Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Fractional Calculus and Applied Analysis

Editor-in-Chief: Kiryakova, Virginia


IMPACT FACTOR 2018: 3.514
5-year IMPACT FACTOR: 3.524

CiteScore 2018: 3.44

SCImago Journal Rank (SJR) 2018: 1.891
Source Normalized Impact per Paper (SNIP) 2018: 1.808

Mathematical Citation Quotient (MCQ) 2017: 0.98

Online
ISSN
1314-2224
See all formats and pricing
More options …
Volume 14, Issue 4

Issues

Fractional calculus of variations for a combined Caputo derivative

Agnieszka Malinowska / Delfim Torres
  • Center for Research and Development in Mathematics and Applications Department of Mathematics, University of Aveiro, 3810-193, Aveiro, Portugal
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2011-09-29 | DOI: https://doi.org/10.2478/s13540-011-0032-6

Abstract

We generalize the fractional Caputo derivative to the fractional derivative C D γα,β, which is a convex combination of the left Caputo fractional derivative of order α and the right Caputo fractional derivative of order β. The fractional variational problems under our consideration are formulated in terms of C D γα,β. The Euler-Lagrange equations for the basic and isoperimetric problems, as well as transversality conditions, are proved.

MSC: Primary 26A33; Secondary 49K05

Keywords: fractional derivatives; Caputo derivatives; fractional variational principles; Euler-Lagrange equations; isoperimetric constraints; transversality conditions

  • [1] O.P. Agrawal, Formulation of Euler-Lagrange equations for fractional variational problems. J. Math. Anal. Appl. 272, No 1 (2002), 368–379. http://dx.doi.org/10.1016/S0022-247X(02)00180-4CrossrefGoogle Scholar

  • [2] O.P. Agrawal, Generalized Euler-Lagrange equations and transversality conditions for FVPs in terms of the Caputo derivative. J. Vib. Control 13, No 9–10 (2007), 1217–1237. http://dx.doi.org/10.1177/1077546307077472CrossrefWeb of ScienceGoogle Scholar

  • [3] O.P. Agrawal, Fractional variational calculus in terms of Riesz fractional derivatives. J. Phys. A 40, No 24 (2007), 6287–6303. http://dx.doi.org/10.1088/1751-8113/40/24/003CrossrefGoogle Scholar

  • [4] R. Almeida, A.B. Malinowska, D.F.M. Torres, A fractional calculus of variations for multiple integrals with application to vibrating string. J. Math. Phys. 51, No 3 (2010), 033503, 12 pp. http://dx.doi.org/10.1063/1.3319559Web of ScienceCrossrefGoogle Scholar

  • [5] R. Almeida, D.F.M. Torres, Calculus of variations with fractional derivatives and fractional integrals. Appl. Math. Lett. 22, No 12 (2009), 1816–1820. http://dx.doi.org/10.1016/j.aml.2009.07.002Web of ScienceCrossrefGoogle Scholar

  • [6] R. Almeida, D.F.M. Torres, Leitmann’s direct method for fractional optimization problems. Appl. Math. Comput. 217, No 3 (2010), 956–962. http://dx.doi.org/10.1016/j.amc.2010.03.085CrossrefWeb of ScienceGoogle Scholar

  • [7] R. Almeida, D.F.M. Torres, Fractional variational calculus for nondifferentiable functions. Comput. Math. Appl. 61, No 10 (2011), 3097–3104. http://dx.doi.org/10.1016/j.camwa.2011.03.098CrossrefWeb of ScienceGoogle Scholar

  • [8] T.M. Atanacković, S. Konjik, S. Pilipović, Variational problems with fractional derivatives: Euler-Lagrange equations. J. Phys. A 41, No 9 (2008), 095201, 12 pp. Google Scholar

  • [9] D. Baleanu, Fractional variational principles in action. Phys. Scripta T136 (2009), Article Number: 014006. Web of ScienceGoogle Scholar

  • [10] D. Baleanu, O.P. Agrawal, Fractional Hamilton formalism within Caputo’s derivative. Czechoslovak J. Phys. 56, No 10–11 (2006), 1087–1092. http://dx.doi.org/10.1007/s10582-006-0406-xCrossrefGoogle Scholar

