Skip to main content
SpringerLink
Log in

Superconvergence Results for Volterra-Urysohn Integral Equations of Second Kind

  • Conference paper
  • First Online:
Mathematics and Computing (ICMC 2017)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 655))

Included in the following conference series:

Abstract

In this paper, we consider the Galerkin method to approximate the solution of Volterra-Urysohn integral equations of second kind with a smooth kernel, using piecewise polynomial bases. We show that the exact solution is approximated with the order of convergence \(\mathcal {O}(h^{r})\) for the Galerkin method, whereas the iterated Galerkin solutions converge with the order \(\mathcal {O}(h^{2r})\) in uniform norm, where h is the norm of the partition and r is the smoothness of the kernel. For improving the accuracy of the approximate solution of the integral equation, the multi-Galerkin method is also discussed here and we prove that the exact solution is approximated with the order of convergence \(\mathcal {O}(h^{3r})\) in uniform norm for iterated multi-Galerkin method. Numerical examples are given to illustrate the theoretical results.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Ahues, M., Largillier, A., Limaye, B.: Spectral Computations for Bounded Operators. CRC Press, Boca Raton (2001)

    Book  MATH  Google Scholar 

  2. Blom, J., Brunner, H.: The numerical solution of nonlinear Volterra integral equations of the second kind by collocation and iterated collocation methods. SIAM J. Sci. Stat. Comput. 8(5), 806–830 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  3. Brunner, H.: Iterated collocation methods and their discretizations for Volterra integral equations. SIAM J. Numer. Anal. 21(6), 1132–1145 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  4. Brunner, H., Houwen, P.: The Numerical Solution of Volterra Equation. North-Holland Publishing Co., Amsterdam (1986)

    MATH  Google Scholar 

  5. Brunner, H., Nørsett, S.P.: Superconvergence of collocation methods for Volterra and Abel integral equations of the second kind. Numer. Math. 36(4), 347–358 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chatelin, F.: Spectral Approximation of Linear Operators ci. SIAM (1983)

    Google Scholar 

  7. Chen, L., Duan, J.: Multistage numerical picard iteration methods for nonlinear Volterra integral equations of the second kind. Adv. Pure Math. 5(11), 672 (2015)

    Article  Google Scholar 

  8. Chen, Z., Long, G., Nelakanti, G.: The discrete multi-projection method for Fredholm integral equations of the second kind. J. Integr. Equ. Appl. 19(2), 143–162 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Day, J.T.: A starting method for solving nonlinear Volterra integral equations. Math. Comput. 21(98), 179–188 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  10. Grammont, L., Kulkarni, R.: A superconvergent projection method for nonlinear compact operator equations. C.R. Math. 342(3), 215–218 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Grammont, L., Kulkarni, R.P., Vasconcelos, P.B.: Modified projection and the iterated modified projection methods for nonlinear integral equations. J. Integr. Equ. Appl. 25(4), 481–516 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kulkarni, R.P.: A superconvergence result for solutions of compact operator equations. Bull. Aust. Math. Soc. 68(3), 517–528 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kumar, S.: Superconvergence of a collocation-type method for Hammerstein equations. IMA J. Num. Anal. 7(3), 313–325 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  14. Long, G., Sahani, M.M., Nelakanti, G.: Polynomially based multi-projection methods for Fredholm integral equations of the second kind. Appl. Math. Comput. 215(1), 147–155 (2009)

    MathSciNet  MATH  Google Scholar 

  15. Maleknejad, K., Sohrabi, S., Rostami, Y.: Numerical solution of nonlinear Volterra integral equations of the second kind by using chebyshev polynomials. Appl. Math. Comput. 188(1), 123–128 (2007)

    MathSciNet  MATH  Google Scholar 

  16. Mohsen, A., El-Gamel, M.: On the numerical solution of linear and nonlinear Volterra integral and integro-differential equations. Appl. Math. Comput. 217(7), 3330–3337 (2010)

    MathSciNet  MATH  Google Scholar 

  17. Rudin, W.: Real and Complex Analysis. Tata McGraw-Hill Education, London (1987)

    MATH  Google Scholar 

  18. Tang, T., Xu, X., Cheng, J.: On spectral methods for Volterra integral equations and the convergence analysis. J. Comput. Math. 26(6), 825–837 (2008)

    MathSciNet  MATH  Google Scholar 

  19. Vainikko, G.M.: Galerkin’s perturbation method and the general theory of approximate methods for non-linear equations. USSR Comput. Math. Math. Phys. 7(4), 1–41 (1967)

    Article  Google Scholar 

  20. Wan, Z., Chen, Y., Huang, Y.: Legendre spectral Galerkin method for second-kind Volterra integral equations. Front. Math. China 4(1), 181–193 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Xie, Z., Li, X., Tang, T.: Convergence analysis of spectral Galerkin methods for Volterra type integral equations. J. Sci. Comput. 53(2), 414–434 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Zhang, S., Lin, Y., Rao, M.: Numerical solutions for second-kind Volterra integral equations by Galerkin methods. Appl. Math. Comput. 45(1), 19–39 (2000)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology, Kharagpur, 721 302, India

    Moumita Mandal & Gnaneshwar Nelakanti

Authors
  1. Moumita Mandal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gnaneshwar Nelakanti
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Moumita Mandal .

Editor information

Editors and Affiliations

  1. Haldia Institute of Technology, Haldia, India

    Debasis Giri

  2. University of Central Florida, Orlando, Florida, USA

    Ram N. Mohapatra

  3. Freie Universität Berlin, Berlin, Germany

    Heinrich Begehr

  4. Fordham University, Bronx, New York, USA

    Mohammad S. Obaidat

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer Nature Singapore Pte Ltd.

About this paper

Cite this paper

Mandal, M., Nelakanti, G. (2017). Superconvergence Results for Volterra-Urysohn Integral Equations of Second Kind. In: Giri, D., Mohapatra, R., Begehr, H., Obaidat, M. (eds) Mathematics and Computing. ICMC 2017. Communications in Computer and Information Science, vol 655. Springer, Singapore. https://doi.org/10.1007/978-981-10-4642-1_31

Download citation

  • DOI: https://doi.org/10.1007/978-981-10-4642-1_31

  • Published:

  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-10-4641-4

  • Online ISBN: 978-981-10-4642-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation