Fractional variational problems on conformable calculus

Yıl 2021, Cilt: 70 Sayı: 2, 719 - 730, 31.12.2021

Süleyman Öğrekçi Serkan Aslıyüce

https://doi.org/10.31801/cfsuasmas.820580

Öz

In this paper, we deal with the variational problems defined by an integral that include fractional conformable derivative. We obtained the optimality results for variational problems with fixed end-point boundary conditions and variable end-point boundary conditions. Then, we studied on the variational problems with integral constraints and holonomic constraints, respectively.

Anahtar Kelimeler

Conformable fractional derivative, calculus of variations, subsidiary conditions

Kaynakça

• Abdeljawad, T., On conformable fractional calculus. J. Comput. Appl. Math., 279 (2015), 57-66. https://doi.org/10.1016/j.cam.2014.10.016
• Agarwal, O. P., Formulation of Euler-Lagrange equations for fractional variational problems. J. Math. Anal. Appl., 272 (2002), 368-379. https://doi.org/10.1016/S0022-247X(02)00180-4
• Agarwal, O. P., Fractional variational calculus and the transversality conditions. J. Phys. A, 39 (33) (2006), 10375-10384. https://doi.org/10.1088/0305-4470/39/33/008
• Agarwal, O. P., Fractional variational calculus in terms of Riesz fractional derivatives. J. Phys. A, 40 (24) (2007), 6287-6303. https://doi.org/10.1088/1751-8113/40/24/003
• Almeida, R., Fractional variational problems with the Riesz-Caputo derivative. Appl. Math. Lett., 25 (2) (2012), 142-148. https://doi.org/10.1016/j.aml.2011.08.003
• Almeida, R., Variational problems involving a Caputo-type fractional derivative. J. Optim. Theory Appl., 174 (1) (2017), 276-294. https://doi.org/10.1007/s10957-016-0883-4
• Bastos, N. R. O., Calculus of variations involving Caputo-Fabrizio fractional differentiation. Stat., Optim. Inf. Comput., 6 (2018), 12-21. https://doi.org/10.19139/soic.v6i1.466
• Batarfi, H., Losada, J., Nieto, J. J., Shammakh, W., Three-point boundary value problems for conformable fractional differential equations. J. Funct. Spaces, 2015, Art. ID 706383, 6 pp. https://doi.org/10.1155/2015/706383
• Chatibi, Y., El Kinani, E. H., Ouhadan, A., Lie symmetry analysis of conformable differential equations. AIMS Math., 4 (4) (2019), 1133--1144. https://doi.org/10.3934/math.2019.4.1133
• Chatibi, Y., El Kinani, E. H., and Ouhadan, A., Variational calculus involving nonlocal fractional derivative with Mittag-Leffler kernel. Chaos, Solitons & Fractals, 118 (2019), 117-121. https://doi.org/10.1016/j.chaos.2018.11.017
• Chatibi, Y., El Kinani, E. H., Ouhadan, A., Lie symmetry analysis and conservation laws for the time fractional Black-Scholes equation. International Journal of Geometric Methods in Modern Physics, 17 (01) (2020), 2050010. https://doi.org/10.1142/S0219887820500103
• Chatibi, Y., El Kinani, E. H., Ouhadan, A., On the discrete symmetry analysis of some classical and fractional differential equations. Math. Methods Appl. Sci., 44 (4) (2021), 2868--2878. https://doi.org/10.1002/mma.6064
• Chung, W. S., Fractional Newton mechanics with conformable fractional derivative. J. Comput. Appl. Math., 290 (2015), 150-158. https://doi.org/10.1016/j.cam.2015.04.049
• Çenesiz, Y., Kurt, A., The solutions of time and space conformable fractional heat equations with conformable Fourier transform. Acta Univ. Sapientiae Math., 7 (2) (2015), 130--140. https://doi.org/10.1515/ausm-2015-0009
• Diethelm, K., The Analysis of Fractional Differential Equations. Springer, 2010.
• Eroğlu, B. B. I., and Yapışkan, D., Generalized conformable variational calculus and optimal control problems with variable terminal conditions. AIMS Mathematics, 5 (2020), 1105-1126. https://doi.org/10.3934/math.2020077
• Gelfand, I. M., and Fomin, S. V., Calculus of variations. Prentice-Hall, Inc., 1963.
• Kadkhoda, N., Jafari, H., An analytical approach to obtain exact solutions of some space-time conformable fractional differential equations. Adv. Difference Equ., 2019, Paper No. 428, 10 pp. https://doi.org/10.1186/s13662-019-2349-0
• Khalil, R., Al Horani, M., Yousef, A., and Sababheh, M., A new definition of fractional derivative. J. Comput. Appl. Math., 264 (2014), 65-70. https://doi.org/10.1016/j.cam.2014.01.002
• Kilbas, A. A., Srivastava, H. M., and Trujillo, J. J., Theory and Applications of Fractional Differential Equations. Elsevier Science, 2006.
• Lazo, M. J., and Torres, D. F. M., Variational calculus with conformable fractional derivatives. IEEE/CAA Journal of Automatica Sinica, 4 (2) (2017), 340-352. https://doi.org/10.1109/JAS.2016.7510160
• Machado, J. T. , Kiryakova, V., and Mainardi, F., Recent history of fractional calculus. Commun. Nonlinear Sci. Numer. Simul., 16 (3) (2011), 1140-1153. https://doi.org/10.1016/j.cnsns.2010.05.027
• Mainardi, F., An historical perspective on fractional calculus in linear viscoelasticity. Fract. Calc. Appl. Anal., 15 (4) (2012), 712-717. https://doi.org/10.2478/s13540-012-0048-6
• Miller, K. S., and Ross, B., An Introduction to Fractional Calculus and Fractional Differential Equations. Wiley Interscience, 1993.
• Riewe, F., Nonconservative Lagrangian and Hamiltonian mechanics. Phys. Rev. E, 53 (2) (1996), 1890-1899. https://doi.org/10.1103/PhysRevE.53.1890
• Riewe, F., Mechanics with fractional derivatives. Phys. Rev. E, 55 (3) (1997), 3581-3592. https://doi.org/10.1103/PhysRevE.55.3581
• Ross., B., A brief history and exposition of the fundamental theory of fractional calculus, pages 1-36. Springer Berlin Heidelberg, Berlin, Heidelberg, 1975.
• Weberszpil, J., and Helayel-Neto, J. A., Variational approach and deformed derivatives. Physica A, 450 (2016), 217-227. https://doi.org/10.1016/j.physa.2015.12.145
• Zhang, J., Ma, X., and Li, L., Optimality conditions for fractional variational problems with Caputo-Fabrizio fractional derivatives. Adv. Differ. Equ., 357 (2017). https://doi.org/10.1186/s13662-017-1388-7