Noether’s theorem for fractional variational problems of variable order

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

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

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

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

[5] R. Caponetto, G. Dongola, L. Fortuna, I. Petras, Fractional Order Systems: Modeling and Control Applications (World Scientific Company, Singapore, 2010) 10.1142/7709Search in Google Scholar

[6] F. Jarad, T. Abdeljawad, D. Baleanu, Nonlinear Anal. Real World Appl. 14, 780 (2013) http://dx.doi.org/10.1016/j.nonrwa.2012.08.00110.1016/j.nonrwa.2012.08.001Search in Google Scholar

[7] D. Baleanu, K. Diethelm, E. Scalas, J. J. Trujillo, Fractional calculus (World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012) 10.1142/8180Search in Google Scholar

[8] M. D. Ortigueira, Fractional calculus for scientists and engineers, Lecture Notes in Electrical Engineering (Springer, Dordrecht, 2011) 84 http://dx.doi.org/10.1007/978-94-007-0747-410.1007/978-94-007-0747-4Search in Google Scholar

[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.2010.05.02710.1016/j.cnsns.2010.05.027Search in Google Scholar

[10] S. G. Samko, B. Ross, Integral Transform. Spec. Funct. 1, 277 (1993) http://dx.doi.org/10.1080/1065246930881902710.1080/10652469308819027Search in Google Scholar

[11] S. G. Samko, Anal. Math. 21, 213 (1995) http://dx.doi.org/10.1007/BF0191112610.1007/BF01911126Search in Google Scholar

[12] B. Ross, S. G. Samko, Internat. J. Math. Math. Sci. 18, 777 (1995) http://dx.doi.org/10.1155/S016117129500100110.1155/S0161171295001001Search in Google Scholar

[13] C. F. M. Coimbra, Ann. Phys. 12, 692 (2003) http://dx.doi.org/10.1002/andp.20031003210.1002/andp.200310032Search in Google Scholar

[14] G. Diaz, C. F. M. Coimbra, Nonlinear Dynam. 56, 145 (2009) http://dx.doi.org/10.1007/s11071-008-9385-810.1007/s11071-008-9385-8Search in Google Scholar

[15] C. F. Lorenzo, T. T. Hartley, Nonlinear Dynam. 29, 57 (2002) http://dx.doi.org/10.1023/A:101658690565410.1023/A:1016586905654Search in Google Scholar

[16] L. E. S. Ramirez, C. F. M. Coimbra, Int. J. Differ. Equ. 2010, 16 (2010) 10.1155/2010/846107Search in Google Scholar

[17] L. E. S. Ramirez, C. F. M. Coimbra, Phys. D 240, 1111 (2011) http://dx.doi.org/10.1016/j.physd.2011.04.00110.1016/j.physd.2011.04.001Search in Google Scholar

[18] F. Riewe, Phys. Rev. E 53, 1890 (1996) http://dx.doi.org/10.1103/PhysRevE.53.189010.1103/PhysRevE.53.1890Search in Google Scholar PubMed

[19] F. Riewe, Phys. Rev. E 55, 3581 (1997) http://dx.doi.org/10.1103/PhysRevE.55.358110.1103/PhysRevE.55.3581Search in Google Scholar

[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.2011.05.00210.1016/j.cnsns.2011.05.002Search in Google Scholar

[21] R. Almeida, D. F. M. Torres, Appl. Math. Lett. 22, 1816 (2009) http://dx.doi.org/10.1016/j.aml.2009.07.00210.1016/j.aml.2009.07.002Search in Google Scholar

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

[23] D. Baleanu, S. I. Muslih, Phys. Scripta 72, 119 (2005) http://dx.doi.org/10.1238/Physica.Regular.072a0011910.1238/Physica.Regular.072a00119Search in Google Scholar

[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.2011.29.41710.3934/dcds.2011.29.417Search in Google Scholar

[25] J. Cresson, J. Math. Phys. 48, 34 (2007) 10.1063/1.2736519Search in Google Scholar

[26] G. S. F. Frederico, D. F. M. Torres, Appl. Math. Comput. 217, 1023 (2010) http://dx.doi.org/10.1016/j.amc.2010.01.10010.1016/j.amc.2010.01.100Search in Google Scholar

[27] M. A. E. Herzallah, D. Baleanu, Nonlinear Dynam. 69, 977 (2012) http://dx.doi.org/10.1007/s11071-011-0319-510.1007/s11071-011-0319-5Search in Google Scholar

[28] D. Mozyrska, D. F. M. Torres, Carpathian J. Math. 26, 210 (2010) Search in Google Scholar

[29] D. Mozyrska, D. F. M. Torres, Signal Process. 91, 379 (2011) http://dx.doi.org/10.1016/j.sigpro.2010.07.01610.1016/j.sigpro.2010.07.016Search in Google Scholar

[30] A. B. Malinowska, D. F. M. Torres, Introduction to the fractional calculus of variations (Imp. Coll. Press, London, 2012) 10.1142/p871Search in Google Scholar

[31] T. M. Atanackovic, S. Pilipovic, Fract. Calc. Appl. Anal. 14, 94 (2011) 10.2478/s13540-011-0007-7Search in Google Scholar

[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 10.1109/CDC.2012.6426489Search in Google Scholar

[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 10.1007/978-3-0348-0516-2_16Search in Google Scholar

[34] J. Jost, X. Li-Jost, Calculus of variations (Cambridge Univ. Press, Cambridge, 1998) Search in Google Scholar

[35] Y. Kosmann-Schwarzbach, The Noether theorems: Invariance and conservation laws in the twentieth century (Springer, New York, 2011) http://dx.doi.org/10.1007/978-0-387-87868-310.1007/978-0-387-87868-3Search in Google Scholar

[36] B. van Brunt, The calculus of variations (Springer, New York, 2004) 10.1007/b97436Search in Google Scholar

[37] T. M. Atanackovic, S. Konjik, S. Pilipovic, S. Simic, Nonlinear Anal. 71, 1504 (2009) http://dx.doi.org/10.1016/j.na.2008.12.04310.1016/j.na.2008.12.043Search in Google Scholar

[38] G. S. F. Frederico, D. F. M. Torres, J. Math. Anal. Appl. 334, 834 (2007) http://dx.doi.org/10.1016/j.jmaa.2007.01.01310.1016/j.jmaa.2007.01.013Search in Google Scholar

[39] G. S. F. Frederico, D. F. M. Torres, Nonlinear Dynam. 53, 215 (2008) http://dx.doi.org/10.1007/s11071-007-9309-z10.1007/s11071-007-9309-zSearch in Google Scholar

[40] P. Ivady, J. Math. Inequal. 3, 227 (2009) http://dx.doi.org/10.7153/jmi-03-2310.7153/jmi-03-23Search in Google Scholar