Superconvergent Nyström Method Based on Spline Quasi-Interpolants for Nonlinear Urysohn Integral Equations
Abstract
:1. Introduction
2. Spline Quasi-Interpolants
3. Superconvergent Nyström Method Based on Spline Quasi-Interpolants
3.1. Construction of the Approximate Solution
3.2. Convergence of the Method
3.3. Iterated Version
4. Numerical Results
- Test 1
- Test 2
- Test 3
5. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
- El-Sayed, W.G.; El-Bary, A.A.; Darwish, M.A. Solvability of Urysohn integral equation. Appl. Math. Comput. 2003, 145, 487–493. [Google Scholar] [CrossRef]
- Jafarian, A.; Esmailzadeh, Z.; Khoshbakhti, L. A numerical method for solving nonlinear integral equations in the urysohn form. Appl. Math. Sci. 2013, 7, 1375–1385. [Google Scholar] [CrossRef]
- Alijani, Z.; Kangro, U. Numerical solution of a linear fuzzy Volterra integral equation of the second kind with weakly singular kernels. Soft Comput. 2022, 26, 12009–12022. [Google Scholar] [CrossRef]
- Atkinson, K.E. A survey of numerical methods for solving nonlinear integral equations. J. Integral Eqns Appl. 1992, 4, 15–46. [Google Scholar] [CrossRef]
- Atkinson, K.E.; Potra, F.A. Projection and iterated projection methods for nonlinear integral equations. SIAM J. Num. Anal. 1987, 24, 1352–1373. [Google Scholar] [CrossRef]
- Grammont, L.; Kulkarni, R.P.; Vasconcelos, P.B. Modified projection and the iterated modified projection methods for non linear integral equations. J. Integral Equ. Appl. 2013, 25, 481–516. [Google Scholar] [CrossRef]
- Grammont, L.; Kulkarni, R.P.; Nidhin, T.J. Modified projection method for Urysohn integral equations with non-smooth kernels. J. Comput. Appl. Math. 2016, 294, 309–322. [Google Scholar] [CrossRef]
- Kulkarni, R.P. A superconvergence result for solutions of compact operator equations. Bull. Austral. Math. Soc. 2003, 68, 517–528. [Google Scholar] [CrossRef] [Green Version]
- Allouch, C.; Sbibih, D.; Tahrichi, M. Superconvergent Nyström and degenerate kernel methods for Hammerstein integral equations. J. Comput. Appl. Math. 2014, 258, 30–41. [Google Scholar] [CrossRef]
- Allouch, C.; Sbibih, D.; Tahrichi, M. Superconvergent Nyström method for Urysohn integral equations. BIT Numer. Math. 2017, 57, 3–20. [Google Scholar] [CrossRef]
- Allouch, C.; Remogna, S.; Sbibih, D.; Tahrichi, M. Superconvergent methods based on quasi-interpolating operators for fredholm integral equations of the second kind. Appl. Math. Comput. 2021, 404, 126227. [Google Scholar] [CrossRef]
- Allouch, C.; Sablonnière, P.; Sbibih, D.; Tahrichi, M. Product integration methods based on discrete spline quasi-interpolants and application to weakly singular integral equations. J. Comput. Appl. Math 2010, 233, 2855–2866. [Google Scholar] [CrossRef]
- Dagnino, C.; Dallefrate, A.; Remogna, S. Spline quasi-interpolating projectors for the solution of nonlinear integral equations. J. Comput. Appl. Math. 2019, 354, 360–372. [Google Scholar] [CrossRef]
- Kumar, S.; Mittal, R.C.; Jiwari, R. A cubic B-spline quasi-interpolation method for solving hyperbolic partial differential equations. Int. J. Comput. Math. 2023, 100, 1580–1600. [Google Scholar] [CrossRef]
- Mittal, R.C.; Kumar, S.; Jiwari, R. A cubic B-spline quasi-interpolation algorithm to capture the pattern formation of coupled reaction-diffusion models. Eng. Comput. 2022, 38, 1375–1391. [Google Scholar] [CrossRef]
