October  2014, 19(8): 2617-2629. doi: 10.3934/dcdsb.2014.19.2617

Generalized fractional isoperimetric problem of several variables

1. 

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

Received  October 2013 Revised  March 2014 Published  August 2014

This work deals with the generalized fractional calculus of variations of several variables. Precisely, we prove a sufficient optimality condition for the fundamental problem and a necessary optimality condition for the isoperimetric problem. Our results cover important particular cases of problems with constant and variable order fractional operators.
Citation: Tatiana Odzijewicz. Generalized fractional isoperimetric problem of several variables. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2617-2629. doi: 10.3934/dcdsb.2014.19.2617
References:
[1]

O. P. Agrawal, Generalized variational problems and Euler-Lagrange equations, Comput. Math. Appl., 59 (2010), 1852-1864. doi: 10.1016/j.camwa.2009.08.029.  Google Scholar

[2]

R. Almeida, Fractional variational problems with the Riesz-Caputo derivative, Appl. Math. Lett., 25 (2012), 142-148. doi: 10.1016/j.aml.2011.08.003.  Google Scholar

[3]

R. Almeida, A. B. Malinowska and D. F. M. Torres, A fractional calculus of variations for multiple integrals with application to vibrating string, J. Math. Phys., 51 (2010), 033503, 12 pp. doi: 10.1063/1.3319559.  Google Scholar

[4]

R. Almeida, A. B. Malinowska and D. F. M. Torres, Fractional Euler-Lagrange differential equations via Caputo derivatives, Fractional Dynamics and Control, 2 (2012), 109-118. doi: 10.1007/978-1-4614-0457-6_9.  Google Scholar

[5]

R. Almeida, S. Pooseh and D. F. M. Torres, Fractional variational problems depending on indefinite integrals, Nonlinear Anal., 75 (2012), 1009-1025. doi: 10.1016/j.na.2011.02.028.  Google Scholar

[6]

D. Baleanu and I. S. Muslih, Lagrangian formulation of classical fields within Riemann-Liouville fractional derivatives, Phys. Scripta, 72 (2005), 119-121. doi: 10.1238/Physica.Regular.072a00119.  Google Scholar

[7]

N. R. O. Bastos, R. A. C. Ferreira and D. F. M. Torres, Necessary optimality conditions for fractional difference problems of the calculus of variations, Discrete Contin. Dyn. Syst., 29 (2011), 417-437. doi: 10.3934/dcds.2011.29.417.  Google Scholar

[8]

L. Bourdin, Existence of a weak solution for fractional Euler-Lagrange equations, J. Math. Anal. Appl., 399 (2013), 239-251. doi: 10.1016/j.jmaa.2012.10.008.  Google Scholar

[9]

J. Cresson, Fractional embedding of differential operators and Lagrangian systems, J. Math. Phys., 48 (2007), 033504, 34 pp. doi: 10.1063/1.2483292.  Google Scholar

[10]

G. S. F. Frederico and D. F. M. Torres, Fractional Noether's theorem in the Riesz-Caputo sense, Appl. Math. Comput., 217 (2010), 1023-1033. doi: 10.1016/j.amc.2010.01.100.  Google Scholar

[11]

U. N. Katugampola, New approach to a generalized fractional integral, Appl. Math. Comput., 218 (2011), 860-865. doi: 10.1016/j.amc.2011.03.062.  Google Scholar

[12]

A. A. Kilbas and M. Saigo, Generalized Mittag-Leffler function and generalized fractional calculus operators, Integral Transform. Spec. Func., 15 (2004), 31-49. doi: 10.1080/10652460310001600717.  Google Scholar

[13]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier, Amsterdam, 2006.  Google Scholar

[14]

M. Klimek, On Solutions of Linear Fractional Differential Equations of a Variational Type, The Publishing Office of Czestochowa University of Technology, Czestochowa, 2009. Google Scholar

[15]

M. Klimek and M. Lupa, Reflection symmetric formulation of generalized fractional variational calculus, Fract. Calc. Appl. Anal., 16 (2013), 243-261. doi: 10.2478/s13540-013-0015-x.  Google Scholar

[16]

A. B. Malinowska, A formulation of the fractional Noether-type theorem for multidimensional Lagrangians, Appl. Math. Lett., 25 (2012), 1941-1946. doi: 10.1016/j.aml.2012.03.006.  Google Scholar

[17]

A. B. Malinowska and D. F. M. Torres, Introduction to the Fractional Calculus of Variations, Imperial College Press, 2012. doi: 10.1142/p871.  Google Scholar

[18]

T. Odzijewicz, Variable order fractional isoperimetric problem of several variables, Advances in the Theory and Applications of Non-integer Order Systems, 257 (2013), 133-139. doi: 10.1007/978-3-319-00933-9_11.  Google Scholar

[19]

T. Odzijewicz, A. B. Malinowska and D. F. M. Torres, Fractional calculus of variations of several independent variables, European Phys. J., 222 (2013), 1813-1826. doi: 10.1140/epjst/e2013-01966-0.  Google Scholar

[20]

T. Odzijewicz, A. B. Malinowska and D. F. M. Torres, Green's theorem for generalized fractional derivative, Fract. Calc. Appl. Anal., 16 (2013), 64-75. doi: 10.2478/s13540-013-0005-z.  Google Scholar

[21]

T. Odzijewicz, A. B. Malinowska and D. F. M. Torres, Generalized fractional calculus with applications to the calculus of variations, Comput. Math. Appl., 64 (2012), 3351-3366. doi: 10.1016/j.camwa.2012.01.073.  Google Scholar

[22]