  • [11] D. Baleanu, A.K. Golmankhaneh, R. Nigmatullin, A.K. Golmankhaneh, Fractional Newtonian mechanics. Cent. Eur. J. Phys. 8, No 1 (2010), 120–125. http://dx.doi.org/10.2478/s11534-009-0085-xCrossrefGoogle Scholar

  • [12] D. Baleanu, S.I. Muslih, Lagrangian formulation of classical fields within Riemann-Liouville fractional derivatives. Phys. Scripta 72, No 2–3 (2005), 119–121. http://dx.doi.org/10.1238/Physica.Regular.072a00119CrossrefGoogle Scholar

  • [13] N.R.O. Bastos, R.A.C. Ferreira, D.F.M. Torres, Discrete-time fractional variational problems. Signal Process. 91, No 3 (2011), 513–524. http://dx.doi.org/10.1016/j.sigpro.2010.05.001CrossrefWeb of ScienceGoogle Scholar

  • [14] R. Brunetti, D. Guido, R. Longo, Modular structure and duality in conformal quantum field theory. Comm. Math. Phys. 156, No 1 (1993), 201–219. http://dx.doi.org/10.1007/BF02096738CrossrefGoogle Scholar

  • [15] J. Cresson, Fractional embedding of differential operators and Lagrangian systems. J. Math. Phys. 48, No 3 (2007), 033504, 34 pp. http://dx.doi.org/10.1063/1.2483292CrossrefGoogle Scholar

  • [16] R.A. El-Nabulsi, D.F.M. Torres, Necessary optimality conditions for fractional action-like integrals of variational calculus with Riemann-Liouville derivatives of order (α, β). Math. Methods Appl. Sci. 30, No 15 (2007), 1931–1939. http://dx.doi.org/10.1002/mma.879CrossrefGoogle Scholar

  • [17] R.A. El-Nabulsi, D.F.M. Torres, Fractional actionlike variational problems. J. Math. Phys. 49, No 5 (2008), 053521, 7 pp. CrossrefGoogle Scholar

  • [18] Fract. Calc. Appl. Anal., p ISSN 1311-0454, e ISSN 1314-2224, Vol. 1 (1998) — Vol. 13 (2010) at http://www.math.bas.bg/~fcaa; Vol. 14 (2011) at http://www.springerlink.com/content/1311-0454. Google Scholar

  • [19] G.S.F. Frederico, D.F.M. Torres, A formulation of Noether’s theorem for fractional problems of the calculus of variations. J. Math. Anal. Appl. 334, No 2 (2007), 834–846. http://dx.doi.org/10.1016/j.jmaa.2007.01.013CrossrefGoogle Scholar

  • [20] G.S.F. Frederico, D.F.M. Torres, Fractional conservation laws in optimal control theory. Nonlinear Dynam. 53, No 3 (2008), 215–222. http://dx.doi.org/10.1007/s11071-007-9309-zWeb of ScienceCrossrefGoogle Scholar

  • [21] G.S.F. Frederico, D.F.M. Torres, Fractional Noether’s theorem in the Riesz-Caputo sense. Appl. Math. Comput. 217, No 3 (2010), 1023–1033. http://dx.doi.org/10.1016/j.amc.2010.01.100Web of ScienceCrossrefGoogle Scholar

  • [22] R. Hilfer, Applications of Fractional Calculus in Physics. World Sci. Publishing, River Edge, NJ (2000). http://dx.doi.org/10.1142/9789812817747CrossrefGoogle Scholar

  • [23] G. Jumarie, Fractional Hamilton-Jacobi equation for the optimal control of nonrandom fractional dynamics with fractional cost function. J. Appl. Math. Comput. 23, No 1–2 (2007), 215–228. http://dx.doi.org/10.1007/BF02831970CrossrefGoogle Scholar

  • [24] G. Jumarie, An approach via fractional analysis to non-linearity induced by coarse-graining in space. Nonlinear Anal. Real World Appl. 11, No 1 (2010), 535–546. http://dx.doi.org/10.1016/j.nonrwa.2009.01.003Web of ScienceCrossrefGoogle Scholar

  • [25] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006). Google Scholar

  • [26] M. Klimek, Stationarity-conservation laws for fractional differential equations with variable coefficients. J. Phys. A 35, No 31 (2002), 6675–6693. http://dx.doi.org/10.1088/0305-4470/35/31/311CrossrefGoogle Scholar

