A novel algorithm to solve nonlinear fractional quadratic integral equations

Younes Talaei; Sanda Micula; Hasan Hosseinzadeh; Samad Noeiaghdam; Younes Talaei; Sanda Micula; Hasan Hosseinzadeh; Samad Noeiaghdam

doi:10.3934/math.2022730

AIMS Mathematics

2022, Volume 7, Issue 7: 13237-13257. doi: 10.3934/math.2022730

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A novel algorithm to solve nonlinear fractional quadratic integral equations

1.
Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran
2.
Department of Mathematics and Computer Science, Babeş-Bolyai University, Cluj-Napoca 400084, Romania
3.
Department of Mathematics, Ardabil Branch, Islamic Azad University, Ardabil 5615883895, Iran
4.
Industrial Mathematics Laboratory, Baikal School of BRICS, Irkutsk National Research Technical University, Irkutsk 664074, Russia
5.
Department of Applied Mathematics and Programming, South Ural State University, Lenin prospect 76, Chelyabinsk 454080, Russia

Received: 01 March 2022 Revised: 25 April 2022 Accepted: 06 May 2022 Published: 12 May 2022
MSC : 33C45, 45E10, 65M70

This paper addresses a new spectral collocation method for solving nonlinear fractional quadratic integral equations. The main idea of this method is to construct the approximate solution based on fractional order Chelyshkov polynomials (FCHPs). To this end, first, we introduce these polynomials and express some of their properties. The operational matrices of fractional integral and product are derived. The spectral collocation method is utilized together with operational matrices to reduce the problem to a system of algebraic equations. Finally, by solving this system, the unknown coefficients are computed. Further, the convergence analysis and numerical stability of the method are investigated. The proposed method is computationally simple and easy to implement in computer programming. The accuracy and applicability of the method is presented by some numerical examples.
- fractional Chelyshkov polynomials,
- nonlinear fractional quadratic integral equations,
- spectral collocation method,
- convergence analysis
Citation: Younes Talaei, Sanda Micula, Hasan Hosseinzadeh, Samad Noeiaghdam. A novel algorithm to solve nonlinear fractional quadratic integral equations[J]. AIMS Mathematics, 2022, 7(7): 13237-13257. doi: 10.3934/math.2022730

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Abstract

This paper addresses a new spectral collocation method for solving nonlinear fractional quadratic integral equations. The main idea of this method is to construct the approximate solution based on fractional order Chelyshkov polynomials (FCHPs). To this end, first, we introduce these polynomials and express some of their properties. The operational matrices of fractional integral and product are derived. The spectral collocation method is utilized together with operational matrices to reduce the problem to a system of algebraic equations. Finally, by solving this system, the unknown coefficients are computed. Further, the convergence analysis and numerical stability of the method are investigated. The proposed method is computationally simple and easy to implement in computer programming. The accuracy and applicability of the method is presented by some numerical examples.

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