A high accurate Rayleigh–Ritz method is developed for solving fractional variational problems (FVPs). The Jacobi poly-fractonomials proposed by Zayernouri and Karniadakis (2013, “Fractional Sturm–Liouville Eigen-Problems: Theory and Numerical Approximation,” J. Comput. Phys., 252(1), pp. 495–517.) are chosen as basis functions to approximate the true solutions, and the Rayleigh–Ritz technique is used to reduce FVPs to a system of algebraic equations. This method leads to exponential decay of the errors, which is superior to the existing methods in the literature. The fractional variational errors are discussed. Numerical examples are given to illustrate the exponential convergence of the method.

References

References
1.
Hairer
,
E.
,
Lubich
,
C.
, and
Wanner
,
G.
,
2006
,
Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations
,
2nd ed.
, Springer Series in Computational Mathematics, Vol.
31
,
Springer-Verlag
,
Berlin, Germany
.
2.
Zhang
,
G.
,
2011
,
Variational Calculus Notes
,
Higher Education Press
,
Beijing, China
.
3.
Riewe
,
F.
,
1996
, “
Nonconservative Lagrangian and Hamiltonian Mechanics
,”
Phys. Rev. E
,
53
(
2
), pp.
1890
1899
.10.1103/PhysRevE.53.1890
4.
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
5.
Agrawal
,
O. P.
,
2007
, “
Generalized Euler–Lagrange Equations and Transversality Conditions for FVPs in Terms of the Caputo Derivative
,”
J. Vib. Control
,
13
(
9–10
), pp.
1217
1237
.10.1177/1077546307077472
6.
Klimek
,
M.
,
2009
,
On Solutions of Linear Fractional Differential Equations of a Variational Type
,
Publishing Office of Czȩstochowa University of Technology
,
Czȩstochowa, Poland
.
7.
Almeida
,
R.
, and
Torres
,
D. F. M.
,
2009
, “
Calculus of Variations With Fractional Derivatives and Fractional Integrals
,”
Appl. Math. Lett.
,
22
(
12
), pp.
1816
1820
.10.1016/j.aml.2009.07.002
8.
Almeida
,
R.
,
Pooseh
,
S.
, and
Torres
,
D. F. M.
,
2012
, “
Fractional Variational Problems Depending on Indefinite Integrals
,”
Nonlinear Anal.
,
75
(
3
), pp.
1009
1025
.10.1016/j.na.2011.02.028
9.
Malinowska
,
A. B.
, and
Torres
,
D. F. M.
,
2012
, “
Multiobjective Fractional Variational Calculus in Terms of a Combined Caputo Derivative
,”
Appl. Math. Comput.
,
218
(
9
), pp.
5099
5111
.10.1016/j.amc.2011.10.075
10.
Idczak
,
D.
, and
Majewski
,
M.
,
2012
, “
Fractional Fundamental Lemma of Order α(n-12,n) With n ∈ N, n2
,”
Dyn. Syst. Appl.
,
21
(
2–3
), pp.
251
268
.
11.
Malinowska
,
A. B.
, and
Torres
,
D. F. M.
,
2012
,
Introduction to the Fractional Calculus of Variations
,
Imperial College Press
,
Singapore
.
12.
Baleanu
,
D.
,
Machado
,
J. A. T.
, and
Luo
,
A. C. J.
,
2012
,
Fractional Dynamics and Control
,
Springer
,
Berlin, Germany
.
13.
Machado
,
J. A. T.
,
2013
, “
Optimal Controllers With Complex Order Derivatives
,”
J. Optim. Theory Appl.
,
156
(
1
), pp.
2
12
.10.1007/s10957-012-0169-4
14.
Frederico
,
G. S. F.
, and
Torres
,
D. F. M.
,
2007
, “
A Formulation of Noether's Theorem for Fractional Problems of the Calculus of Variations
,”
J. Math. Anal. Appl.
,
334
(
2
), pp.
834
846
.10.1016/j.jmaa.2007.01.013
15.
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
16.
Bourdin
,
L.
,
Cresson
,
J.
, and
Greff
,
I.
,
2013
, “
A Continuous/Discrete Fractional Noether's Theorem
,”
Commun. Nonlinear Sci. Numer. Simul.
,
18
(
4
), pp.
878
887
.10.1016/j.cnsns.2012.09.003
17.
Malinowska
,
A. B.
,
2012
, “
A Formulation of the Fractional Noether-Type Theorem for Multidimensional Lagrangians
,”
Appl. Math. Lett.
,
25
(
11
), pp.
1941
1946
.10.1016/j.aml.2012.03.006
18.
Klimek
,
M.
,
Odzijewicz
,
T.
, and
Malinowska
,
A. B.
,
2014
, “
Variational Methods for the Fractional Sturm–Liouville Problem
,”
J. Math. Anal. Appl.
,
416
(
1
), pp.
402
426
.10.1016/j.jmaa.2014.02.009
19.
Bourdin
,
L.
