A reliable numerical approach for analyzing fractional variational problems with subsidiary conditions
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A reliable numerical approach for analyzing fractional variational problems with subsidiary conditions K. Sayevand1 · M. R. Rostami1,2 Received: 3 September 2018 / Revised: 8 January 2019 / Accepted: 29 January 2019 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019
Abstract This paper obtains the necessary optimality conditions for a new class of isoperimetric fractional variational problems depending on indefinite integrals (IFVPDI). A new condition is added to the IFVPDI and a modified direct numerical approach based on the Epsilon–Ritz method and polynomial basis functions is applied. The optimization problem is reduced to the problem of optimizing a real value function. The convergence and error analysis of the proposed approach are also assessed. Illustrative examples are included demonstrating the validity and applicability of the method. Keywords Fractional calculus · Fractional variational problem · Isoperimetric problem · Epsilon method · Ritz method Mathematics Subject Classification 78M30 · 49K05 · 49M30 · 33F05
1 Introduction Fractional calculus is defined by integrals and derivatives of non-integer order. This subject has various applications in mathematics and other fields (Dehghan et al. 2016; Dehghan and Abbaszadeh 2017; Ferreira et al. 2008; Magin 2004; Hilfer 2000; Oustaloup et al. 2003; Diethelm and Freed 1999; Hilfer 2000). Recently, fractional calculus attracted attention of a considerable number of studies (see Yang et al. 2016; Yang and Machado 2017; Yangab et al. 2017). The calculus of variations is an old problem in applied mathematics and optimization problems. Recently, many researchers were motivated to study this problem using fractional
Communicated by José Tenreiro Machado.
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K. Sayevand [email protected] M. R. Rostami [email protected]
1
Faculty of Mathematical Sciences, Malayer University, P. O. Box 16846-13114, Malayer, Iran
2
Faculty of Science, Mahallat Institute of Higher Education, Mahallat, Iran
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K. Sayevand and M. R. Rostami
derivatives. It is clear that FVP and isoperimetric fractional variational problems (IFVP) are two important different types of fractional optimization. The isoperimetric variational problems are applied in astronomy, physics, geometry, algebra and analysis (see Almeida and Torres 2009a; Blasjo 2005). Isoperimetric problems were investigated in the scope of engineering applications by Curtis (2004). General optimality conditions have been developed for FVP and IFVP. For instance in Agrawal (2002), Agrawal achieved the necessary optimality conditions for IFVP with Riemann–Liouville derivatives. In Almeida and Torres (2011), Almeida and Torres present necessary and sufficient optimality conditions for a class of FVP with respect to the Caputo fractional derivative. Optimality conditions for FVP with functionals including both fractional derivatives and integrals are presented in Almeida and Torres (2009b). Agrawal (2010) discusses about a general form of FVP and claims that the derived expressions
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