Numerical solutions for fractional optimal control problems by using generalised fractional-order Chebyshev wavelets

Abstract

This paper proposes a new numerical method for solving fractional optimal control problems (FOCPs). The method is based on generalised fractional-order Chebyshev wavelets (GFOCW). The exact value of the Riemann–Liouville fractional integral operator of the GFOCW is given by applying the incomplete beta function. By using the properties of GFOCW and the collocation method, the FOCP is reduced to a parameter optimisation problem. The last problem is solved by known algorithms. Six numerical examples are given. One of them is an application example in a cancer model. Through these numerical examples, we will show that for some cases of our examples, we will get the exact solutions. These solutions were not obtained previously in the literature. In addition, our method gives more accurate results in comparison with the existing methods.

Keywords:

• Fractional optimal control
• fractional-order Chebyshev wavelet
• numerical method
• beta function
• fractional integral

Acknowledgments

The authors wish to express their sincere thanks to the anonymous Associate Editor and the referees for valuable suggestions that improved the final manuscript.

Data availability statement

Data sharing is not applicable to this article as no new data were created or analysed in this study.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Notes on contributors

Ghodsieh Ghanbari

Ghodsieh Ghanbari received her B.Sc. and M.Sc. degrees from the Department of Mathematics and Statistics from Ferdowsi University in Mashhad, Iran, in 2010 and 2013, respectively. Currently, she is a Ph.D. student at the Department of Mathematics and Statistics at Mississippi State University. Her research interests include fractional calculus, optimal control problems, and calculus of variations.

Mohsen Razzaghi

Mohsen Razzaghi received his B.Sc. degree in Mathematics and a Ph.D. degree in Applied Mathematics, both from the University of Sussex in England. Since 1986, he has been with the Department of Mathematics and Statistics at Mississippi State University, where he is currently a Professor and Department Head. He was named a W.L. Giles Distinguished Professor in 2019. Dr. Razzaghi received two Fulbright scholar programs from 2011–2012, another in 2015–2016, and one Fulbright Specialist in 2019 in Romania. His current area of research centers on orthogonal functions, optimal control, wavelets, fractional calculus, and their applications in mathematical modeling and engineering. He has over 225 refereed journal publications in mathematics, mathematical physics, and engineering. One of his papers co-authored with one of his Ph.D. students has been cited over 750 times.