main-content

Erschienen in:

01.12.2020

Discrete projection methods for Hammerstein integral equations on the half-line

verfasst von: Nilofar Nahid, Gnaneshwar Nelakanti

Erschienen in: Calcolo | Ausgabe 4/2020

Einloggen, um Zugang zu erhalten

Abstract

In this paper, we study discrete projection methods for solving the Hammerstein integral equations on the half-line with a smooth kernel using piecewise polynomial basis functions. We show that discrete Galerkin/discrete collocation methods converge to the exact solution with order $${\mathcal {O}}(n^{-min\{r, d\}}),$$ whereas iterated discrete Galerkin/iterated discrete collocation methods converge to the exact solution with order $${\mathcal {O}}(n^{-min\{2r, d\}}),$$ where $$n^{-1}$$ is the maximum norm of the graded mesh and r denotes the order of the piecewise polynomial employed and $$d-1$$ is the degree of precision of quadrature formula. We also show that iterated discrete multi-Galerkin/iterated discrete multi-collocation methods converge to the exact solution with order $${\mathcal {O}}(n^{-min\{4r, d\}})$$. Hence by choosing sufficiently accurate numerical quadrature rule, we show that the convergence rates in discrete projection and discrete multi-projection methods are preserved. Numerical examples are given to uphold the theoretical results.
Literatur
1.
Allouch, C., Sbibih, D., Tahrichi, M.: Legendre superconvergent Galerkin-collocation type methods for Hammerstein equations. J. Comput. Appl. Math. 353, 253–264 (2019)
2.
Allouch, C., Sbibih, D., Tahrichi, M.: Numerical solutions of weakly singular Hammerstein integral equations. Appl. Math. Comput. 329, 118–128 (2018)
3.
Allouch, C., Sbibih, D., Tahrichi, M.: Superconvergent Nyström and degenerate kernel methods for Hammerstein integral equations. J. Comput. Appl. Math. 258, 30–41 (2014)
4.
Amini, S., Sloan, I.H.: Collocation methods for the second kind integral equations with non-compact operators. J. Integral Equ. Appl. 2, 1–30 (1989) CrossRef
5.
Anselone, P.M., Lee, J.W.: Nonlinear integral equations on the half-line. J. Integral Equ. Appl. 4, 1–14 (1992)
6.
Anselone, P.M., Sloan, I.H.: Numerical solutions of integral equations on the half-line. The Wiener-Hopf case. University of NSW, Sydney (1988) MATH
7.
Assari, P.: The numerical solution of Fredholm-Hammerstein integral equations by combining the collocation method and radial basis functions. Filomat 33, 667–682 (2019)
8.
Assari, P., Dehghan, M.: A meshless discrete Galerkin method based on the free shape parameter radial basis functions for solving Hammerstein integral equations. Numer. Math. Theory Methods Appl. 11, 540–568 (2018)
9.
Assari, P.: A meshless local Galerkin method for the numerical solution of Hammerstein integral equations based on the moving least squares technique. J. Appl. Anal. Comput. 9, 75–104 (2019) MathSciNet
10.
Assari, P.: Thin plate spline Galerkin scheme for numerically solving nonlinear weakly singular Fredholm integral equations. Appl. Anal. 98, 2064–2084 (2019)
11.
Assari, P., Asadi-Mehregan, F.: Local multiquadric scheme for solving two-dimensional weakly singular Hammerstein integral equations. Int. J. Numer. Model. Electron. Networks Devices Fields 32, e2488 (2019) CrossRef
12.
Atkinson, K.E., Bogomolny, A.: The discrete Galerkin method for integral equations. Math. Comput. 48, 595–616 (1987)
13.
Atkinson, K.E., Flores, J.: The discrete collocation method for nonlinear integral equations. IMA J. Numer. Anal. 13, 195–213 (1993)
14.
Atkinson, K.E., Potra, F.: The discrete Galerkin method for nonlinear integral equations. J. Integral Equ. Appl. 1, 17–54 (1988)
15.
Browder, F. E.: Nonlinear functional analysis and nonlinear integral equations of Hammerstein and Urysohn type. Contributions to Nonlinear Functional Analysis, pp. 425–500 (1971)
