AccScience Publishing / IJOCTA / Online First / DOI: 10.36922/IJOCTA026330215
Submit to IJOCTA
Cite this article
12
Download
101
Views
Related Info Links
Google Scholar
More by Authors Links
A. M. Nagy
Journal Browser
Volume | Year
Issue
Search
News and Announcements
View All
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
Show Less
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/ )
Download PDF
XML
Cite
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.
References
  1. Atangana A. Fractal-fractional differentiation and integration: connecting fractal calculus and fractional calculus to predict complex systems. Chaos Solitons Fractals. 2017;102:396-406. https://doi.org/10.1016/j.chaos.2017.04.027
  2. Khalil R, Al Horani M, Yousef A, Sababheh M. A new definition of fractional derivative. J Comput Appl Math. 2014;264:65-70. https://doi.org/10.1016/j.cam.2014.01.002
  3. Samko SG, Kilbas AA, Marichev OI. Fractional Integrals and Derivatives: Theory and Applications. Philadelphia, PA: Gordon and Breach; 1993. Accessed June 1, 2012. https://scispace.com/pdf/fractional-integrals-and-derivatives-theory-and-applications-3mo1ouv89m.pdf
  4. Oldham KB, Spanier J. The Fractional Calculus. New York, NY: Academic Press; 1974. Accessed January 1, 2012. https://ia801900.us.archive.org/11/items/in.ernet.dli.2015.134320/2015.134320.The-Fractional-Calculus.pdf
  5. Podlubny I. Fractional Differential Equations. San Diego, CA: Academic Press; 1999. https://doi.org/10.1016/S0076-5392(99)X8001-5
  6. Zafar ZUA, Sene N, Rezazadeh H, Esfandian N. Tangent nonlinear equation in context of fractal fractional operators with nonsingular kernel. Math Sci. 2021;16(2):121-131. https://doi.org/10.1007/s40096-021-00403-7
  7. Katugampola UN. New approach to a generalized fractional integral. Appl Math Comput. 2011;218(3):860-865. https://doi.org/10.1016/j.amc.2011.03.062
  8. Katugampola UN. A new approach to generalized fractional derivative. Bull Math Anal Appl. 2014;6(4):1-15. Accessed May 1, 2017. http://www.bmathaa.org/repository/docs/BMAA6-4-1.pdf
  9. Erturk VS, Kumar P. Solution of a COVID-19 model via new generalized Caputo-type fractional derivatives. Chaos Solitons Fractals. 2020;139:110280. https://doi.org/10.1016/j.chaos.2020.110280
  10. Ben Makhlouf A, Nagy AM. Finite-time stability of linear Caputo-Katugampola fractional-order time delay systems. Asian J Control. 2020;22(1):297-306. https://doi.org/10.1002/asjc.1880
  11. Abdeljawad T, Madjidi F, Jarad F, Sene N. On dynamic systems in the frame of singular function dependent kernel fractional derivatives. Mathematics. 2019;7(10):946. https://doi.org/10.3390/math7100946
  12. Nagy AM. Numerical solutions for nonlinear multi-term fractional differential equations via Dickson operational matrix. Int J Comput Math. 2022;99(7):1505-1515. https://doi.org/10.1080/00207160.2021.1986214
  13. Sweilam NH, Nagy AM, Al-Ajami TM. Numerical solutions of fractional optimal control with Caputo-Katugampola derivative. Adv Differ Equ. 2021;2021:425. https://doi.org/10.1186/s13662-021-03580-w
  14. Alipour M, Soradi-Zeid S. Optimal control governed by nonlinear variable-order fractional integro-differential equations through generalized Dickson polynomial expansions. J Comput Appl Math. 2026;473:116917. https://doi.org/10.1016/j.cam.2025.116917
  15. Dehestani H, Ordokhani Y. A numerical study on fractional optimal control problems described by Caputo-Fabrizio fractional integro-differential equation. Optim Control Appl Methods. 2023;44(4):1873-1892. https://doi.org/10.1002/oca.2957
  16. Mohammadi F, Moradi L, Machado JAT. A discrete polynomials approach for optimal control of fractional Volterra integro-differential equations. J Vib Control. 2022;28(1-2):72-82. https://doi.org/10.1177/1077546320971156
  17. Pirouzeh Z, Skandari MHN, Pirbazari KN, Shateyi S. A pseudo-spectral approach for optimal control problems of variable-order fractional integro-differential equations. AIMS Math. 2024;9(9):23692-23710. https://doi.org/10.3934/math.20241151
  18. Singh A, Kanaujiya A, Mohapatra J. Euler wavelets method for optimal control problems of fractional integro-differential equations. J Comput Appl Math. 2025;454:116178. https://doi.org/10.1016/j.cam.2024.116178
