Legendre Spectral Projection Methods for Hammerstein Integral Equations with Weakly Singular Kernel

  • Bijaya Laxmi PanigrahiEmail author
Original Paper


In this paper, we consider the Legendre Galerkin and Legendre collocation methods for solving the Fredholm–Hammerstein integral equation with weakly singular kernels. We evaluate the convergence rates for both the methods in both \(L^2\) and infinity-norm. To improve the convergence rates, iterated Legendre Galerkin and iterated Legendre collocation methods have been considered. We prove that iterated Legendre Galerkin methods converge faster than Legendre Galerkin methods in both \(L^2\) and infinity-norm. Numerical examples are presented to validate the theoretical estimate.


Hammerstein integral equations Weakly singular kernels Spectral method Galerkin method Collocation method Legendre polynomials 



The author takes this opportunity to thank the Reviewers for their valuable suggestions which improve the version of the paper.


  1. 1.
    Atkinson, K.E.: The Numerical Solution of Integral Equations of the Second Kind. Cambridge University Press, Cambridge (1997)CrossRefGoogle Scholar
  2. 2.
    Brezis, H., Browder, F.E.: Some new results about Hammerstein equations. Bull. Am. Math. Soc. 80, 567–572 (1974)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Brezis, H., Browder, F.E.: Existence theorems for nonlinear integral equations of Hammerstein type. Bull. Am. Math. Soc. 81, 73–78 (1975)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Brezis, H., Browder, F.E.: Nonlinear integral equations and systems of Hammerstein type. Bull. Am. Math. Soc. 82, 115–147 (1976)CrossRefGoogle Scholar
  5. 5.
    Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods: Fundamentals in Single Domains. Springer, Berlin (2006)zbMATHGoogle Scholar
  6. 6.
    Chidumea, C.E., Djitté, N.: Iterative approximation of solutions of nonlinear equations of Hammerstein type. Nonlinear Anal. 70, 4086–4092 (2009)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Cianciaruso, F., Infante, G., Pietramala, P.: Solutions of perturbed Hammerstein integral equations with applications. Nonlinear Anal. Real World Appl. 33, 317–347 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Das, P., Sahani, M.M., Nelakanti, G., Long, G.: Legendre spectral projection methods for Fredholm Hammerstein integral equations. J. Sci. Comput. 68, 213–230 (2016)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Ezquerro, J.A., Hernández-Verón, M.A.: The majorant principle applied to Hammerstein integral equations. Appl. Math. Lett. 75, 50–58 (2018)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Guo, B.: Spectral Methods and Their Applications. World Scientific, Singapore (1998)CrossRefGoogle Scholar
  11. 11.
    Huang, Q., Xie, H.: Superconvergence of Galerkin solutions for Hammerstein equations. Int. J. Numer. Anal. Model. 6(4), 696–710 (2009)MathSciNetGoogle Scholar
  12. 12.
    Huang, Q., Zhang, S.: Superconvergence of interpolated collocation solutions for Hammerstein equations. Numer. Methods Partial Differ. Equ. 26(2), 290–304 (2010)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Kaneko, H., Noren, R.D., Padilla, P.A.: Superconvergence of the iterated collocation methods for Hammerstein equations. J. Comput. Appl. Math. 80(2), 335–349 (1997)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kaneko, H., Xu, Y.: Superconvergence of the iterated Galerkin methods for Hammerstein equations. SIAM J. Numer. Anal. 33(3), 1048–1064 (1996)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Kaneko, H., Noren, R., Xu, Y.: Regularity of the solution of Hammerstein equations with weakly singular kernels. Integral Equ. Oper. Theory 13(5), 660–670 (1990)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kumar, S.: The numerical solution of Hammerstein equations by a method based on polynomial collocation. J. Aust. Math. Soc. Ser. B 31(3), 319–329 (1990)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Kumar, S.: Superconvergence of a collocation-type method for Hammerstein equations. IMA J. Numer. Anal. 7(3), 313–325 (1987)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Minjibir, M.S., Mohammed, I.: Iterative algorithms for solutions of Hammerstein integral inclusions. Appl. Math. Comput. 320, 389–399 (2018)MathSciNetGoogle Scholar
  19. 19.
    Panigrahi, B.L., Nelakanti, G.: Legendre Galerkin method for weakly singular Fredholm integral equations and the corresponding eigenvalue problem. J. Appl. Math. Comput. 43, 175–197 (2013)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Vainikko, G.M.: A perturbed Galerkin method and the general theory of approximate methods for non-linear equations. USSR Comput. Math. Phys. 7(4), 1–41 (1967)CrossRefGoogle Scholar
  21. 21.
    Panigrahi, B.L.: Error analysis of Jacobi spectral collocation methods for Fredholm-Hammerstein integral equations with weakly singular kernel. Int J Comput Math (2018).

Copyright information

© Springer Nature India Private Limited 2018

Authors and Affiliations

  1. 1.Department of MathematicsSambalpur UniversitySambalpurIndia

Personalised recommendations