Legendre spectral Galerkin and multi-Galerkin methods for nonlinear Volterra integral equations of Hammerstein type

  • Moumita MandalEmail author
  • Gnaneshwar Nelakanti
Original Research Paper


In this paper, we discuss the superconvergence of the Galerkin solutions for second kind nonlinear integral equations of Volterra–Hammerstein type with a smooth kernel. Using Legendre polynomial bases, we obtain order of convergence \({\mathcal{O}}(n^{-r})\) for the Legendre Galerkin method in both \(L^2\)-norm and infinity norm, where n is the highest degree of the Legendre polynomial employed in the approximation and r is the smoothness of the kernel. The iterated Legendre Galerkin solutions converge with the order \({\mathcal{O}}(n^{-2r}),\) whose convergence order is the same as that of the multi-Galerkin solutions. We also prove that iterated Legendre multi-Galerkin method has order of convergence \({\mathcal{O}}(n^{-3r})\) in both \(L^2\)-norm and infinity norm. Numerical examples are given to demonstrate the efficacy of Galerkin and multi-Galerkin methods.


Volterra–Hammerstein integral equations Smooth kernels Legendre polynomial Galerkin method Multi-Galerkin method Superconvergence rates 

Mathematics Subject Classification

45B05 45G10 65R20 


Compliance with ethical standards

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.


  1. 1.
    Brunner, H. 1992. Implicitly linear collocation methods for nonlinear Volterra equations. Applied Numerical Mathematics 9 (3): 235–247.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Brunner, H. 2004. Collocation methods for Volterra integral and related functional differential equations. Cambridge: Cambridge University Press.CrossRefzbMATHGoogle Scholar
  3. 3.
    Brunner, H., and P.J. Houwen. 1986. The numerical solution of Volterra equations. Amsterdam: Elsevier Science Ltd.zbMATHGoogle Scholar
  4. 4.
    Burton, A.J., G.F. Miller., and Hardy J. Wilkinson. 1971. The application of integral equation methods to the numerical solution of some exterior boundary-value problems. In Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, vol. 323.
  5. 5.
    Canuto, C., M.Y. Hussaini, A. Quarteroni, and T.A. Zang. 2006. Spectral methods: Fundamentals in single domains. Berlin: Springer.zbMATHGoogle Scholar
  6. 6.
    Chen, Z., J. Li, and Y. Zhang. 2011. A fast multiscale solver for modified Hammerstein equations. Applied Mathematics and Computation 218: 3057–3067.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Das, P., and G. Nelakanti. 2016. Error analysis of polynomial-based multi-projection methods for a class of nonlinear Fredholm integral equations. Journal of Applied Mathematics and Computing 56: 1–24.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Das, P., M.M. Sahani, G. Nelakanti, and G. Long. 2016. Legendre spectral projection methods for Fredholm–Hammerstein integral equations. Journal of Scientific Computing 68 (1): 213–230.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Elnagar, G.N., and M. Kazemi. 1996. Chebyshev spectral solution of nonlinear Volterra–Hammerstein integral equations. Journal of Computational and Applied Mathematics 76 (5): 147–158.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Guo, B.Y. 1998. Spectral methods and their applications. Singapore: World Scientific.CrossRefzbMATHGoogle Scholar
  11. 11.
    Ghoreishi, F., and M. Hadizadeh. 2009. Numerical computation of the tau approximation for the Volterra–Hammerstein integral equations. Numerical Algorithms 52 (4): 541–559.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Jerri, A. 1999. Introduction to integral equations with applications. Hoboken: Wiley.zbMATHGoogle Scholar
  13. 13.
    Kumar, S. 1987. Superconvergence of a collocation-type method for Hammerstein equations. IMA Journal of Numerical Analysis 7 (3): 313–325.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kumar, S., and I.H. Sloan. 1987. A new collocation-type method for Hammerstein integral equations. Mathematics of Computation 48: 585–593.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Mario, A., L. Alain, and L. Balmohan. 2001. Spectral computations for bounded operators. Boca Raton: CRC Press.zbMATHGoogle Scholar
  16. 16.
    Maleknejad, K., S. Sohrabi, and Y. Rostami. 2007. Numerical solution of nonlinear Volterra integral equations of the second kind by using Chebyshev polynomials. Applied Mathematics and Computation 188 (1): 123–128.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Maleknejad, K., and P. Torabi. 2012. Application of fixed point method for solving nonlinear Volterra–Hammerstein integral equation. University “Politehnica” of Bucharest Scientific Bulletin, Series A: Applied Mathematics and Physics 74 (1): 45–56.MathSciNetzbMATHGoogle Scholar
  18. 18.
    Mandal, M., and G. Nelakanti. 2017. Superconvergence of Legendre spectral projection methods for Fredholm–Hammerstein integral equations. Journal of Computational and Applied Mathematics 319: 423–439.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Mandal, M., and G. Nelakanti. 2017. Superconvergence results for linear second-kind Volterra integral equations. Journal of Applied Mathematics and Computing 56: 1–14.zbMATHGoogle Scholar
  20. 20.
    Mandal, M., and G. Nelakanti. 2017. Superconvergence results for Volterra–Urysohn integral equations of second kind. In International Conference on Mathematics and Computing, 358–379. Berlin: SpringerGoogle Scholar
  21. 21.
    Mandal, M., and G. Nelakanti. 2017. Superconvergence results of Legendre spectral projection methods for Volterra integral equations of second kind. Journal of Applied Mathematics and Computing 56: 1–16.zbMATHGoogle Scholar
  22. 22.
    Shen, J., T. Tang, and L.L. Wang. 2011. Spectral methods: Algorithms, analysis and applications, vol. 41. Berlin: Springer Science and Business Media.zbMATHGoogle Scholar
  23. 23.
    Tang, T., X. Xu, and J. Cheng. 2008. On spectral methods for Volterra integral equations and the convergence analysis. Journal of Computational Mathematics 26: 825–837.MathSciNetzbMATHGoogle Scholar
  24. 24.
    Vainikko, G.M. 1967. Galerkin’s perturbation method and the general theory of approximate methods for non-linear equations. USSR Computational Mathematics and Mathematical Physics 7 (4): 1–41.CrossRefGoogle Scholar
  25. 25.
    Wan, Z., Y. Chen, and Y. Huang. 2009. Legendre spectral Galerkin method for second-kind Volterra integral equations. Frontiers of Mathematics in China 4 (1): 181–193.MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Weeen, F.Vander. 1982. Application of the boundary integral equation method to Reissner’s plate model. International Journal for Numerical Methods in Engineering 18 (1): 1–10.CrossRefzbMATHGoogle Scholar

Copyright information

© Forum D'Analystes, Chennai 2019

Authors and Affiliations

  1. 1.Mathematics Department, SASVIT UniversityVelloreIndia
  2. 2.Mathematics DepartmentIIT KharagpurKharagpurIndia

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