In this paper, the fractional variational integrators for fractional variational problems depending on indefinite integrals in terms of the Caputo derivative are developed. The corresponding fractional discrete Euler–Lagrange equations are derived. Some fractional variational integrators are presented based on the Grünwald–Letnikov formula. The fractional variational errors are discussed. Some numerical examples are given to illustrate these results.

References

References
1.
Podlubny
,
I.
,
1999
,
Fractional Differential Equations
,
Academic Press
,
San Diego CA
.
2.
Hilfer
,
R.
,
1999
,
Applications of Fractional Calculus in Physics
,
World Scientific
,
Singapore
.
3.
West
,
B. J.
,
Bologna
,
M.
, and
Grigolini
,
P.
,
2003
,
Physics of Fractal Operators
,
Springer
,
New York
.
4.
Kilbas
,
A. A.
,
Srivastava
,
H. M.
, and
Trujillo
,
J. J.
,
2006
,
Theory and Applications of Fractional Differential Equations
,
Elsevier
,
Amsterdam
.
5.
Tarasov
,
V. E.
,
2010
,
Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media
,
Springer
,
New York
.
6.
Riewe
F.
,
1997
, “
Mechanics With Fractional Derivatives
,”
Phys. Rev. E.
,
55
, pp.
3581
3592
.10.1103/PhysRevE.55.3581
7.
Agrawal
O. P.
,
2002
, “
Formulation of Euler-Lagrange Equations for Fractional Variational Problems
,”
J. Math. Anal. Appl.
,
272
(
1
), pp.
368
379
.10.1016/S0022-247X(02)00180-4
8.
Agrawal
O. P.
,
2007
, “
Generalized Euler-Lagrange Equations and Transversality Conditions for FVPs in Terms of the Caputo Derivatives
,”
J. Vib. Control
,
13
(
9–10
), pp.
1217
1237
.10.1177/1077546307077472
9.
Almeida
,
R.
,
Pooseh
,
S.
, and
Torres
,
D. F. M.
,
2012
, “
Fractional Variational Problems Depending on Indefinite Integrals
,”
Nonlinear Anal. Theory, Methods Appl.
,
75
, pp.
1009
1025
.10.1016/j.na.2011.02.028
10.
Martins
,
N.
, and
Torres
,
D. F. M.
,
2011
, “
Generalizing the Variational Theory on Time Scales to Include the Delta Indefinite Integral
,”
Comput. Math. Appl.
,
61
(
9
), pp.
2424
2435
.10.1016/j.camwa.2011.02.022
11.
Almeida
,
R.
, and
Torres
,
D. F. M.
,
2011
, “
Necessary and Sufficient Conditions for the Fractional Calculus of Variations With Caputo Derivatives
,”
Commun. Nonlinear Sci. Numer. Simul.
,
16
(
3
), pp.
1490
1500
.10.1016/j.cnsns.2010.07.016
12.
Almeida
,
R.
,
2012
, “
Fractional Variational Problems With the Riesz–Caputo Derivative
,”
Appl. Math. Lett.
,
25
, pp.
142
148
.10.1016/j.aml.2011.08.003
13.
Frederico
,
G. S. F.
, and
Torres
,
D. F. M.
,
2010
, “
Fractional Noether’s Theorem in the Riesz–Caputo Sense
,”
Appl. Math. Comput.
,
217
(
3
), pp.
1023
1033
.10.1016/j.amc.2010.01.100
14.
Atanacković
,
T. M.
,
Konjik
,
S.
,
Pilipović
,
S.
, and
Simić
,
S.
,
2009
, “
Variational Problems With Fractional Derivatives: Invariance Conditions and Noether’s Theorem
,”
Nonlinear Anal. Theor, Methods Appl.
,
71
, pp.
1504
1517
.10.1016/j.na.2008.12.043
15.
Malinowska
,
A. B.
,
2012
, “
A Formulation of the Fractional Noether-Type Theorem for Multidimensional Lagrangians
,”
Appl. Math. Lett.
,
25
, pp.
1941
1946
.10.1016/j.aml.2012.03.006
16.
Malinowska
,
A. B.
, and
Torres
,
D. F. M.
,
2012
,
Introduction to the Fractional Calculus of Variations
,
World Scientific
,
Singapore
.
17.
Almeida
,
R.
,
Malinowska
,
A. B.
, and
Torres
,
D. F. M.
,
2010
, “
A Fractional Calculus of Variations for Multiple Integrals With Application to Vibrating String
,”
J. Math. Phys.
,
51
(
3
),
033503
.10.1063/1.3319559
18.
Odzijewicz
,
T.
,
Malinowska
,
A. B.
, and
Torres
,
D. F. M.
,
2012
, “
Fractional Calculus of Variations in Terms of a Generalized Fractional Integral With Applications to Physics
,”
Abstr. Appl. Anal.
, Paper No. 871912.
19.
Jumarie
,
G.
,
2008
, “
Stock Exchange Fractional Dynamics Defined as Fractional Exponential Growth Driven by (Usual) Gaussian White Noise. Application to Fractional Black-Scholes Equations
,”
Insur. Math. Econ.
,
42
(
1
), pp.
271
287
.10.1016/j.insmatheco.2007.03.001
20.
Muslih
,
S. I.
, and
Baleanu
,
D.
