A new local non-integer derivative and its application to optimal control problems

Xingfa Yang; Yin Yang; M. H. Noori Skandari; Emran Tohidi; Stanford Shateyi; Xingfa Yang; Yin Yang; M. H. Noori Skandari; Emran Tohidi; Stanford Shateyi

doi:10.3934/math.2022915

AIMS Mathematics

2022, Volume 7, Issue 9: 16692-16705. doi: 10.3934/math.2022915

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Research article

A new local non-integer derivative and its application to optimal control problems

1.
College of Mechanical and Electrical Engineering, Changsha University, Changsha 410022, China
2.
School of Mathematics and Computational Science, Xiangtan University, National Center for Applied Mathematics in Hunan, Hunan International Scientific and Technological Innovation Cooperation Base of Computational Science, Xiangtan 411105, Hunan, China
3.
Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran
4.
Department of Mathematics, Kosar University of Bojnord, P. O. Box 9415615458, Bojnord, Iran
5.
Department of Mathematics and Applied Mathematics, University of Venda, P. Bag X5050, Thohoyandu 950, South Africa

Received: 28 February 2022 Revised: 03 April 2022 Accepted: 14 April 2022 Published: 12 July 2022
MSC : 26A33, 34A08, 49M37, 49M25

Here, a new local non-integer derivative is defined and is shown that it coincides to classical derivative when the order of derivative be integer. We call this derivative, adaptive derivative and present some of its important properties. Also, we gain and state Rolle's theorem and mean-value theorem in the sense of this new derivative. Moreover, we define the optimal control problems governed by differential equations including adaptive derivative and apply the Legendre spectral collocation method to solve this type of problems. Finally, some numerical test problems are presented to clarify the applicability of new defined non-integer derivative with high accuracy. Through these examples, one can see the efficiency of this new non-integer derivative as a tool for modeling real phenomena in different branches of science and engineering that described by differential equations.
- adaptive derivative,
- adaptive optimal control problems,
- Legendre spectral collocation method
Citation: Xingfa Yang, Yin Yang, M. H. Noori Skandari, Emran Tohidi, Stanford Shateyi. A new local non-integer derivative and its application to optimal control problems[J]. AIMS Mathematics, 2022, 7(9): 16692-16705. doi: 10.3934/math.2022915

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Abstract

Here, a new local non-integer derivative is defined and is shown that it coincides to classical derivative when the order of derivative be integer. We call this derivative, adaptive derivative and present some of its important properties. Also, we gain and state Rolle's theorem and mean-value theorem in the sense of this new derivative. Moreover, we define the optimal control problems governed by differential equations including adaptive derivative and apply the Legendre spectral collocation method to solve this type of problems. Finally, some numerical test problems are presented to clarify the applicability of new defined non-integer derivative with high accuracy. Through these examples, one can see the efficiency of this new non-integer derivative as a tool for modeling real phenomena in different branches of science and engineering that described by differential equations.

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