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Fractional-Order Legendre Functions and Their Application to Solve Fractional Optimal Control of Systems Described by Integro-differential Equations

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Abstract

In this paper, we introduce a set of functions called fractional-order Legendre functions (FLFs) to obtain the numerical solution of optimal control problems subject to the linear and nonlinear fractional integro-differential equations. We consider the properties of these functions to construct the operational matrix of the fractional integration. Also, we achieved a general formulation for operational matrix of multiplication of these functions to solve the nonlinear problems for the first time. Then by using these matrices the mentioned fractional optimal control problem is reduced to a system of algebraic equations. In fact the functions of the problem are approximated by fractional-order Legendre functions with unknown coefficients in the constraint equations, performance index and conditions. Thus, a fractional optimal control problem converts to an optimization problem, which can then be solved numerically. The convergence of the method is discussed and finally, some numerical examples are presented to show the efficiency and accuracy of the method.

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Acknowledgements

We would like to thank the referees for their helpful suggestions to improve the earlier version of this article.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Mathematical Sciences, Alzahra University, Tehran, Iran

    Kobra Rabiei & Yadollah Ordokhani

  2. Department of Computer Science, Faculty of Mathematical Sciences and Computer, Kharazmi University, Tehran, Iran

    Esmaeil Babolian

Authors
  1. Kobra Rabiei
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  2. Yadollah Ordokhani
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  3. Esmaeil Babolian
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Correspondence to Yadollah Ordokhani.

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Rabiei, K., Ordokhani, Y. & Babolian, E. Fractional-Order Legendre Functions and Their Application to Solve Fractional Optimal Control of Systems Described by Integro-differential Equations. Acta Appl Math 158, 87–106 (2018). https://doi.org/10.1007/s10440-018-0175-0

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