F. Riewe, Nonconservative Lagrangian and Hamiltonian mechanics, Phys. Rev. E (3), 53 (1996), 1890-1899. doi: 10.1103/PhysRevE.53.1890.  Google Scholar

[23]

F. Riewe, Mechanics with fractional derivatives, Phys. Rev. E (3), 55 (1997), 3581-3592. doi: 10.1103/PhysRevE.55.3581.  Google Scholar

show all references

References:
[1]

O. P. Agrawal, Generalized variational problems and Euler-Lagrange equations, Comput. Math. Appl., 59 (2010), 1852-1864. doi: 10.1016/j.camwa.2009.08.029.  Google Scholar

[2]

R. Almeida, Fractional variational problems with the Riesz-Caputo derivative, Appl. Math. Lett., 25 (2012), 142-148. doi: 10.1016/j.aml.2011.08.003.  Google Scholar

[3]

R. Almeida, A. B. Malinowska and D. F. M. Torres, A fractional calculus of variations for multiple integrals with application to vibrating string, J. Math. Phys., 51 (2010), 033503, 12 pp. doi: 10.1063/1.3319559.  Google Scholar

[4]

R. Almeida, A. B. Malinowska and D. F. M. Torres, Fractional Euler-Lagrange differential equations via Caputo derivatives, Fractional Dynamics and Control, 2 (2012), 109-118. doi: 10.1007/978-1-4614-0457-6_9.  Google Scholar

[5]

R. Almeida, S. Pooseh and D. F. M. Torres, Fractional variational problems depending on indefinite integrals, Nonlinear Anal., 75 (2012), 1009-1025. doi: 10.1016/j.na.2011.02.028.  Google Scholar

[6]

D. Baleanu and I. S. Muslih, Lagrangian formulation of classical fields within Riemann-Liouville fractional derivatives, Phys. Scripta, 72 (2005), 119-121. doi: 10.1238/Physica.Regular.072a00119.  Google Scholar

[7]

N. R. O. Bastos, R. A. C. Ferreira and D. F. M. Torres, Necessary optimality conditions for fractional difference problems of the calculus of variations, Discrete Contin. Dyn. Syst., 29 (2011), 417-437. doi: 10.3934/dcds.2011.29.417.  Google Scholar

[8]

L. Bourdin, Existence of a weak solution for fractional Euler-Lagrange equations, J. Math. Anal. Appl., 399 (2013), 239-251. doi: 10.1016/j.jmaa.2012.10.008.  Google Scholar

[9]

J. Cresson, Fractional embedding of differential operators and Lagrangian systems, J. Math. Phys., 48 (2007), 033504, 34 pp. doi: 10.1063/1.2483292.  Google Scholar

[10]

G. S. F. Frederico and D. F. M. Torres, Fractional Noether's theorem in the Riesz-Caputo sense, Appl. Math. Comput., 217 (2010), 1023-1033. doi: 10.1016/j.amc.2010.01.100.  Google Scholar

[11]

U. N. Katugampola, New approach to a generalized fractional integral, Appl. Math. Comput., 218 (2011), 860-865. doi: 10.1016/j.amc.2011.03.062.  Google Scholar

[12]

A. A. Kilbas and M. Saigo, Generalized Mittag-Leffler function and generalized fractional calculus operators, Integral Transform. Spec. Func., 15 (2004), 31-49. doi: 10.1080/10652460310001600717.  Google Scholar

[13]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier, Amsterdam, 2006.  Google Scholar

[14]

M. Klimek, On Solutions of Linear Fractional Differential Equations of a Variational Type, The Publishing Office of Czestochowa University of Technology, Czestochowa, 2009. Google Scholar

[15]

M. Klimek and M. Lupa, Reflection symmetric formulation of generalized fractional variational calculus, Fract. Calc. Appl. Anal., 16 (2013), 243-261. doi: 10.2478/s13540-013-0015-x.  Google Scholar

[16]

A. B. Malinowska, A formulation of the fractional Noether-type theorem for multidimensional Lagrangians, Appl. Math. Lett., 25 (2012), 1941-1946. doi: 10.1016/j.aml.2012.03.006.  Google Scholar

[17]

A. B. Malinowska and D. F. M. Torres, Introduction to the Fractional Calculus of Variations, Imperial College Press, 2012. doi: 10.1142/p871.  Google Scholar

[18]

T. Odzijewicz, Variable order fractional isoperimetric problem of several variables, Advances in the Theory and Applications of Non-integer Order Systems, 257 (2013), 133-139. doi: 10.1007/978-3-319-00933-9_11.  Google Scholar

[19]

T. Odzijewicz, A. B. Malinowska and D. F. M. Torres, Fractional calculus of variations of several independent variables, European Phys. J., 222 (2013), 1813-1826. doi: 10.1140/epjst/e2013-01966-0.  Google Scholar

[20]

T. Odzijewicz, A. B. Malinowska and D. F. M. Torres, Green's theorem for generalized fractional derivative, Fract. Calc. Appl. Anal., 16 (2013), 64-75. doi: 10.2478/s13540-013-0005-z.  Google Scholar

[21]

T. Odzijewicz, A. B. Malinowska and D. F. M. Torres, Generalized fractional calculus with applications to the calculus of variations, Comput. Math. Appl., 64 (2012), 3351-3366. doi: 10.1016/j.camwa.2012.01.073.  Google Scholar

[22]

F. Riewe, Nonconservative Lagrangian and Hamiltonian mechanics, Phys. Rev. E (3), 53 (1996), 1890-1899. doi: 10.1103/PhysRevE.53.1890.  Google Scholar

[23]

F. Riewe, Mechanics with fractional derivatives, Phys. Rev. E (3), 55 (1997), 3581-3592. doi: 10.1103/PhysRevE.55.3581.  Google Scholar

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