Approximate technique for solving fractional variational problems
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Approximate technique for solving fractional variational problems HALEH TAJADODI1 , NEMATOLLAH KADKHODA2 , HOSSEIN JAFARI3,4 and MUSTAFA INC5,6 ,∗ 1 Department
of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran of Mathematics, Faculty of Basic Sciences, Bozorgmehr University of Qaenat, Qaenat, Iran 3 Department of Mathematics, University of Mazandaran, Babolsar, Iran 4 Department of Mathematical Sciences, University of South Africa (UNISA), P.O. Box 392, Pretoria 0003, South Africa 5 Department of Mathematics, Firat University, Elazig 23119, Turkey 6 Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan ∗ Corresponding author. E-mail: [email protected] 2 Department
MS received 28 January 2020; revised 12 May 2020; accepted 1 July 2020 Abstract. The purpose of this paper is to suggest a numerical technique to solve fractional variational problems (FVPs). These problems are based on Caputo fractional derivatives. Rayleigh–Ritz method is used in this technique. First we approximate the objective function by the trapezoidal rule. Then, the unknown function is expanded in terms of the Bernstein polynomials. By this method, a system of algebraic equations is driven. We provide examples to show the effectiveness of this technique, which is considered in the current study. Keywords. Fractional variational problems; Bernstein polynomials; Rayleigh–Ritz method. PACS Nos 02.60.–x; 02.70.–c; 02.90.+p
1. Introduction
FVPs. A new approach based on operational matrix for this type of problems was carried out by Eldien [20]. In Fractional calculus is a generalisation of classical 2017, Eldein et al [21] published a paper in which they calculus, and it has a long mathematical history. Recently, derived numerical solutions of FVPs using shifted Legthere has been a huge interest in the field of fractional endre polynomials. A numerical scheme is utilised to calculus. There are a wide range of applications in sci- achieve the solutions of FVPs by Lotfi and Yousefi [22]. ences like biology, chemistry, fluid mechanics, signal Another numerical method was introduced by Khader processing and control theory [1–4]. Various types of and Hendy [23] to solve these problems. The modiapproximate methods have been developed for solving fied Jacobi polynomials for the solution of a class of fractional differential equations over the years [5–12]. fractional variational and optimal control problems was It is worth pointing out that the fractional calculus of applied by Dehghan et al [24]. variations and the fractional optimal control problems There are many different methods for obtaining the have essential roles in science and engineering [13–16]. solution of FVPs. The purpose of this study is to use a Hence, this subject attracts the attention of a number numerical technique of the following fractional variaof scientists and engineers. Riewe [17,18] introduced tional problems: 1 fractional calculus of variations. The problem of variational c
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