  • [27] A.B. Malinowska, D.F.M. Torres, On the diamond-alpha Riemann integral and mean value theorems on time scales. Dynam. Systems Appl. 18, No 3–4 (2009), 469–481. Google Scholar

  • [28] A.B. Malinowska, D.F.M. Torres, Generalized natural boundary conditions for fractional variational problems in terms of the Caputo derivative. Comput. Math. Appl. 59, No 9 (2010), 3110–3116. http://dx.doi.org/10.1016/j.camwa.2010.02.032CrossrefGoogle Scholar

  • [29] A.B. Malinowska, D.F.M. Torres, Natural boundary conditions in the calculus of variations. Math. Methods Appl. Sci. 33, No 14 (2010), 1712–1722. http://dx.doi.org/10.1002/mma.1289CrossrefGoogle Scholar

  • [30] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993). Google Scholar

  • [31] D. Mozyrska, D.F.M. Torres, Minimal modified energy control for fractional linear control systems with the Caputo derivative. Carpathian J. Math. 26, No 2 (2010), 210–221. Google Scholar

  • [32] T. Odzijewicz, D.F.M. Torres, Fractional calculus of variations for double integrals. Balkan J. Geom. Appl. 16, No 2 (2011), 102–113. Google Scholar

  • [33] K.B. Oldham, J. Spanier, The Fractional Calculus. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York (1974). Google Scholar

  • [34] A.Yu. Plakhov, D.F.M. Torres, Newton’s aerodynamic problem in media of chaotically moving particles. Mat. Sb. 196, No 6 (2005), 111–160 (In Russian); transl. in EN: Sb. Math. 196, No 5–6 (2005), 885–933. Google Scholar

  • [35] I. Podlubny, Fractional Differential Equations. Academic Press, San Diego, CA (1999). Google Scholar

  • [36] E.M. Rabei, B.S. Ababneh, Hamilton-Jacobi fractional mechanics. J. Math. Anal. Appl. 344, No 2 (2008), 799–805. http://dx.doi.org/10.1016/j.jmaa.2008.03.011CrossrefGoogle Scholar

  • [37] E.M. Rabei, K.I. Nawafleh, R.S. Hijjawi, S.I. Muslih, D. Baleanu, The Hamilton formalism with fractional derivatives. J. Math. Anal. Appl. 327, No 2 (2007), 891–897. http://dx.doi.org/10.1016/j.jmaa.2006.04.076CrossrefGoogle Scholar

  • [38] F. Riewe, Nonconservative Lagrangian and Hamiltonian mechanics. Phys. Rev. E (3) 53, No 2 (1996), 1890–1899. http://dx.doi.org/10.1103/PhysRevE.53.1890CrossrefGoogle Scholar

  • [39] B. Ross, S.G. Samko, E.R. Love, Functions that have no first order derivative might have fractional derivatives of all orders less than one. Real Anal. Exchange 20, No 1 (1994/95), 140–157. Google Scholar

  • [40] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives. Transl. from the 1987 Russian original, Gordon and Breach, Yverdon (1993). Google Scholar

  • [41] M.R. Sidi Ammi, R.A.C. Ferreira, D.F.M. Torres, Diamond-α Jensen’s inequality on time scales. J. Inequal. Appl. 2008, Art. ID 576876 (2008), 13 pp. Google Scholar

  • [42] V.E. Tarasov, Fractional vector calculus and fractional Maxwell’s equations. Ann. Physics 323, No 11 (2008), 2756–2778. http://dx.doi.org/10.1016/j.aop.2008.04.005Web of ScienceCrossrefGoogle Scholar

  • [43] J.L. Troutman, Variational Calculus and Optimal Control. Second Ed., Springer, New York (1996). http://dx.doi.org/10.1007/978-1-4612-0737-5CrossrefGoogle Scholar

  • [44] B. van Brunt, The Calculus of Variations. Springer, New York (2004). Google Scholar

About the article

Published Online: 2011-09-29

Published in Print: 2011-12-01


Citation Information: Fractional Calculus and Applied Analysis, Volume 14, Issue 4, Pages 523–537, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: https://doi.org/10.2478/s13540-011-0032-6.

