Geodesic
Constructing conservation laws for fractional-order integro-differential equations
Teoretičeskaâ i matematičeskaâ fizika, Tome 184 (2015) no. 2, pp. 179-199 Cet article a éte moissonné depuis la source Math-Net.Ru

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In a class of functions depending on linear integro-differential fractional-order variables, we prove an analogue of the fundamental operator identity relating the infinitesimal operator of a point transformation group, the Euler–Lagrange differential operator, and Noether operators. Using this identity, we prove fractional-differential analogues of the Noether theorem and its generalizations applicable to equations with fractional-order integrals and derivatives of various types that are Euler–Lagrange equations. In explicit form, we give fractional-differential generalizations of Noether operators that gives an efficient way to construct conservation laws, which we illustrate with three examples.
Keywords: integro-differential fractional-order equation, symmetry, conservation law, fundamental operator identity, Noether theorem. 
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S. Yu. Lukashchuk. Constructing conservation laws for fractional-order integro-differential equations. Teoretičeskaâ i matematičeskaâ fizika, Tome 184 (2015) no. 2, pp. 179-199. http://geodesic.mathdoc.fr/item/TMF_2015_184_2_a0/

[1] S. G. Samko, A. A. Kilbas, O. I. Marichev, Integraly i proizvodnye drobnogo poryadka i nekotorye ikh prilozheniya, Nauka i tekhnika, Minsk, 1987 | MR | Zbl

[2] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier, Amsterdam, 2006 | MR | Zbl

[3] R. Metzler, J. Klafter, Phys. Rep., 339:1 (2000), 1–77 | DOI | MR | Zbl

[4] R. Klages, G. Radons, I. M. Sokolov (eds.), Anomalous Transport: Foundations and Applications, Wiley-VCH, Berlin, 2008

[5] J. Klafter, S. C. Lim, R. Metzler (eds.), Fractional Dynamics: Recent Advances, World Sci., Singapore, 2011 | MR

[6] V. V. Uchaikin, Fractional Derivatives for Physicists and Engineers, Springer, Singapore, 2013 | MR | MR | Zbl

[7] L. V. Ovsyannikov, Gruppovoi analiz differentsialnykh uravnenii, Nauka, M., 1978 | MR

[8] N. Kh. Ibragimov, Gruppy preobrazovanii v matematicheskoi fizike, Nauka, M., 1983 | MR | Zbl

[9] R. K. Gazizov, A. A. Kasatkin, S. Yu. Lukaschuk, Vestn. UGATU, 9:3 (2007), 125–135

[10] R. K. Gazizov, A. A. Kasatkin, S. Yu. Lukashchuk, Phys. Scr., 2009:T136 (2009), 014016, 5 pp. | DOI

[11] R. K. Gazizov, A. A. Kasatkin, S. Yu. Lukaschuk, Ufimsk. matem. zhurn., 4:4 (2012), 54–68 | MR | Zbl

[12] E. Noether, Nachr. d. König. Gesellsch. d. Wiss. zu Göttingen, Math-phys. Klasse, 2 (1918), 235–257 | Zbl

[13] N. Kh. Ibragimov, TMF, 1:3 (1969), 350–359 | DOI | MR

[14] S.-H. Zhang, D.-Y. Chen, J.-L. Fu, Chinese Phys. B, 21:10 (2012), 100202, 8 pp. | DOI

[15] T. M. Atanackovic, S. Konjik, S. Pilipovic, S. Simic, Nonlinear Anal., 71:5–6 (2009), 1504–1517 | DOI | MR | Zbl

[16] G. S. F. Frederico, D. F. M. Torres, “Fractional Noether's theorem with classical and Riemann–Liouville derivatives”, Decision and Control (CDC), Proceedings of 51st IEEE Conference on Decision and Control (Maui, Hawaii, December 10–13, 2012), IEEE, New York, 2012, 6885–6890 | DOI

[17] G. S. F. Frederico, T. Odzijewicz, D. F. M. Torres, Appl. Anal., 96:1 (2014), 153–170 | DOI | MR

[18] M. Klimek, J. Phys. A: Math. Theor., 34:31 (2001), 6167–6184 | DOI | MR | Zbl

[19] S. W. Wheatcraft, M. M. Meerschaert, Adv. Water Resour., 31:10 (2008), 1377–1381 | DOI

[20] V. E. Tarasov, Phys. Plasmas, 20:10 (2013), 102110, 10 pp., arXiv: 1307.4930 | DOI

[21] G. S. F. Frederico, D. F. M. Torres, J. Math. Anal. Appl., 334:2 (2007), 834–846 | DOI | MR | Zbl

[22] A. B. Malinowska, Appl. Math. Lett., 25:11 (2012), 1941–1946 | DOI | MR | Zbl

[23] L. Bourdin, J. Cresson, I. Greff, Commun. Nonlinear Sci. Numer. Simulat., 18:4 (2013), 878–887 | DOI | MR | Zbl

[24] N. H. Ibragimov, Elementary Lie Group Analysis and Ordinary Differential Equations, Wiley Series in Mathematical Methods in Practice, 4, John Wiley Sons, Chichester, 1999 | MR

[25] L. E. Elsgolts, Differentsialnye uravneniya i variatsionnoe ischislenie, Nauka, M., 1969 | MR

[26] O. P. Agrawal, J. Phys. A: Math. Theor., 40:21 (2007), 5469–5477 | DOI | MR | Zbl

[27] D. Baleanu, J. J. Trujillo, Nonlin. Dynam., 52:4 (2008), 331–335 | DOI | MR | Zbl

[28] P. Paradisi, R. Cesari, F. Mainardi, F. Tampieri, Physica A, 293:1–2 (2001), 130–142 | DOI | Zbl

[29] D. A. Benson, S. W. Wheatcraft, M. M. Meerschaert, Water Resources Research, 36:6 (2000), 1403–1412 | DOI

[30] V. M. Goloviznin, P. S. Kondratenko, L. V. Matveev, I. A. Korotkin, I. L. Dranikov, Anomalnaya diffuziya radionuklidov v silnoneodnorodnykh geologicheskikh formatsiyakh, Nauka, M., 2010

[31] T. Odzijewicz, A. B. Malinowska, D. F. M. Torres, Europ. Phys. J., 222:8 (2013), 1813–1826

[32] N. H. Ibragimov, J. Phys. A: Math. Theor., 44:43 (2011), 432002, 8 pp. | DOI | Zbl

[33] N. Kh. Ibragimov, E. D. Avdonina, UMN, 68:5(413) (2013), 111–146 | DOI | MR | Zbl