,
2013
, “
Existence of a Weak Solution for Fractional Euler–Lagrange Equations
,”
J. Math. Anal. Appl.
,
399
(
1
), pp.
239
251
.10.1016/j.jmaa.2012.10.008
20.
Wang
,
D.
, and
Xiao
,
A.
,
2012
, “
Fractional Variational Integrators for Fractional Variational Problems
,”
Commun. Nonlinear Sci. Numer. Simul.
,
17
(
2
), pp.
602
610
.10.1016/j.cnsns.2011.06.028
21.
Wang
,
D.
, and
Xiao
,
A.
,
2013
, “
Fractional Variational Integrators for Fractional Euler–Lagrange Equations With Holonomic Constraints
,”
Commun. Nonlinear Sci. Numer. Simul.
,
18
(
4
), pp.
905
914
.10.1016/j.cnsns.2012.08.025
22.
Wang
,
D.
, and
Xiao
,
A.
,
2013
, “
Numerical Methods for Fractional Variational Problems Depending on Indefinite Integrals
,”
ASME J. Comput. Nonlinear Dyn.
,
8
(
2
), p.
021018
.10.1115/1.4007858
23.
Bourdin
,
L.
,
Cresson
,
J.
,
Greff
,
I.
, and
Inizan
,
P.
,
2013
, “
Variational Integrator for Fractional Euler–Lagrange Equations
,”
Appl. Numer. Math.
,
71
, pp.
14
23
.10.1016/j.apnum.2013.03.003
24.
Atanacković
,
T. M.
,
Janev
,
M.
,
Konjik
,
S.
,
Pilipović
,
S.
, and
Zorica
,
D.
,
2014
, “
Expansion Formula for Fractional Derivatives in Variational Problems
,”
J. Math. Anal. Appl.
,
409
(
2
), pp.
911
924
.10.1016/j.jmaa.2013.07.071
25.
Lotfi
,
A.
, and
Yousefi
,
S. A.
,
2013
, “
A Numerical Technique for Solving a Class of Fractional Variational Problems
,”
J. Comput. Appl. Math.
,
237
(
1
), pp.
633
643
.10.1016/j.cam.2012.08.005
26.
Khader
,
M. M.
, and
Hendy
,
A. S.
,
2013
, “
A Numerical Technique for Solving Fractional Variational Problems
,”
Math. Methods Appl. Sci.
,
36
(
10
), pp.
1281
1289
.10.1002/mma.2681
27.
Zayernouri
,
M.
, and
Em Karniadakis
,
G.
,
2013
, “
Fractional Sturm–Liouville Eigen-Problems: Theory and Numerical Approximation
,”
J. Comput. Phys.
,
252
(
1
), pp.
495
517
.10.1016/j.jcp.2013.06.031
28.
Podlubny
,
I.
,
1999
,
Fractional Differential Equations
,
Academic
,
New York
.
29.
Kilbas
,
A. A.
,
Srivastava
,
H. M.
, and
Trujillo
,
J. J.
,
2006
,
Theory and Applications of Fractional Differential Equations
,
Elsevier, Amsterdam
,
The Netherlands
.
30.
Agrawal
,
O. P.
,
2008
, “
A General Finite Element Formulation for Fractional Variational Problems
,”
J. Math. Anal. Appl.
,
337
(
1
), pp.
1
12
.10.1016/j.jmaa.2007.03.105
31.
Almeida
,
R.
,
Khosravian-Arab
,
H.
, and
Shamsi
,
M.
,
2013
, “
A Generalized Fractional Variational Problem Depending on Indefinite Integrals: Euler–Lagrange Equation and Numerical Solution
,”
J. Vib. Control
,
19
(
14
), pp.
2177
2186
.10.1177/1077546312458818
32.
Zayernouri
,
M.
, and
Em Karniadakis
,
G.
,
2014
, “
Exponentially Accurate Spectral and Spectral Element Methods for Fractional ODEs
,”
J. Comput. Phys.
,
257
(
15
), pp.
460
480
.10.1016/j.jcp.2013.09.039
33.
Zayernouri
,
M.
, and
Em Karniadakis
,
G.
,
2014
, “
Fractional Spectral Collocation Method
,”
SIAM J. Sci. Comput.
,
36
(
1
), pp.
A40
A62
.10.1137/130933216
34.
Marsden
,
J. E.
, and
West
,
M.
,
2001
, “
Discrete Mechanics and Variational Integrators
,”
Acta Numer.
,
10
, pp.
357
514
.10.1017/S096249290100006X
35.
Lin
,
Y.
, and
Xu
,
C.
,
2007
, “
Finite Difference/Spectral Approximations for the Time-Fractional Diffusion Equation
,”
J. Comput. Phys.
,
225
(
2
), pp.
1533
1552
.10.1016/j.jcp.2007.02.001
36.
Li
,
X.
, and
Xu
,
C.
,
2009
, “
A Space–Time Spectral Method for the Time Fractional Diffusion Equation
,”
SIAM J. Numer. Anal.
,
47
(
3
), pp.
2108
2131
.10.1137/080718942
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