Jump to ContentJump to Main Navigation

Open Physics

formerly Central European Journal of Physics

1 Issue per year


IMPACT FACTOR increased in 2014: 1.085

SCImago Journal Rank (SJR) 2014: 0.372
Source Normalized Impact per Paper (SNIP) 2014: 0.650
Impact per Publication (IPP) 2014: 1.000

Open Access
VolumeIssuePage

Issues

Noether’s theorem for fractional variational problems of variable order

1Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193, Aveiro, Portugal

2Faculty of Computer Science, Bialystok University of Technology, 15-351, Białystok, Poland

© 2013 Versita Warsaw. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. (CC BY-NC-ND 3.0)

Citation Information: Open Physics. Volume 11, Issue 6, Pages 691–701, ISSN (Online) 2391-5471, DOI: 10.2478/s11534-013-0208-2, October 2013

Publication History

Published Online:
2013-10-09

Abstract

We prove a necessary optimality condition of Euler-Lagrange type for fractional variational problems with derivatives of incommensurate variable order. This allows us to state a version of Noether’s theorem without transformation of the independent (time) variable. Considered derivatives of variable order are defined in the sense of Caputo.

Keywords: variable order fractional integrals; variable order fractional derivatives; fractional variational analysis; Euler-Lagrange equations; Noether’s theorem

  • [1] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations (Elsevier, Amsterdam, 2006) 204

  • [2] M. Klimek, On solutions of linear fractional differential equations of a variational type (The Publishing Office of Czestochowa University of Technology, Czestochowa, 2009)

  • [3] I. Podlubny, Fractional differential equations, Mathematics in Science and Engineering (Academic Press, San Diego, CA, 1999) 198

  • [4] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives (Gordon and Breach, Yverdon, 1993)

  • [5] R. Caponetto, G. Dongola, L. Fortuna, I. Petras, Fractional Order Systems: Modeling and Control Applications (World Scientific Company, Singapore, 2010)

  • [6] F. Jarad, T. Abdeljawad, D. Baleanu, Nonlinear Anal. Real World Appl. 14, 780 (2013) http://dx.doi.org/10.1016/j.nonrwa.2012.08.001 [CrossRef]

  • [7] D. Baleanu, K. Diethelm, E. Scalas, J. J. Trujillo, Fractional calculus (World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012)

  • [8] M. D. Ortigueira, Fractional calculus for scientists and engineers, Lecture Notes in Electrical Engineering (Springer, Dordrecht, 2011) 84 http://dx.doi.org/10.1007/978-94-007-0747-4 [CrossRef]

  • [9] J. A. Tenreiro Machado, V. Kiryakova, F. Mainardi, Commun. Nonlinear Sci. Numer. Simul. 16, 1140 (2011) http://dx.doi.org/10.1016/j.cnsns.2010.05.027 [CrossRef]

  • [10] S. G. Samko, B. Ross, Integral Transform. Spec. Funct. 1, 277 (1993) http://dx.doi.org/10.1080/10652469308819027 [CrossRef]

  • [11] S. G. Samko, Anal. Math. 21, 213 (1995) http://dx.doi.org/10.1007/BF01911126 [CrossRef]

  • [12] B. Ross, S. G. Samko, Internat. J. Math. Math. Sci. 18, 777 (1995) http://dx.doi.org/10.1155/S0161171295001001 [CrossRef]

  • [13] C. F. M. Coimbra, Ann. Phys. 12, 692 (2003) http://dx.doi.org/10.1002/andp.200310032 [CrossRef]

  • [14] G. Diaz, C. F. M. Coimbra, Nonlinear Dynam. 56, 145 (2009) http://dx.doi.org/10.1007/s11071-008-9385-8 [CrossRef]

  • [15] C. F. Lorenzo, T. T. Hartley, Nonlinear Dynam. 29, 57 (2002) http://dx.doi.org/10.1023/A:1016586905654 [CrossRef]

  • [16] L. E. S. Ramirez, C. F. M. Coimbra, Int. J. Differ. Equ. 2010, 16 (2010)

  • [17] L. E. S. Ramirez, C. F. M. Coimbra, Phys. D 240, 1111 (2011) http://dx.doi.org/10.1016/j.physd.2011.04.001 [CrossRef]

  • [18] F. Riewe, Phys. Rev. E 53, 1890 (1996) http://dx.doi.org/10.1103/PhysRevE.53.1890 [CrossRef]

  • [19] F. Riewe, Phys. Rev. E 55, 3581 (1997) http://dx.doi.org/10.1103/PhysRevE.55.3581 [CrossRef]