,
2005
, “
Hamiltonian Formulation of Systems With Linear Velocities Within Riemann–Liouville Fractional Derivatives
,”
J. Math. Anal. Appl.
,
304
(
2
), pp.
599
606
.10.1016/j.jmaa.2004.09.043
21.
Momani
,
S.
, and
Odibat
,
Z.
,
2007
, “
Numerical Comparison of Methods for Solving Linear Differential Equations of Fractional Order
,”
Chaos, Solitons Fractals
,
31
, pp.
1248
1255
.10.1016/j.chaos.2005.10.068
22.
Agrawal
,
O. P.
, and
Kumar
,
P.
,
2007
, “
Comparison of Five Numerical Schemes for Fractional Differential Equations
,”
Advances in Fractional Calculus
,
J.
Sabatier
, ed.,
Springer
,
New York
, pp.
33
75
.
23.
Agrawal
,
O. P.
,
2008
, “
A General Fnite Element Formulation for Fractional Variational Problems
,”
J. Math. Anal. Appl.
,
337
, pp.
1
12
.10.1016/j.jmaa.2007.03.105
24.
Lubich
,
C.
,
1985
, “
Fractional Linear Multistep Methods for Abel–Volterra Integral Equations of the Second Kind
,”
Math. Comput.
,
45
(
172
), pp.
463
469
.10.1090/S0025-5718-1985-0804935-7
25.
Lubich
,
C.
,
1986
, “
Discrete Fractional Calculus
,”
SIAM J. Math. Anal.
,
17
(
3
), pp.
704
719
.10.1137/0517050
26.
Garrappa
,
R.
, and
Popolizio
,
M.
,
2011
, “
Generalized Exponential Time Differencing Methods for Fractional Order Problems
,”
Comput. Math. Appl.
,
62
, pp.
876
890
.10.1016/j.camwa.2011.04.054
27.
He
,
J. H.
,
2006
, “
Some Asymptotic Methods for Strongly Nonlinear Equations
,”
Int. J. Mod. Phys. B
,
20
(
10
), pp.
1141
1199
.10.1142/S0217979206033796
28.
Ferreira
,
R. A. C.
, and
Torres
,
D. F. M.
,
2011
, “
Fractional h-Difference Equations Arising From the Calculus of Variations
,”
Appl. Anal. Discrete Math.
,
5
(
1
), pp.
110
121
.10.2298/AADM110131002F
29.
Klimek
,
M.
,
2008
, “
G-Meijer Functions Series as Solutions for Certain Fractional Variational Problem on a Finite Time Interval
,”
J. Eur. Syst. Autom.
,
42
, pp.
653
664
.10.3166/jesa.42.653-664
30.
Cresson
,
J.
, and
Inizan
,
P.
,
2012
, “
Variational Formulations of Differential Equations and Asymmetric Fractional Embedding
,”
J. Math. Anal. Appl.
,
385
, pp.
975
997
.10.1016/j.jmaa.2011.07.022
31.
Cresson
,
J.
,
Malinowska
,
A. B.
, and
Torres
,
D. F. M.
,
2012
, “
Time Scale Differential, Integral and Variational Embeddings of Lagrangian Systems
,”
Comput. Math. Appl.
,
64
(
7
), pp.
2294
2301
.10.1016/j.camwa.2012.03.003
32.
Marsden
,
J. E.
, and
West
,
M.
,
2001
, “
Discrete Mechanics and Variational Integrators
,”
Acta Numerica
,
10
, pp.
1
158
.10.1017/S096249290100006X
33.
Leyendecker
,
S.
,
Marsden
,
J. E.
, and
Ortiz
,
M.
,
2008
, “
Variational Integrators for Constrained Dynamical Systems
,”
ZAMM
,
88
(
9
), pp.
677
708
.10.1002/zamm.200700173
34.
Wang
,
D.
, and
Xiao
,
A.
,
2012
, “
Fractional Variational Integrators for Fractional Variational Problems
,”
Commun. Nonlinear Sci. Numer. Simul.
,
17
, pp.
602
610
.10.1016/j.cnsns.2011.06.028
35.
Wang
,
D.
, and
Xiao
,
A.
,
2012
, “
Fractional Variational Integrators for Fractional Euler-Lagrange Equations With Holonomic Constraints
,”
Commun. Nonlinear Sci. Numer. Simul.
(in press).
36.
Bastos
,
N. R. O.
,
Ferreira
,
R. A. C.
, and
Torres
,
D. F. M.
,
2011
, “
Necessary Optimality Conditions for Fractional Difference Problems of the Calculus of Variations
,”
Discrete Contin. Dyn. Syst.
,
29
(
2
), pp.
417
437
.10.3934/dcds.2011.29.417
37.
Bastos
,
N. R. O.
,
Ferreira
,
R. A. C.
, and
Torres
,
D. F. M.
,
2011
, “
Discrete-Time Fractional Variational Problems
,”
Signal Process
,
91
(
3
), pp.
513
524
.10.1016/j.sigpro.2010.05.001
38.
Müller
,
S.
, and
Ortiz
,
M.
,
2004
, “
On the Gamma-Convergence of Discrete Dynamics and Variational Integrators
,”
J. Nonlinear Sci.
,
14
(
4
), pp.
153
212
.10.1007/js00332-004-0585-1
You do not currently have access to this content.