  19. Soradi-Zeid S, Alipour M. A new method for solving optimal control problems of nonlinear variable-order fractional Volterra-Fredholm integro-differential equations. Iran J Sci Technol Trans Electr Eng. 2025;49:1561-1577. https://doi.org/10.1007/s40998-025-00854-x
  20. Modanli M, Murad MAS, Abdulazeez ST. A new computational method-based integral transform for solving time-fractional equation arises in electromagnetic waves. Z Angew Math Phys. 2023;74(5):186. https://doi.org/10.1007/s00033-023-02076-9
  21. Johnson M, Raja MM, Vijayakumar V, Shukla A, Nisar KS, Jahanshahi H. Optimal control results for impulsive fractional delay integrodifferential equations of order 1<r<2 via sectorial operator. Nonlinear Anal Model Control. 2023;28(3):468-490. https://doi.org/10.15388/namc.2023.28.31721
  22. Dhawan K, Vats RK, Nain AK, Shukla A. Well-posedness and Ulam-Hyers stability of Hilfer fractional differential equations of order (1,2] with nonlocal boundary conditions. Bull Sci Math. 2024;191:103401. https://doi.org/10.1016/j.bulsci.2024.103401
  23. Sene N. Introduction to the fractional-order chaotic system under fractional operator in Caputo sense. Alex Eng J. 2021;60(4):3997-4014. https://doi.org/10.1016/j.aej.2021.02.056
  24. Diethelm K. The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Berlin, Germany: Springer-Verlag; 2010. https://doi.org/10.1007/978-3-642-14574-2
  25. Almeida R, Malinowska AB, Odzijewicz T. Fractional differential equations with dependence on the Caputo-Katugampola derivative. J Comput Nonlinear Dyn. 2016;11(6):061017. https://doi.org/10.1115/1.4034432
  26. Jarad F, Abdeljawad T, Baleanu D. On the generalized fractional derivatives and their Caputo modification. J Nonlinear Sci Appl. 2017;10(5):2607-2619. https://doi.org/10.22436/jnsa.010.05.27
  27. Gambo YY, Jarad F, Baleanu D, Abdeljawad T. On Caputo modification of the Hadamard fractional derivatives. Adv Differ Equ. 2014;2014:10. https://doi.org/10.1186/1687-1847-2014-10
  28. Sene N. Generalized Mittag-Leffler input stability of the fractional-order electrical circuits. IEEE Open J Circuits Syst. 2020;1:233-242. https://doi.org/10.1109/OJCAS.2020.3032546
  29. Shen J, Tang T, Wang LL. Spectral Methods: Algorithms, Analysis and Applications. Berlin, Germany: Springer; 2011. https://doi.org/10.1007/978-3-540-71041-7
  30. Hesthaven JS, Gottlieb S, Gottlieb D. Spectral Methods for Time-Dependent Problems. Cambridge, UK: Cambridge University Press; 2007. https://doi.org/10.1017/CBO9780511618352
  31. Bhrawy AH, Assas LM, Tohidi E, Alghamdi MA. A Legendre-Gauss collocation method for neutral functional-differential equations with proportional delays. Adv Differ Equ. 2013;2013:63. https://doi.org/10.1186/1687-1847-2013-63
  32. Atkinson KE. An Introduction to Numerical Analysis. 2nd ed. New York, NY: John Wiley & Sons; 1989.
  33. Fornberg B. A Practical Guide to Pseudospectral Methods. Cambridge, UK: Cambridge University Press; 1996. https://doi.org/10.1017/CBO9780511626357
  34. Sweilam NH, Al-Ajami TM. Legendre spectral-collocation method for solving some types of fractional optimal control problems. J Adv Res. 2015;6(3):393-403. https://doi.org/10.1016/j.jare.2014.05.004
  35. Benjemaa M. Taylor's formula involving generalized fractional derivatives. Appl Math Comput. 2018;335:182-195. https://doi.org/10.1016/j.amc.2018.04.040
  36. Khanduzi R, Ebrahimzadeh A, Peyghami MR. A modified teaching-learning-based optimization for optimal control of Volterra integral systems. Soft Comput. 2018;22:5889-5899. https://doi.org/10.1007/s00500-017-2933-8
  37. Maleknejad K, Ebrahimzadeh A. Optimal control of Volterra integro-differential systems based on Legendre wavelets and collocation method. Int J Math Comput Sci. 2014;8(7):1057-1061. Accessed July 1, 2024. https://zenodo.org/records/1093708
Share
Back to top
An International Journal of Optimization and Control: Theories & Applications, Electronic ISSN: 2146-5703 Print ISSN: 2146-0957, Published by AccScience Publishing
We use cookies on our website to ensure you get the best experience.
Read more about our cookies here
Accept