International Journal of Nonlinear Analysis and Applications

Legendre Kantorovich methods for Uryshon integral equations

Document Type : Research Paper

Authors

1 University Mohammed I, FPN, MSC Team, LAMAO Laboratory, Nador, Morocco

2 University Mohammed I, ESTO, ANAA Team, ANO Laboratory, Oujda, Morocco

Abstract

In this paper, the Kantorovich method for the numerical solution of nonlinear \emph{Uryshon} equations with a smooth kernel is considered. The approximating operator is chosen to be either the orthogonal projection or an interpolatory projection using a Legendre polynomial basis. 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.

Keywords

[1] C. Allouch, D. Sbibih and M. Tahrichi, Legendre superconvergent Galerkin-collocation type methods for Hammerstein equations, J. Comp. Appl. Math, 353 (2019) 253–264.
[2] C. Allouch, D. Sbibih and M. Tahrichi, Superconvergent Nystr¨om method for Urysohn integral equations, BIT Numerical Math. 57 (2017) 3–20.
[3] K. E. Atkinson and J. Flores, The discrete collocation method for nonlinear integral equations, IMA J. Numerical Anal. 13 (1993) 195–213.
[4] K.E. Atkinson, The Numerical Solution of Integral Equations of the Second Kind, Cambridge University Press, Cambridge, 1997.
[5] M. Golberg and C. Chen, Discrete Projection Methods for Integral Equations, Computational Mechanics Publications, 1997.
[6] K. Atkinson, F. Potra, Projection and iterated projection methods for nonlinear integral equations, SIAM J. Numer. Anal. 24 (1987) 1352–1373.
[7] P. Das, M. Sahani, G. Nelakanti and G. Long, Legendre spectral projection methods for Fredholm-Hammerstein integral equations, J. Sci. Comput. 68 (2016) 213–230.
[8] P. Das, G. Nelakanti and G. Long, Discrete Legendre spectral Galerkin method for Urysohn integral equations, Int. J. Comput. Math. 95 (2018) 465–489.
[9] P. Das, G. Nelakanti and G. Long, Discrete Legendre spectral projection methods for Fredholm-Hammerstein integral equations, J. Comput. Appl. Math. 278 (2015) 293–305.
[10] S. Kumar, The numerical solution of Hammerstein equations by a method based on polynomial collocation, Aust. Math. Soc. J. Ser. B Appl. Math. 31 (1990) 319–329.
[11] M. Golberg, Improved convergence rates for some discrete Galerkin methods, Meth. J. Int. Eqns. Appl. 8 (1996) 307- 335.
[12] I.H. Sloan, Polynomial interpolation and hyper interpolation over general regions, J. App. Theory 83 (1995) 238–254.
[13] L. Grammont, R.P. Kulkarni and P.B. Vasconcelos, the iterated modified projection methods for nonlinear integral equations, J. Int. Eq. Appl. 25 (2013) 481–516.
[14] Q. Lin, I.H. Sloan and R. Xie, Extrapolation of the iterated-collocation method for integral equations of the second kind, SIAM J. Numer. Anal. 6 (1990) 1535–1541.
[15] E. Schock, Galerkin-like methods for equations of the second kind, J. Int. Eqns. Appl. 4 (1982) 361-364.
[16] I.H. Sloan, Error analysis for a class of degenerate kernel methods, Numer. Math. 25 (1976) 231–238.
[17] F. Riesz and B.S. Nagy, Functional Analysis, Frederick Ungar Pub, New York, 1955.
[18] G.M. Vainikko, Galerkin’s perturbation method and the general theory of approximate methods for non-linear equations, USSR Comput. Math. Math. Phys. 7 (1967) 1–41.
International Journal of Nonlinear Analysis and Applications
Volume 13, Issue 1
March 2022
Pages 143-157
  • Receive Date: 23 January 2021
  • Revise Date: 15 March 2021
  • Accept Date: 26 May 2021