# An indirect spectral shifted Gegenbauer collocation method for discretizing fractional optimal control problems

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## Keywords:

Keywords: Riemann-Liouville fractional derivative, Gegenbauer polynomials, Optimal control problem## Abstract

Many best properties of the shifted Gegenbauer functions were used to obtain a new

closed formula of the left and right Riemann-Liouville fractional derivative. The new formulas

have been used to approximate the solution of the fractional optimal control problems. The

indirect spectral-shifted Gegenbauer collocation method is applied to discretizing FOCPs with a

dynamic fractional differential equation. The FOCPs were reduced to the system of algebraic

equations. Special attention is given to studying the convergence analysis and estimating an

error upper bound of the presented formulas. Illustrative numerical examples are integrated to

show the truth and applicability of this new technique.

## References

References

H.M. El-Hawary, M.S. Salim, H.S. Hussien, An optimal ultraspherical approximation

of integrals, Int. J. Comput. Math. 76 (2000) 219–237.

K.T. Elgindy, K.A. Smith-Miles, Optimal Gegenbauer quadrature over arbitrary integration nodes, 2012 (submitted for publication).

Li, Changpin, and Fanhai Zeng. Numerical methods for fractional calculus. Vol. 24.

CRC Press, 2015.

Pooseh S, Almeida R, Torres DFM. A numerical scheme to solve fractional optimal

control problems. In: Conference Papers in Mathematics, 2013; 2013. 10p [Article ID:165298].

Saleh Ali, Mushtaq, and Mohammed K. Almoaeet. "An Indirect Spectral Collocation

Method Based on Shifted Jacobi Functions for Solving Some Class of Fractional Optimal Control

Problems." Journal of Physics: Conference Series. Vol. 1818. No. 1. IOP Publishing, 2021.

Kareem T. Elgindy, Kate A. Smith-Miles Optimal Gegenbauer quadrature over

arbitrary integration nodes, Journal of Computational and Applied Mathematics 242 (2013)

–106

E-Gindy, T., H. Ahmed, and Marina Melad. Shifted Gegenbauer operational

matrix and its applications for solving fractional differential equations. Journal of the Egyptian

Mathematical Society 26.1 (2018): 72-90.

Hafez, R. M., and Y. H. Youssri. Shifted Gegenbauer–-Gauss collocation method

for solving fractional neutral functional differential equations with proportional delays.

Kragujevac Journal of Mathematics 46.6 (2022): 981-996.

K.T. Elgindy, Kate A. Smith-Miles, Solving boundary value problems, integral, and

integrodifferential equations using Gegenbauer integration matrices, Journal of Computational

and Applied Mathematics 237 (2013) 307–325

J.J.Trujillo, On Riemann-Liouvill Generalized Taylor's Formula, Joranl Of

Mathematical Analysis and Applications,231,255-265(1999).

Kanti B Datta, M Mohan Orthogonal functions in systems and control book, World

Scientific 1995.

B. Spain, M.G. Smith Functions of mathematical physics, Van Nostrand

Reinhold Company, London, 1970. Chapter 10 deals with Laguerre polynomials.[13] Kreyszig, Erwin. Introductory functional analysis with applications. Vol. 17. John

Wiley and Sons, 1991

S. S. Bayin, Mathematical Methods in Science and Engineering, Wiley,(2006),

Chapter 3.

Shakoor Pooseh,Ricardo Ameida and Delfim F. M. Torres , Numerical

approximations to fractional problems of the calculus of variations and optimal control, Chapter

V, Fractional calculus in analysis, Dynamics and optimal control (Editor: Jacky Cresson), Series:

Mathematics Research Developments, Nova Science Publishers, New York, 2014.

O.P.Agrawa, Aquadratic numerical scheme for fractional optimal control

problems, Trans.ASME, J.Dyn.Syst.Meas. control 130 (2008), No.1,011010-011016.

A. Lotfi, S.A. Yousefi, M.Dehghan Numerical solution of a class of fractional

optimal control problems via the Legender orthonormal basis combined with the operational

matrix and the Gauss quadrature rule, J.comput.Appl.Math.,250, pp.143-160,2013

Elgindy, Kareem T. High–order, stable, and efficient pseudospectral method

using barycentric Gegenbauer quadratures, Applied Numerical Mathematics, 113 (2017): 1-25.

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*Results in Nonlinear Analysis*,

*7*(3), 177–193. Retrieved from https://nonlinear-analysis.com/index.php/pub/article/view/351

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