Discrete Legendre modified projection-type methods for Hammerstein integral equations
Анотація
We investigate discrete modified projection-type methods for the numerical approximation of nonlinear Hammerstein integral equations with sufficiently smooth kernels. Such equations arise in various applications and require efficient numerical techniques for their accurate resolution. The proposed approach is based on Legendre polynomial bases, which provide a suitable framework for constructing approximate solutions in appropriate function spaces. By combining these bases with sufficiently accurate numerical quadrature rules, we derive discrete formulations of modified Galerkin-type and modified collocation-type methods.
These methods are designed to improve the accuracy of classical projection techniques while maintaining computational efficiency. A comprehensive convergence analysis is performed for both approximate and iterated approximate solutions.
Under suitable regularity assumptions on the kernel and the exact solution, we establish superconvergence results, showing that the proposed methods achieve higher-order accuracy compared to standard approaches.
Moreover, we provide a rigorous error analysis that highlights the role of discretization and quadrature in the overall approximation process. The obtained theoretical estimates demonstrate that the use of Legendre-based discretization leads to significant improvements in convergence behavior.
These results will contribute to the development of efficient numerical approaches for solving nonlinear Hammerstein-type integral equations.
Посилання
M. Ahues, A. Largillier, B. Limaye, Spectral Computations for Bounded Operators, CRC Press, New York, 2001.
C. Allouch, M. Arrai, H. Bouda, M. Tahrichi, Legendre Superconvergent Degenerate Kernel and Nystrom Methods for Nonlinear Integral Equations, Ukr. Math. J. 75 (2023), 663–681. https://doi.org/10.1007/s11253-023-02222-6
C. Allouch, D. Sbibih, M. Tahrichi, Legendre Superconvergent Galerkin-Collocation Type Methods for Hammerstein Equations, J. Comp. Appl. Math. 353 (2019), 253–264. https://doi.org/10.1016/j.cam.2018.12.040
M. Arrai, C. Allouch, H. Bouda, Fast Discrete Solvers for Nonlinear Hammerstein Equations, Proc. Inst. Math. Mech. 49 (2) (2023), 193–209. https://doi.org/10.30546/2409-4994.2023.49.2.193
M. Arrai, C. Allouch, H. Bouda, M. Tahrichi, Legendre-Kantorovich Method for Fredholm Integral Equations of the Second Kind, Math. Mod. Comp. 9 (3) (2022), 471–482. https://doi.org/10.23939/mmc2022.03.471
P. Assari, The Numerical Solution of Fredholm-Hammerstein Integral Equations by Combining the Collocation Method and Radial Basis Functions, Filomat 33 (2019), 667–682. https://doi.org/10.2298/FIL1903667A
P. Assari, A Meshless Local Galerkin Method for the Numerical Solution of Hammerstein Integral Equations Based on the Moving Least Squares Technique, J. Appl. Anal. Comp. 9 (2019), 75–104. https://doi.org/10.11948/2019.75
K.E. Atkinson, The Numerical Solution of Integral Equations of the Second Kind, Cambridge University Press, Cambridge, 1997. https://doi.org/10.1017/CBO9780511626340
K.E. Atkinson, J. Flores, The Discrete Collocation Method for Nonlinear Integral Equations, IMA Journal of Numerical Analysis 13 (1993), 195–213. https://doi.org/10.1093/IMANUM/13.2.195
H. Bouda, C. Allouch, K. Kant, Z. El Allali, Error Analysis of a Jacobi Modified Projection-Type Method for Weakly Singular Volterra-Hammerstein Integral Equations, Electron. Trans. Numer. Anal. 60 (2024), 446–470. https://doi.org/10.1553/etna_vol60s446
M. Golberg, C. Chen, Discrete Projection Methods for Integral Equations, Computational Mechanics Publications, 1997.
M. Golberg, Improved Convergence Rates for Some Discrete Galerkin Methods, J. Int. Eq. Appl. 8 (1996), 307–335. https://doi.org/10.1216/jiea/1181075955
M. Mandal, K. Kant, G. Nelakanti, Discrete Legendre Spectral Methods for Hammerstein Type Weakly Singular Nonlinear Fredholm Integral Equations, Inter. J. Comp. Math. 98 (2021), 2251–2267. https://doi.org/10.1080/00207160.2021.1891225
N. Nahid, G. Nelakanti, Discrete Projection Methods for Hammerstein Integral Equations on the Half-Line, Calcolo 57 (2020). https://doi.org/10.1007/s10092-020-00386-2
P. Das, G. Nelakanti, G. Long, Discrete Legendre Spectral Galerkin Method for Urysohn Integral Equations, Inter. J. Comp. Math. 95 (2018), 465–489. https://doi.org/10.1080/00207160.2017.1294687
P. Das, G. Nelakanti, Error Analysis of Discrete Legendre Multi-Projection Methods for Nonlinear Fredholm Integral Equations, Numerical Functional Analysis and Optimization 38 (2017), 549–574. https://doi.org/10.1080/01630563.2016.1248563
P. Das, G. Nelakanti, G. Long, Discrete Legendre Spectral Projection Methods for Fredholm-Hammerstein Integral Equations, J. Comp. App. Math. 278 (2015), 293–305. https://doi.org/:10.1016/j.cam.2014.10.012
H. Kaneko, Y. Xu, Superconvergence of the Iterated Galerkin Methods for Hammerstein Equations, SIAM J. Numer. Anal. 33 (1996), 1048–1064. https://doi.org/10.2307/2158494
S. Kumar, I.H. Sloan, A New Collocation-Type Method for Hammerstein Equations, Math. Comp. 48 (1987), 585–593. https://doi.org/10.2307/2007829
R.P. Kulkarni, G.R. Rakshit, Discrete Modified Projection Methods for Urysohn Integral Equations With Green’s Function Type Kernels, Math. Mod. Anal. 25 (2020), 421–440. https://doi.org/10.3846/mma.2020.11093
S. Kumar, Superconvergence of a Collocation-Type Method for Hammerstein Equations, IMA J. Numer. Anal. 7 (1987), 313–325. https://doi.org/10.1093/imanum/7.3.313
R.P. Kulkarni, G. Rakshit, Discrete Modified Projection Method for Urysohn Integral Equations With Smooth Kernels, Appl. Num. Math. 126 (2018), 180–198. https://doi.org/10.1016/j.apnum.2017.12.008
I.H. Sloan, Polynomial Interpolation and Hyperinterpolation Over General Regions, Journal of Approximation Theory 83 (1995), 238–254. https://doi.org/10.1006/jath.1995.1119
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. https://doi.org/10.1016/0041-5553(67)90140-1
Авторське право (c) 2026 H. Bouda, C. Allouch, M. Arrai
Ця робота ліцензується відповідно до Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Matematychni Studii is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) license.
