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Calcolo
Published in: Calcolo 2/2023

01-06-2023

Spectral approximation methods for nonlinear integral equations with non-smooth kernels

Authors: C. Allouch, D. Sbibih, M. Tahrichi

Published in: Calcolo | Issue 2/2023

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Abstract

In this paper, polynomially based projection and modified projection methods for approximating the solution of Uryshon integral equations with a kernel of Green’s function type are proposed. The projection is either an orthogonal projection or an interpolatory projection using Legendre polynomial basis. The orders of convergence of these methods and the superconvergence of their iterated versions are analyzed. A numerical example is given to illustrate the theoretical results.
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Literature
1.
Ahues, M., Largillier, A., Limaye, B.: Spectral Computations for Bounded Operators. CRC Press, Boca Raton (2001)CrossRefMATH
2.
Allouch, C., Sbibih, D., Tahrichi, M.: Legendre superconvergent Galerkin-collocation type methods for Hammerstein equations. J. Comp. Appl. Math. 353, 253–264 (2018)MathSciNetCrossRefMATH
3.
Allouch, C., Sbibih, D., Tahrichi, M.: Superconvergent projection methods for Hammestein integral equations. J. Int. Equ. Appl. 31(1), 1–28 (2019)MATH
4.
Atkinson, K., Potra, F.: Projection and iterated projection methods for nonlinear integral equations. SIAM J. Numer. Anal. 24, 1352–1373 (1987)MathSciNetCrossRefMATH
5.
Atkinson, K., Potra, F.: The discrete Galerkin method for nonlinear integral equations. J. Int. Equ. Appl. 1(1), 17–54 (1988)MathSciNetMATH
6.
Chatelin, F., Lebbar, R.: Superconvergence results for the iterated projection method applied to a Fredholm integral equation of the second kind and the corresponding eigenvalue problem. J. Int. Equ. Appl. 6, 71–91 (1984)MathSciNetMATH
7.
Das, P., Nelakanti, G.: Error analysis of discrete legendre multi-projection methods for nonlinear Fredholm integral equations. Numer. Funct. Anal. Optim. 38(6), 1–26 (2017)MathSciNetMATH
8.
Das, P., Sahani, M., Nelakanti, G.: Legendre spectral projection methods for Urysohn integral equations. J. Comp. Appl. Math. 263, 88–102 (2014)MathSciNetCrossRefMATH
9.
Das, P., Sahani, M., Nelakanti, G., Long, G.: Legendre spectral projection methods for Fredholm–Hammerstein integral equations. J. Sci. Comput. 68, 213–230 (2016)MathSciNetCrossRefMATH
10.
Golberg, M.A.: Discrete polynomial-based Galerkin methods for Fredholm integral equations. J. Int. Equ. Appl. 6, 197–211 (1994)MathSciNetMATH
11.
Golberg, M.A., Chen, C.S.: Discrete Projection Methods for Integral Equations. Computational Mechanics Publications, Boston (1997)MATH
12.
Grammont, L., Kulkarni, R.P., Vasconcelos, P.B.: Modified projection and the iterated modified projection methods Afor nonlinear integral equations. J. Int. Equ. Appl. 25(4) (2013)
13.
Grammont, L., Kulkarni, R.P., Nidhin, T.J.: Modified projection method for Urysohn integral equations with non-smooth kernels. J. Comp. Appl. Math. 294, 309–322 (2016)MathSciNetCrossRefMATH
14.
Kulkarni, R.P., Nidhin, T.J.: Approximate solution of Uryshon integral equations with non-smooth kernels. J. Int. Equ. Appl. 28(2), 221–261 (2016)MATH
15.
Kulkarni, R.P., Rakshit, G.: Discrete modified projection methods for Urysohn integral equations with Green’s function type kernels. Math. Model. Anal. 25(3), 421–440 (2020)MathSciNetCrossRefMATH
16.
17.
Kumar, S.: The numerical solution of Hammestein equations by a method based on polynomial collocation. J. Austral. Math. Soc. Ser. B 31, 319–329 (1990)MathSciNetCrossRef
18.
19.
Mandal, M., Nelakanti, G.: Superconvergence of Legendre spectral projection methods for Fredholm–Hammerstein integral equations. J. Comp. Appl. Math. 319, 423–439 (2017)MathSciNetCrossRefMATH
20.
Mandal, M., Nelakanti, G.: Superconvergence results of Legendre spectral projection methods for weakly singular Fredholm–Hammerstein integral equations. J. Comp. Appl. Math. 349, 114–131 (2019)MathSciNetCrossRefMATH
21.
Panigrahi, B.L.: Error analysis of Jacobi spectral collocation methods for Fredholm–Hammerstein integral equations with weakly singular kernel. Int. J. Comput. Math. 96, 1230–1253 (2019)MathSciNetCrossRefMATH
22.
23.
Riesz, F., Nagy, B.S.: Functional Analysis. Frederick Ungar Pub, New York (1955)MATH
24.
Schock, E.: Galerkin-like methods for equations of the second kind. J. Int. Equ. Appl. 4, 361–364 (1982)MathSciNetMATH
25.
Schumaker, L.L.: Spline Functions: Basic Theory. John Wiley & Sons, New York (1981)MATH
26.
27.
28.
Tang, T., Xu, X., Cheng, J.: On spectral methods for Volterra type integral equations and the convergence analysis. J. Comput. Math. 26(6), 825–837 (2008)MathSciNetMATH
Metadata
Title
Spectral approximation methods for nonlinear integral equations with non-smooth kernels
Authors
C. Allouch
D. Sbibih
M. Tahrichi
Publication date
01-06-2023
Publisher
Springer International Publishing
Published in
Calcolo / Issue 2/2023
Print ISSN: 0008-0624
Electronic ISSN: 1126-5434
DOI
https://doi.org/10.1007/s10092-023-00519-3

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