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RESEARCH ARTICLE

Legendre spectral-collocation approximation for an optimal control problem governed by generalized Caputo integro-differential equations

A. M. Nagy1,2* ,  T. M. Al-Ajami3 ,  N. H. Sweilam3
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1 Department of Mathematics, College of Science, Kuwait University, Safat , Kuwait
2 Department of Mathematics, Faculty of Science, Benha University, Benha , Egypt
3 Department of Mathematics, Faculty of Science, Cairo University, Giza , Egypt
IJOCTA, 026330215 https://doi.org/10.36922/IJOCTA026330215
Received: 14 August 2026 | Revised: 18 September 2026 | Accepted: 22 September 2026 | Published online: 13 October 2026
© 2026 by the Author(s). This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution -Noncommercial 4.0 International License (CC-by the license) ( https://creativecommons.org/licenses/by-nc/4.0/ )
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Abstract

This work investigates a class of optimal control problems governed by generalized Caputo fractional integro-differential equations. The numerical solution is based on a Legendre spectral-collocation method combined with the Gauss-Legendre quadrature rule. To reduce the problem to a finite-dimensional optimization problem, the Rayleigh-Ritz method is applied, and the resulting nonlinear system is solved using Newton’s iterative procedure. We derive an approximation to the generalized Caputo fractional derivative. Additionally, we present an error bound for this approximation and for the proposed method. The accuracy and efficiency of the developed method are verified by numerical experiments.

Keywords
Generalized Caputo fractional derivative
Legendre polynomials
Fractional Integro-differential equation
Spectral methods
Optimal control
Funding
None.
Conflict of interest
The authors confirm they have no conflicts of interest.
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An International Journal of Optimization and Control: Theories & Applications, Electronic ISSN: 2146-5703 Print ISSN: 2146-0957, Published by AccScience Publishing
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