Boundary Element Methods. WIT Press, Southampton (1998)

  102. Sobolev, S.L.: Applications of Functional Analysis in Mathematical Physics, vol. 7 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, USA (1963)

  103. Tanaka, M., Sladek, V., Sladek, J.: Regularization techniques applied to boundary element methods. Appl. Mech. Rev. 47(10), 457–499 (1994)

    Google Scholar 

  104. Tomioka, S., Nishiyama, S.: Analytical regularization of hypersingular integral for Helmholtz equation in boundary element method. Engineering Analysis with Boundary Elements 34, 393–404 (2010)

    MATH  MathSciNet  Google Scholar 

  105. Tsepoura, K.G., Tsinopoulos, S., Polyzos, D., Beskos, D.E.: A boundary element method for solving 2-D and 3-D static gradient elastic problems. Part II: Numerical implementation. Comput. Methods Appl. Mech. Eng. 192, 2875–2907 (2003)

    MATH  MathSciNet  Google Scholar 

  106. Vainikko, G.: Multidimensional Weakly Singular Integral Equations, p 169. Springer-Verlag, Berlin Heidelberg (1993)

    MATH  Google Scholar 

  107. Wendland, W.L., Stephan, E.P.: A hypersingular boundary integral method for two-dimensional screen and crack problems. Arch. Ration. Mech. Anal. 112, 363–390 (1990)

    MATH  MathSciNet  Google Scholar 

  108. Young, A.: Improved numerical method for the Traction BIE by application of Stokes’ theorem. Int. J. Numer. Methods Eng. 40, 3141–3161 (1997)

    MATH  Google Scholar 

  109. Zemanian, A.H.: Distribution Theory and Transform Analysis. Dover, New York (1987)

    MATH  Google Scholar 

  110. Zozulya, V.V.: Integrals of Hadamard type in dynamic problem of the crack theory. Doklady Academii Nauk. Ukrainskoy SSR, Ser. A. Physical Mathematical & Technical Sciences 2, 19–22 (1991). (in Ukrainian)

    MathSciNet  Google Scholar 

  111. Zozulya, V.V.: Contact interaction between the edges of a crack and an infinite plane under a harmonic loading. Int. Appl. Mech. 28(1), 61–65 (1992)

    MathSciNet  Google Scholar 

  112. Zozulya, V.V.: Regularization of the divergent integrals. I. General consideration. Electron. J. Bound. Elem. 4(2), 49–57 (2006)

    MathSciNet  Google Scholar 

  113. Zozulya, V.V.: Regularization of the divergent integrals. II. Application in Fracture Mechanics. Electron. J. Bound. Elem. 4(2), 58–56 (2006)

    MathSciNet  Google Scholar 

  114. Zozulya, V.V. In: Skerget, P., Brebbia, C.A. (eds.) : Regularization of the hypersingular integrals in 3-D problems of fracture mechanics. In: Boundary Elements and Other Mesh Reduction Methods XXX, pp 219–228. WIT Press, Southampton (2008)

  115. Zozulya, V.V.: Divergent Integrals in Elastostatics. Regularization in 2-D case. Electron. J. Bound. Elem. 7(2), 50–88 (2009)

    MathSciNet  Google Scholar 

  116. Zozulya, V.V.: Regularization of hypersingular integrals in 3-D fracture mechanics: triangular BE, and piecewise-constant and piecewise-linear approximations. Engineering Analysis with Boundary Elements 34(2), 105–113 (2010)

    MATH  MathSciNet  Google Scholar 

  117. Zozulya, V.V. In: Constanda, C., Perez, M.E. (eds.) : Regularization of the divergent integrals in boundary integral equations for elastostatics. In: Integral Methods in Science and Engineering. Vol. 1. Analytic Methods, pp 333–347, Birkhäuser (2010)

  118. Zozulya V.V. Regularization of the divergent integrals in boundary integral equations. In: Advances in Boundary Element Techniques. In: Zhang, Ch., Aliabadi, M.H., Schanz, M. (eds.) , pp 561–568. Published by EC, Ltd, UK (2010)

  119. Zozulya, V.V.: Divergent integrals in elastostatics: regularization in 3-D case. CMES: Comput. Model. Eng. Sci. 70(3), 253–349 (2010)

    MATH  MathSciNet  Google Scholar 

  120. Zozulya, V.V.: Divergent integrals in elastostatics: general considerations. ISRN Applied Mathematics. Article ID 726402, 25 (2011)

    Google Scholar 

  121. Zozulya, V.V.: An approach based on generalized functions to regularize divergent integrals, Engineering Analysis with Boundary Elements 40, 162–180 (2014). (submitted)

  122. Zozulya, V.V., Gonzalez-Chi, P.I.: Weakly singular, singular and hypersingular integrals in elasticity and fracture mechanics. J. Chin. Inst. Eng. 22(6), 763–775 (1999)

    MathSciNet  Google Scholar 

  123. Zozulya, V.V., Lukin, A.N.: Solution of three-dimensional problems of fracture mechanics by the method of integral boundary equations. Int. Appl. Mech. 34(6), 544–551 (1998)

    MathSciNet  Google Scholar 

  124. Zozulya, V.V., Men’shikov, V.A.: Solution of three dimensional problems of the dynamic theory of elasticity for bodies with cracks using hypersingular integrals. Int. Appl. Mech. 36(1), 74–81 (2000)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Centro de Investigacion Cientifica de Yucatan A.C., Calle 43, No 130, Colonia: Chuburná de Hidalgo, C.P. 97200, Mérida, Yucatán, México

    V. V. Zozulya

Authors
  1. V. V. Zozulya
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. V. Zozulya.

Additional information

Communicated by: Leslie Greengard

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zozulya, V.V. Regularization of divergent integrals: A comparison of the classical and generalized-functions approaches. Adv Comput Math 41, 727–780 (2015). https://doi.org/10.1007/s10444-014-9399-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10444-014-9399-3

Keywords

Mathematics Subject Classification (2010)

Search

Navigation