Skip to main content
SpringerLink
Log in

Optimal and Approximate Solutions of Singular Integral Equations by Means of Reproducing Kernels

  • Published:
Complex Analysis and Operator Theory Aims and scope Submit manuscript

Abstract

A reproducing kernel method is proposed to obtain the optimal and approximate solutions of Carleman singular integral equations. Therefore, we will be mostly interested in singular integral equations with a Cauchy type kernel and whose coefficients are real or complex valued functions. The new method and corresponding concepts allow the analysis of associated discrete singular integral equations and corresponding inverse source problems in appropriate frameworks.

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. Abdou, M.A., Nasr, A.A.: On the numerical treatment of the singular integral equation of the second kind. Appl. Math. Comput. 146, 373–380 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Atkinson, K.E.: The numerical evaluation of singular integrals of Cauchy type. SIAM J. Numer. Anal. 9, 284–299 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  3. Atkinson, K.E.: The Numerical Solution of Integral Equations of the Second Kind. Cambridge University Press, Cambridge (1997)

    Book  MATH  Google Scholar 

  4. Bogveradze, G., Castro, L.P.: Toeplitz plus Hankel operators with infinite index. Integral Equ. Oper. Theory 62, 43–63 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Carleman, T.: Sur la résolution de certaines équations intégrales. Arkiv. Mat. Astron. Fys. 16(26), 1–19 (1922)

    Google Scholar 

  6. Castro, L.P., Chen, Q., Saitoh, S.: Source inversion of heat conduction from a finite number of observation data. Appl. Anal. 89, 801–813 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Castro, L.P., Rojas, E.M.: Reduction of singular integral operators with flip and their Fredholm property. Lobachevskii J. Math. 29(3), 119–129 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Castro, L.P., Rojas, E.M.: Similarity transformation methods for singular integral operators with reflection on weighted Lebesgue spaces. Int. J. Mod. Math. 3, 295–313 (2008)

    MathSciNet  MATH  Google Scholar 

  9. Castro, L.P., Rojas, E.M.: A collocation method for singular integral operators with reflection. RIMS Kôkyûroku 1719, 155–167 (2010)

    Google Scholar 

  10. Castro, L.P., Saitoh, S., Sawano, Y., Simões, A.M.: General inhomogeneous discrete linear partial differential equations with constant coefficients on the whole spaces. Complex Anal. Oper. Theory 6, 307–324 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Castro, L.P., Saitoh, S., Sawano, Y., Silva, A.S.: Discrete linear differential equations. Analysis 32, 18 (2012). doi:10.1524/anly.2012.1104

    Article  MathSciNet  Google Scholar 

  12. Cotterell, B., Rice, J.R.: Slightly curved or kinked cracks. Int. J. Fract. 16, 155–169 (1980)

    Article  Google Scholar 

  13. Estrada, R., Kanwal, R.P.: Singular Integral Equations. Birkhäuser, Boston (2000)

    Book  MATH  Google Scholar 

  14. Fujiwara, H.: High-accurate numerical method for integral equation of the first kind under multiple-precision arithmetic. Theor. Appl. Mech. Jpn. 52, 192–203 (2003)

    Google Scholar 

  15. Fujiwara, H., Matsuura, T., Saitoh, S., Sawano, S.Y.: Numerical Real Inversion of the Laplace Transform by Using a High-Accuracy Numerical Method. Further Progress in Analysis, pp. 574–583. World Sci. Publ., Hackensack (2009)

  16. Fujiwara, H.: Numerical Real Inversion of the Laplace Transform by Reproducing Kernel and Multiple-Precision Arithmetic. Progress in Analysis and Its Applications. Proceedings of the 7th International ISAAC Congress, pp. 289–295. World Scientific, Singapore (2010)

  17. Gerasoulis, A.: Piecewise-polynomial quadratures for Cauchy singular integrals. SIAM J. Numer. Anal. 23, 891–901 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  18. Higgins, J.R.: A Sampling Principle Associated with Saitoh’s Fundamental Theory of Linear Transformations. Analytic Extension Formulas and Their Applications (International Society for Analysis, Applications and Computing, vol. 9), pp. 73–86. Kluwer, Dordrecht (2001)

