Legendre–Kantorovich method for Fredholm integral equations of the second kind

In the present paper, we consider polynomially based Kantorovich method for the numerical solution of Fredholm integral equation of the second kind with a smooth kernel.  The used projection is  either the orthogonal projection or an interpolatory projection using Legendre polynomial bases.  The order of convergence of the proposed method and those of superconvergence of the iterated versions are established.  We show that these orders of convergence are valid in the corresponding discrete methods obtained by replacing the integration by a quadrature rule.  Numerical examples are given to illustrate the theoretical estimates.

інтегральне рівняння Фредгольма
projection operator
Legendre polynomial
superconvergence
quadrature rule
discrete method
  1. Atkinson K. E.  The Numerical Solution of Integral Equations of the Second Kind.  Cambridge University Press, Cambridge (1997).
  2. Atkinson K. E., Han W.  Theoretical numerical analysis.  Springer Verlag, Berlin (2005).
  3. Kantorovich L., Krylov V.  Approximate Methods of Higher Analysis.  Noordhoff, Groningen, The Netherlands (1964).
  4. Golberg M.  Discrete Polynomial-Based Galerkin Methods for Fredholm Integral Equations.  Journal of Integral Equations and Applications. 6 (2), 197–211 (1994).
  5. Kulkarni R. P., Nelakanti G.  Iterated discrete polynomially based Galerkin methods.  Applied Mathematics and Computation. 146 (1), 153–165 (2003).
  6. Nelakanti G., Panigrahi B. L.  Legendre Galerkin method for weakly singular Fredholm integral equations and the corresponding eigenvalue problem.  Journal of Applied Mathematics and Computing. 43, 175–197 (2013).
  7. Long G., Nelakanti G., Sahani M. M.  Polynomially based multi-projection methods for Fredholm integral equations of the second kind.  Applied Mathematics and Computation. 215 (1), 147–155 (2009).
  8. Das P., Nelakanti G.  Convergence analysis of discrete Legendre spectral projection methods for Hammerstein integral equations of mixed type.  Applied Mathematics and Computation. 265, 574–601 (2015).
  9. Das P., Nelakanti G.  Error analysis of discrete legendre multi-projection methods for nonlinear Fredholm integral equations.  Numerical Functional Analysis and Optimization. 38 (5), 549–574 (2017).
  10. Das P., Long G., Nelakanti G.  Discrete Legendre spectral projection methods for Fredholm–Hammerstein integral equations.  Journal of Computational and Applied Mathematics. 278, 293–305 (2015).
  11. Chen C., Golberg M.  Discrete projection methods for integral equations.  Computational Mechanics Publications (1997).
  12. Sloan I. H.  Four variants of the Gaterkin method for Integral equations of the second kind.  IMA Journal of Numerical Analysis. 4 (1), 9–17 (1984).
  13. Golberg M.  Improved convergence rates for some discrete Galerkin methods.  Journal of Integral Equations and Applications. 8 (3), 307–335 (1996).
  14. Sloan I. H.  Polynomial interpolation and hyperinterpolation over general regions.  Journal of Approximation Theory. 83 (2), 238–254 (1995).