Chebyshev polynomials to Volterra-Fredholm integral equations of the first kind
DOI:
https://doi.org/10.35819/remat2024v10i1id6699Keywords:
Chebyshev polynomials, Volterra-Fredholm integral equations, ill-posed problems, perturbed equationsAbstract
Numerous methods have been studied and discussed for solving ill-posed Volterra integral equations and ill-posed Fredholm integral equations, but rarely for both simultaneously. In this study, we focus on numerically solving the ill-posed Volterra-Fredholm integral equation of the first kind by replacing it with its perturbed counterpart. We employ Chebyshev polynomials of the first kind to solve the perturbed equation. Our findings suggest that this technical approach is superior to the regularization method of Tikhonov. It is simpler, less cumbersome, and this simplicity is demonstrated through various examples.
Downloads
References
HADAMARD, J. Lectures on Cauchy's problem in linear partial differential equations. New Haven: Yale University Press, 1923. Available in: https://archive.org/details/lecturesoncauchy00hadauoft. Access at: February 5, 2024.
KUMAR, J.; MANCHANDA, P.; POOJA. Numerical solution of Fredholm integral equations of the first kind using Legendre wavelets collocation method. International Journal of Pure and Applied Mathematics. v. 117, n. 1, p. 33-43, 2017. DOI: https://doi.org/10.12732/ijpam.v117i1.4.
LAKHAL, A.; NADIR, M.; NADIR, M. N. Application of Chebyshev polynomials to Volterra-Fredholm integral equations. Australian Journal of Mathematical Analysis and Applications. v. 19, n. 2, p. 1-8, 2022. Available in: https://ajmaa.org/searchroot/files/pdf/v19n2/v19i2p8.pdf. Access at: February 5, 2024.
LAMM, P. K. A Survey of Regularization Methods for First-Kind Volterra Equations. Vienna, New York: Springer, 2000, p. 53-82. Available in: https://users.math.msu.edu/users/lamm/Preprints/Mt_Holyoke_Survey/index.html. Access at: February 5, 2024.
MALEKNEJAD, K.; KAJANI, M. T.; MAHMOUDI, Y. Numerical solution of linear Fredholm and Volterra integral equations of the second kind using Legendre wavelets. Journal of Sciences, Islamic Republic of Iran. v. 13, n. 2, p. 161-166, 2002. Available in: https://journal.ut.ac.ir/article_31744_3af2254cc9e8b974559cf3ec796e9692.pdf. Access at: February 5, 2024.
NADIR, M.; BENDJABRI, N. On the invertibility of the Cauchy singular integral. International Journal of Mathematics and Computation. v. 29, n. 2, p. 113-118, 2018. Available in: http://www.ceser.in/ceserp/index.php/ijmc/article/view/5496. Access at: February 5, 2024.
NADIR, M.; DJAIDJA, N. Approximation method for Volterra integral equation of the first kind. International Journal of Mathematics and Computation. v. 29, n. 4, p. 67-72, 2018. Available in: http://www.ceser.in/ceserp/index.php/ijmc/article/view/5677. Access at: February 5, 2024.
NADIR, M.; DJAIDJA, N. Comparison between Taylor and perturbed method for Volterra integral equation of the first kind. Numerical Algebra, Control and Optimization. v. 11, n. 4, p. 487-493, 2021. DOI: https://doi.org/10.3934/naco.2020039.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2024 REMAT: Revista Eletrônica da Matemática
This work is licensed under a Creative Commons Attribution 4.0 International License.
REMAT retains the copyright of published articles, having the right to first publication of the work, mention of first publication in the journal in other published media and distribution of parts or of the work as a whole in order to promote the magazine.
This is an open access journal, which means that all content is available free of charge, at no cost to the user or his institution. Users are permitted to read, download, copy, distribute, print, search or link the full texts of the articles, or use them for any other legal purpose, without requesting prior permission from the magazine or the author. This statement is in accordance with the BOAI definition of open access.