Export Citation

© 2011 Diogenes Co., Sofia. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Chuan-Jing Song and Yi Zhang
Fractional Calculus and Applied Analysis, 2018, Volume 21, Number 2, Page 509
[3]
Shao-Kai Luo, Yun Dai, Ming-Jing Yang, and Xiao-Tian Zhang
International Journal of Theoretical Physics, 2017
[4]
Dina Tavares, Ricardo Almeida, and Delfim Torres
Discrete and Continuous Dynamical Systems - Series S, 2017, Volume 11, Number 1, Page 143
[5]
Shao-Kai Luo, Yun Dai, Xiao-Tian Zhang, and Ming-Jing Yang
International Journal of Non-Linear Mechanics, 2017
[6]
Agnieszka B Malinowska
Journal of Vibration and Control, 2013, Volume 19, Number 8, Page 1161
[7]
Mehdi Dehghan, Ehsan-Allah Hamedi, and Hassan Khosravian-Arab
Journal of Vibration and Control, 2016, Volume 22, Number 6, Page 1547
[8]
Dina Tavares, Ricardo Almeida, and Delfim F.M. Torres
Journal of Computational and Applied Mathematics, 2017
[9]
Shao-Fang Wen, Yong-Jun Shen, Xiao-Na Wang, Shao-Pu Yang, and Hai-Jun Xing
Chaos: An Interdisciplinary Journal of Nonlinear Science, 2016, Volume 26, Number 8, Page 084309
[10]
Shaofang Wen, Yongjun Shen, Xianghong Li, and Shaopu Yang
International Journal of Non-Linear Mechanics, 2016, Volume 84, Page 130
[11]
F. Ghomanjani
Journal of the Egyptian Mathematical Society, 2016, Volume 24, Number 4, Page 638
[12]
Yong-Jun Shen, Shao-Fang Wen, Xiang-Hong Li, Shao-Pu Yang, and Hai-Jun Xing
Nonlinear Dynamics, 2016, Volume 85, Number 3, Page 1457
[13]
Samer S. Ezz-Eldien, Ramy M. Hafez, Ali H. Bhrawy, Dumitru Baleanu, and Ahmed A. El-Kalaawy
Journal of Optimization Theory and Applications, 2017, Volume 174, Number 1, Page 295
[14]
Rami Ahmad El-Nabulsi
Nonlinear Dynamics, 2013, Volume 74, Number 1-2, Page 381
[15]
Hossein Jafari, Hale Tajadodi, and Dumitru Baleanu
Fractional Calculus and Applied Analysis, 2013, Volume 16, Number 1
[16]
Rami Ahmad El-Nabulsi
International Journal of Theoretical Physics, 2012, Volume 51, Number 12, Page 3978
[17]
Rami Ahmad El-Nabulsi
Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas, 2013, Volume 107, Number 2, Page 419
[18]
Gastão S.F. Frederico and Delfim F.M. Torres
Reports on Mathematical Physics, 2013, Volume 71, Number 3, Page 291
[19]
Tatiana Odzijewicz, Agnieszka B. Malinowska, and Delfim F.M. Torres
Computers & Mathematics with Applications, 2012, Volume 64, Number 10, Page 3351
[20]
Agnieszka Malinowska and Delfim Torres
Fractional Calculus and Applied Analysis, 2012, Volume 15, Number 3
[21]
Małgorzata Klimek and Maria Lupa
Fractional Calculus and Applied Analysis, 2013, Volume 16, Number 1
[22]
Dina Tavares, Ricardo Almeida, and Delfim F.M. Torres
Optimization, 2015, Volume 64, Number 6, Page 1381
[23]
F. Bahrami, H. Fazli, and A. Jodayree Akbarfam
Communications in Nonlinear Science and Numerical Simulation, 2015, Volume 23, Number 1-3, Page 39
[24]
Tomasz Blaszczyk and Mariusz Ciesielski
Fractional Calculus and Applied Analysis, 2014, Volume 17, Number 2
[25]
Yongjun Shen, Peng Wei, Chuanyi Sui, and Shaopu Yang
Mathematical Problems in Engineering, 2014, Volume 2014, Page 1
[26]
Hui Wang and Tie-Cheng Xia
Journal of Mathematical Physics, 2013, Volume 54, Number 4, Page 043505

Comments (0)

Please log in or register to comment.
Log in