  • [20] O. P. Agrawal, S. I. Muslih, D. Baleanu, Commun. Nonlinear Sci. Numer. Simul. 16, 4756 (2011) http://dx.doi.org/10.1016/j.cnsns.2011.05.002 [CrossRef]

  • [21] R. Almeida, D. F. M. Torres, Appl. Math. Lett. 22, 1816 (2009) http://dx.doi.org/10.1016/j.aml.2009.07.002 [CrossRef]

  • [22] D. Baleanu, T. Avkar, Nuovo Cimento Soc. Ital. Fis. B 119, 73 (2004)

  • [23] D. Baleanu, S. I. Muslih, Phys. Scripta 72, 119 (2005) http://dx.doi.org/10.1238/Physica.Regular.072a00119 [CrossRef]

  • [24] N. R. O. Bastos, R. A. C. Ferreira, D. F. M. Torres, Discrete Contin. Dyn. Syst. 29, 417 (2011) http://dx.doi.org/10.3934/dcds.2011.29.417 [CrossRef]

  • [25] J. Cresson, J. Math. Phys. 48, 34 (2007)

  • [26] G. S. F. Frederico, D. F. M. Torres, Appl. Math. Comput. 217, 1023 (2010) http://dx.doi.org/10.1016/j.amc.2010.01.100 [CrossRef]

  • [27] M. A. E. Herzallah, D. Baleanu, Nonlinear Dynam. 69, 977 (2012) http://dx.doi.org/10.1007/s11071-011-0319-5 [CrossRef]

  • [28] D. Mozyrska, D. F. M. Torres, Carpathian J. Math. 26, 210 (2010)

  • [29] D. Mozyrska, D. F. M. Torres, Signal Process. 91, 379 (2011) http://dx.doi.org/10.1016/j.sigpro.2010.07.016 [CrossRef]

  • [30] A. B. Malinowska, D. F. M. Torres, Introduction to the fractional calculus of variations (Imp. Coll. Press, London, 2012)

  • [31] T. M. Atanackovic, S. Pilipovic, Fract. Calc. Appl. Anal. 14, 94 (2011)

  • [32] T. Odzijewicz, A. B. Malinowska, D. F. M. Torres, Variable order fractional variational calculus for double integrals, Proc. 51st IEEE Conference on Decision and Control, Dec. 10–13, 2012, Maui, Hawaii, art. no. 6426489, 6873

  • [33] T. Odzijewicz, A. B. Malinowska, D. F. M. Torres, In: A. Almeida, L. P. Castro, F.-O. Speck (Eds.), Fractional variational calculus of variable order, Advances in Harmonic Analysis and Operator Theory (Birkhäuser, Basel, 2013) 291

  • [34] J. Jost, X. Li-Jost, Calculus of variations (Cambridge Univ. Press, Cambridge, 1998)

  • [35] Y. Kosmann-Schwarzbach, The Noether theorems: Invariance and conservation laws in the twentieth century (Springer, New York, 2011) http://dx.doi.org/10.1007/978-0-387-87868-3 [CrossRef]

  • [36] B. van Brunt, The calculus of variations (Springer, New York, 2004)

  • [37] T. M. Atanackovic, S. Konjik, S. Pilipovic, S. Simic, Nonlinear Anal. 71, 1504 (2009) http://dx.doi.org/10.1016/j.na.2008.12.043 [CrossRef]

  • [38] G. S. F. Frederico, D. F. M. Torres, J. Math. Anal. Appl. 334, 834 (2007) http://dx.doi.org/10.1016/j.jmaa.2007.01.013 [CrossRef]

  • [39] G. S. F. Frederico, D. F. M. Torres, Nonlinear Dynam. 53, 215 (2008) http://dx.doi.org/10.1007/s11071-007-9309-z [CrossRef]

  • [40] P. Ivady, J. Math. Inequal. 3, 227 (2009) http://dx.doi.org/10.7153/jmi-03-23 [CrossRef]

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]
Dina Tavares, Ricardo Almeida, and Delfim F.M. Torres
Optimization, 2015, Volume 64, Number 6, Page 1381
[2]
Ricardo Almeida and Delfim F.M. Torres
Applied Mathematics and Computation, 2015, Volume 257, Page 74
[3]
Stanislav Yu. Lukashchuk
Nonlinear Dynamics, 2015, Volume 80, Number 1-2, Page 791
[4]
Ricardo Almeida and Delfim F. M. Torres
The Scientific World Journal, 2013, Volume 2013, Page 1
[5]
Matheus J. Lazo and Delfim F.M. Torres
Optimization, 2014, Volume 63, Number 8, Page 1157

Comments (0)

Please log in or register to comment.