Abstract
We prove multidimensional integration by parts formulas for generalized fractional derivatives and integrals. The new results allow us to obtain optimality conditions for multidimensional fractional variational problems with Lagrangians depending on generalized partial integrals and derivatives. A generalized fractional Noether’s theorem, a formulation of Dirichlet’s principle and an uniqueness result are given.
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N.H. Abel, Euvres completes de Niels Henrik Abel (Christiana: Imprimerie de Grondahl and Son; New York and London: Johnson Reprint Corporation. VIII, 1965), p. 621
O.P. Agrawal, Comput. Math. Appl. 59, 1852 (2010)
R. Almeida, A.B. Malinowska, D.F.M. Torres, J. Math. Phys. 51, 033503 (2010)
R. Almeida, D.F.M. Torres, Appl. Math. Lett. 22, 1816 (2009)
D. Baleanu, S. Muslih, Physica Scripta 72, 119 (2005)
L. Bourdin, T. Odzijewicz, D.F.M. Torres, Adv. Dyn. Syst. Appl. 8, 3 (2013)
J. Cresson, J. Math. Phys. 48, 033504 (2007)
L.C. Evans, Partial Differential Equations (Gruaduate Studies in Mathematics, American Mathematical Society, United States of America, 1997)
G.M. Ewing, Calculus of variations with applications (Courier Dover Publications, New York, 1985)
G.S.F. Frederico, D.F.M. Torres, J. Math. Anal. Appl. 334, 834 (2007)
G.S.F. Frederico, D.F.M. Torres, Appl. Math. Comput. 217, 1023 (2010)
J. Jost, X. Li-Jost, Calculus of variations (Cambridge Univ. Press, Cambridge, 1998)
A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and applications of fractional differential equations (North-Holland Mathematics Studies, 204, Elsevier, Amsterdam, 2006)
V. Kiryakova, Generalized fractional calculus and applications (Longman Sci. Tech., Harlow, 1994)
M. Klimek, On solutions of linear fractional differential equations of a variational type (The Publishing Office of Czenstochowa University of Technology, Czestochowa, 2009)
M. Klimek, M. Lupa, Fract. Calc. Appl. Anal. 16, 243 (2013)
M.J. Lazo, D.F.M. Torres, J. Optim. Theory Appl. 156, 56 (2013)
A.B. Malinowska, Appl. Math. Lett. 25, 1941 (2012)
A.B. Malinowska, J. Vib. Control 19, 1161 (2013)
A.B. Malinowska, D.F.M. Torres, Introduction to the fractional calculus of variations (Imperial College Press, London & World Scientific Publishing, Singapore, 2012)
E. Noether, Gött. Nachr., 235 (1918)
T. Odzijewicz, A.B. Malinowska, D.F.M. Torres, Nonlinear Anal. 75, 1507 (2012)
T. Odzijewicz, A.B. Malinowska, D.F.M. Torres, Comput. Math. Appl. 64, 3351 (2012)
T. Odzijewicz, A.B. Malinowska, D.F.M. Torres, Abstr. Appl. Anal., ID 871912 (2012)
T. Odzijewicz, A.B. Malinowska, D.F.M. Torres, Fract. Calc. Appl. Anal. 16, 64 (2013)
T. Odzijewicz, D.F.M. Torres, Balkan J. Geom. Appl. 16, 102 (2011)
I. Podlubny, Fractional differential equations (Academic Press, San Diego, CA, 1999)
F. Riewe, Phys. Rev. E (3) 53, 1890 (1996)
F. Riewe, Phys. Rev. E (3) 55, 3581 (1997)
S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional integrals and derivatives (Gordon and Breach, Yverdon, 1993)
V.E. Tarasov, Ann. Phys. 323, 2756 (2008)
D.F.M. Torres, Eur. J. Control 8, 56 (2002)
D.F.M. Torres, Commun. Pure Appl. Anal. 3, 491 (2004)
B. van Brunt, The calculus of variations (Universitext, Springer, New York, 2004)
R. Weinstock, Calculus of variations with applications to physics and engineering (McGraw Hill Book Company Inc., 1952)
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Odzijewicz, T., Malinowska, A.B. & Torres, D.F.M. Fractional calculus of variations of several independent variables. Eur. Phys. J. Spec. Top. 222, 1813–1826 (2013). https://doi.org/10.1140/epjst/e2013-01966-0
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DOI: https://doi.org/10.1140/epjst/e2013-01966-0