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Superconvergence Results for Volterra-Urysohn Integral Equations of Second Kind

  • Moumita MandalEmail author
  • Gnaneshwar Nelakanti
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 655)

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.

Keywords

Volterra-Urysohn integral equations Smooth kernels Galerkin method Multi-Galerkin method Piecewise polynomials Superconvergence rates 

References

  1. 1.
    Ahues, M., Largillier, A., Limaye, B.: Spectral Computations for Bounded Operators. CRC Press, Boca Raton (2001)CrossRefzbMATHGoogle Scholar
  2. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brunner, H.: Iterated collocation methods and their discretizations for Volterra integral equations. SIAM J. Numer. Anal. 21(6), 1132–1145 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brunner, H., Houwen, P.: The Numerical Solution of Volterra Equation. North-Holland Publishing Co., Amsterdam (1986)zbMATHGoogle Scholar
  5. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chatelin, F.: Spectral Approximation of Linear Operators ci. SIAM (1983)Google Scholar
  7. 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)CrossRefGoogle Scholar
  8. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Day, J.T.: A starting method for solving nonlinear Volterra integral equations. Math. Comput. 21(98), 179–188 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Grammont, L., Kulkarni, R.: A superconvergent projection method for nonlinear compact operator equations. C.R. Math. 342(3), 215–218 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kulkarni, R.P.: A superconvergence result for solutions of compact operator equations. Bull. Aust. Math. Soc. 68(3), 517–528 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kumar, S.: Superconvergence of a collocation-type method for Hammerstein equations. IMA J. Num. Anal. 7(3), 313–325 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 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)MathSciNetzbMATHGoogle Scholar
  15. 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)MathSciNetzbMATHGoogle Scholar
  16. 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)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Rudin, W.: Real and Complex Analysis. Tata McGraw-Hill Education, London (1987)zbMATHGoogle Scholar
  18. 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)MathSciNetzbMATHGoogle Scholar
  19. 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)CrossRefGoogle Scholar
  20. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 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)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of TechnologyKharagpurIndia

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