16.
Chandler, G.A., Graham, I.G.: The convergence of Nyström methods for wiener-hopf equations. Numer. Math. 52, 345–364 (1987) CrossRef
17.
Chen, Z., Long, G., Nelakanti, G.: The discrete multi-projection method for Fredholm integral equations of the second kind. J. Integral Equ. Appl. 19, 143–162 (2007)
18.
Corduneanu, C.: Integral equations and stability of feedback systems. Academic Press Inc, Cambridge (1973) MATH
19.
Das, P., Nelakanti, G.: Error analysis of discrete legendre multi-projection methods for nonlinear Fredholm integral equations. Numer. Funct. Anal. Optim. 38, 549–574 (2017)
20.
Das, P., Nelakanti, G.: Discrete legendre spectral Galerkin method for Urysohn integral equations. Int. J. Comput. Math. 95, 465–489 (2018)
21.
Das, P., Nelakanti, G., Long, G.: Discrete Legendre spectral projection methods for Fredholm–Hammerstein integral equations. J. Comput. Appl. Math. 278, 293–305 (2015)
22.
Das, P., Nelakanti, G.: Superconvergence results for the iterated discrete legendre Galerkin method for Hammerstein integral equations. J. Comput. Sci. Comput. Math. 5, 75–83 (2015) CrossRef
23.
Eggermont, P.P.B.: On noncompact Hammerstein integral equations and a nonlinear boundary value problem for the heat equation. J. Integral Equ. Appl. 4, 47–68 (1992)
24.
Finn, G.: Studies in spectral line formation: I. formulation and simple applications. J. Quant. Spectrosc. Radiat. Transfer 8, 1675–1703 (1968) CrossRef
25.
Ganesh, M., Joshi, M.: Numerical solutions of nonlinear integral equations on the half-line. Numer. Funct. Anal. Optim. 10, 1115–1138 (1989)
26.
Graham, I.G., Mendes, W.R.: Nyström-product integration for wiener-hopf equations with applications to radiative transfer. IMA J. Numer. Anal. 9, 261–284 (1989)
27.
Guenther, R.B., Lee, J.W., O’Regan, D.: Boundary value problems on infinite intervals and semiconductor devices. J. Math. Anal. Appl. 116, 335–348 (1986)
28.
Golberg, M.A., Chen, C.S.: Discrete Projection Methods for Integral Equations. Computational Mechanics Publications, Southampton (1997) MATH
29.
Golberg, M., Bowman, H.: Optimal convergence rates for some discrete projection methods. Appl. Math. Comput. 96, 237–271 (1998)
30.
Kaneko, H., Noren, R., Padilla, P.: Superconvergence of the iterated collocation methods for Hammerstein equations. J. Comput. Appl. Math. 80, 335–349 (1997)
31.
Kumar, S.: A discrete collocation-type method for Hammerstein equations. SIAM J. Numer. Anal. 25, 328–341 (1988)
32.
Kulkarni, R.P., Gnaneshwar, N.: Iterated discrete polynomially based Galerkin methods. Appl. Math. Comput. 146, 153–165 (2003)
33.
Moré, J.J., Cosnard, M.Y.: Algorithm 554: Brentm, a fortran subroutine for the numerical solution of nonlinear equations [c5]. Trans. Math. Softw. (TOMS) 6, 240–251 (1980) CrossRef
34.
Michael, G.: Improved convergence rates for some discrete Galerkin methods. J. Integral Equ. Appl. 8, 307–335 (1996)
35.
Nahid, N., Das, P., Nelakanti, G.: Projection and multi projection methods for nonlinear integral equations on the half-line. J. Comput. Appl. Math. 359, 119–144 (2019)
36.
Noble, B.: Certain dual integral equations. Stud. Appl. Math. 37, 128–136 (1958)
37.
Vainikko, G.M.: Galerkin’s perturbation method and the general theory of approximate methods for nonlinear equations. USSR Comput. Math. Math. Phys. 7, 1–41 (1967) CrossRef
Titel
Discrete projection methods for Hammerstein integral equations on the half-line
verfasst von
Nilofar Nahid
Gnaneshwar Nelakanti
Publikationsdatum
01.12.2020
Verlag
Springer International Publishing
Erschienen in
Calcolo / Ausgabe 4/2020
Print ISSN: 0008-0624
Elektronische ISSN: 1126-5434
DOI
https://doi.org/10.1007/s10092-020-00386-2

Zur Ausgabe