  19. Hills, D.A., Nowell, D., Sackfield, A.: Mechanics of Elastic Contacts. Butterworth-Heinemann, Oxford (1993)

    Google Scholar 

  20. Hills, D.A., Kelly, P.A., Dai, D.N., Korsunsky, A.M.: Solution of Crack Problems: The Distributed Dislocation Technique. Kluwer, Dordrecht (1996)

    Book  MATH  Google Scholar 

  21. Hochstadt, H.: Integral Equations. Wiley, New York (1973)

    MATH  Google Scholar 

  22. Itou, H., Saitoh, S.: Analytical and numerical solutions of linear singular integral equations. Int. J. Appl. Math. Stat. 12, 76–89 (2007)

    MathSciNet  MATH  Google Scholar 

  23. Jin, X.Q., Keer, L.M., Wang, Q.: A practical method for singular integral equations of the second kind. Eng. Fract. Mech. 75, 1005–1014 (2008)

    Article  Google Scholar 

  24. Karelin, A.A.: Applications of operator equalities to singular integral operators and to Riemann boundary value problems. Math. Nachr. 280, 1108–1117 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kurtz, R.D., Farris, T.N., Sun, C.T.: The numerical solution of Cauchy singular integral equations with application to fracture. Int. J. Fract. 66, 139–154 (1994)

    Article  Google Scholar 

  26. Litvinchuk, G.S.: Solvability Theory of Boundary Value Problems and Singular Integral Equations with Shift. Kluwer, Dordrecht (2000)

    Book  MATH  Google Scholar 

  27. MacCamy, R.C.: Pairs of Carleman-type integral equations. Proc. Am. Math. Soc. 16, 871–875 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  28. Mandal, B.N., Bera, G.H.: Approximate solution of a class of singular integral equations of second kind. J. Comput. Appl. Math. 206, 189–195 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  29. Mikhlin, S.G., Prössdorf, S.: Singular Integral Operators. Springer, Berlin (1986)

    Book  Google Scholar 

  30. Muskhelishvili, N.I.: Singular Integral Equations. Noordhoff, Groningen (1972)

    Google Scholar 

  31. Orton, M.: Distributions and singular integral equations of Carleman type. Appl. Anal. 9, 219–231 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  32. Saitoh, S.: Integral Transforms, Reproducing Kernels and their Applications. Pitman Research Notes in Mathematics Series, vol. 369. Addison Wesley Longman Ltd, UK (1997)

  33. Saitoh, S.: Hilbert spaces induced by Hilbert space valued functions. Proc. Am. Math. Soc. 89, 74–78 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  34. Saitoh, S.: Theory of reproducing kernels: applications to approximate solutions of bounded linear operator functions on Hilbert spaces. AMS Trans. Ser. 2 230, 107–134 (2010)

    MathSciNet  Google Scholar 

  35. Stenger, F.: Numerical Methods Based on Sinc and Analytic Functions. Springer Series in Computational Mathematics, vol. 20. Springer, New York (1993)

    Book  Google Scholar 

  36. Tricomi, F.G.: Integral Equations (reprint of the 1957 original). Dover, New York (1985)

    Google Scholar 

Download references

Acknowledgments

This work was supported in part by FEDER funds through COMPETE-Operational Programme Factors of Competitiveness (“Programa Operacional Factores de Competitividade”) and by Portuguese funds through the Center for Research and Development in Mathematics and Applications and the Portuguese Foundation for Science and Technology (“FCT-Fundação para a Ciência e a Tecnologia”), within project PEst-C/MAT/UI4106/2011 with COMPETE number FCOMP-01-0124-FEDER-022690. S. Saitoh is supported in part by the Grant-in-Aid for the Scientific Research (C)(2) (No. 24540113).

Author information

Authors and Affiliations

  1. Department of Mathematics, Center for Research and Development in Mathematics and Applications, University of Aveiro, 3810-193 , Aveiro, Portugal

    L. P. Castro & S. Saitoh

Authors
  1. L. P. Castro
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Saitoh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to L. P. Castro.

Additional information

Communicated by Daniel Aron Alpay.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Castro, L.P., Saitoh, S. Optimal and Approximate Solutions of Singular Integral Equations by Means of Reproducing Kernels. Complex Anal. Oper. Theory 7, 1839–1851 (2013). https://doi.org/10.1007/s11785-012-0254-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11785-012-0254-6

Keywords

Mathematics Subject Classification (2000)

Search

Navigation