- de Boor, C. A Practical Guide to Splines, Revised ed.; Springer: New York, NY, USA, 2001. [Google Scholar]
- Sablonnière, P. Univariate spline quasi-interpolants and applications to numerical analysis. Rend. Sem. Mat. Univ. Pol. Torino 2005, 63, 107–118. [Google Scholar]
- Vainikko, G. Galerkin’s perturbation method and the general theory of approximate methods for nonlinear equations. USSR Comput. Math. Math. Phys. 1967, 7, 1–41. [Google Scholar] [CrossRef]
- Potra, F.A.; Pták, V. Nondiscrete Induction and Iterative Processes; Pitman Advanced Publishing Program: Boston, MA, USA, 1984. [Google Scholar]
n | n | ||||||||
---|---|---|---|---|---|---|---|---|---|
Methods Based on | Methods Based on | ||||||||
40 | 2.84 (−07) | 9.10 (−09) | 40 | 3.31 (−05) | 3.04 (−07) | ||||
80 | 3.99 (−08) | 2.8 | 1.04 (−09) | 3.1 | 80 | 4.60 (−07) | 6.2 | 6.79 (−09) | 5.5 |
160 | 2.98 (−10) | 7.1 | 5.44 (−12) | 7.6 | 160 | 5.86 (−10) | 9.6 | 1.49 (−11) | 8.8 |
320 | 2.27 (−12) | 7.0 | 2.42 (−14) | 7.8 | 320 | 6.93 (−12) | 6.4 | 1.97 (−13) | 6.2 |
n | n | ||||||||
---|---|---|---|---|---|---|---|---|---|
Methods Based on | Methods Based on | ||||||||
8 | 1.15 (−10) | 2.54 (−12) | 8 | 1.67 (−10) | 3.96 (−11) | ||||
16 | 1.09 (−12) | 6.8 | 9.77 (−15) | 8.0 | 16 | 1.27 (−12) | 7.0 | 4.03 (−13) | 6.6 |
32 | 1.07 (−14) | 6.7 | - | - | 32 | 8.88 (−15) | 7.2 | 1.20 (−14) | - |
64 | - | - | - | - | 64 | - | - | - | - |
128 | - | - | - | - | 128 | - | - | - | - |
n | n | ||||||||
---|---|---|---|---|---|---|---|---|---|
Methods Based on | Methods Based on | ||||||||
8 | 3.97 (−09) | 2.71 (−11) | 8 | 2.25 (−08) | 5.72 (−10) | ||||
16 | 3.61 (−11) | 6.8 | 1.57 (−13) | 7.4 | 16 | 1.95 (−10) | 6.9 | 1.02 (−11) | 5.8 |
32 | 2.93 (−13) | 6.9 | 1.78 (−15) | - | 32 | 9.60 (−13) | 7.7 | 6.39 (−14) | 7.3 |
64 | 5.77 (−15) | - | - | - | 64 | 3.33 (−15) | 6.2 | 1.89 (−15) | - |
128 | - | - | - | - | 128 | - | - | - | - |
n | n | ||||||||
---|---|---|---|---|---|---|---|---|---|
Methods Based on | Methods Based on | ||||||||
8 | 2.55 (−10) | 4.69 (−12) | 8 | 4.34 (−10) | 2.33 (−11) | ||||
16 | 2.28 (−12) | 6.8 | 2.66 (−14) | 7.5 | 16 | 5.69 (−12) | 6.3 | 1.71 (−13) | 7.1 |
32 | 1.60 (−14) | 7.2 | 1.78 (−15) | - | 32 | 1.42 (−14) | 8.6 | 1.78 (−15) | - |
64 | - | - | - | - | 64 | - | - | - | |
128 | - | - | - | - | 128 | - | - | - |
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Remogna, S.; Sbibih, D.; Tahrichi, M. Superconvergent Nyström Method Based on Spline Quasi-Interpolants for Nonlinear Urysohn Integral Equations. Mathematics 2023, 11, 3236. https://doi.org/10.3390/math11143236
Remogna S, Sbibih D, Tahrichi M. Superconvergent Nyström Method Based on Spline Quasi-Interpolants for Nonlinear Urysohn Integral Equations. Mathematics. 2023; 11(14):3236. https://doi.org/10.3390/math11143236
Chicago/Turabian StyleRemogna, Sara, Driss Sbibih, and Mohamed Tahrichi. 2023. "Superconvergent Nyström Method Based on Spline Quasi-Interpolants for Nonlinear Urysohn Integral Equations" Mathematics 11, no. 14: 3236. https://doi.org/10.3390/math11143236
APA StyleRemogna, S., Sbibih, D., & Tahrichi, M. (2023). Superconvergent Nyström Method Based on Spline Quasi-Interpolants for Nonlinear Urysohn Integral Equations. Mathematics, 11(14), 3236. https://doi.org/10.3390/math11143236