Fractional derivatives (FDs) or derivatives of arbitrary order have been used in many applications, and it is envisioned that in the future they will appear in many functional minimization problems of practical interest. Since fractional derivatives have such properties as being non-local, it can be extremely challenging to find analytical solutions for fractional parametric optimization problems, and in many cases, analytical solutions may not exist. Therefore, it is of great importance to develop numerical methods for such problems. This paper presents a numerical scheme for a linear functional minimization problem that involves FD terms. The FD is defined in terms of the Riemann-Liouville definition; however, the scheme will also apply to Caputo derivatives, as well as other definitions of fractional derivatives. In this scheme, the spatial domain is discretized into several subdomains and 2-node one-dimensional linear elements are adopted to approximate the solution and its fractional derivative at point within the domain. The fractional optimization problem is converted to an eigenvalue problem, the solution of which leads to fractional orthogonal functions. Convergence study of the number of elements and error analysis of the results ensure that the algorithm yields stable results. Various fractional orders of derivative are considered, and as the order approaches the integer value of 1, the solution recovers the analytical result for the corresponding integer order problem.

References

References
1.
Ross
,
B.
, 1977, “
The Development of Fractional Calculus, 1695-1900
,”
Hist. Math.
,
4
, pp.
75
89
.
2.
Podlubny
,
I.
, 1999,
Fractional Differential Equations
,
Academic Press
,
San Diego, CA
.
3.
Kilbas
,
A. A.
,
Srivastava
,
H. M.
, and
Trujillo
,
J. J.
, 2006,
Theory and Applications of Fractional Differential Equations: North-Holland Mathematics Studies
,
Elsevier
,
Amsterdam
, Vol.
204
.
4.
Agrawal
,
O. P.
, 2002, “
Formulation of Euler–Lagrange Equations for Fractional Variational Problems
,”
J. Math. Anal. Appl.
,
272
, pp.
368
379
.
5.
Riewe
,
F.
, 1996, “
Nonconservative Lagrangian and Hamiltonian Mechanics
,”
Phys. Rev. E
,
53
, pp.
1890
1899
.
6.
Riewe
,
F.
, 1997, “
Mechanics with Fractional Derivatives
,”
Phys. Rev. E
,
55
(
3
), pp.
3582
3592
.
7.
Agrawal
,
O. P.
, 2006, “
Fractional Variational Calculus and the Transversality Conditions
,”
J. Phys. A: Math. Gen.
,
39
, pp.
10375
10384
.
8.
Klimek
,
M.
, 2001, “
Fractional Sequential Mechanics - Models with Symmetric Fractional Derivative
,”
Czech. J. Phys.
,
51
, pp.
1348
1354
.
9.
Klimek
,
M.
, 2002, “
Stationary Conservation Laws for Fractional Differential Equations with Variable Coefficients
,”
J. Phys. A
,
35
, pp.
6675
6693
.
10.
Agrawal
,
O. P.
, 2007, “
Fractional Variational Calculus in Terms of Riesz Fractional Derivatives
,”
J. Phys. A: Math. Theor.
,
40
, pp.
6287
6303
.
11.
Agrawal
,
O. P.
, 2010, “
Generalized Variationalal Problems and Euler-Lagrange Equations
,”
Comput. Math. Appl.
,
59
, pp.
1852
1864
.
12.
Baleanu
,
D.
, and
Avkar
,
T.
, 2004, “
Lagrangians with Linear Velocities within Riemann–Liouville Fractional Derivatives
,”
Nuovo Cimento Soc. Ital. Fis. B
,
119
, pp.
73
79
.
13.
Muslih
,
S. I.
, and
Baleanu
,
D.
, 2005, “
Formulation of Hamiltonian Equations for Fractional Variational Problems
,”
Czech. J. Phys.
,
55
, pp.
633
642
.
14.
Herzallah
,
M. A. E.
, and
Baleanu
,
D.
, 2009, “
Fractional-Order Euler–Lagrange Equations and Formulation of Hamiltonian Equations
,”
Nonlinear Dyn.
,
58
, pp.
385
391
.
15.
Almeida
,
R.
, and
Torres
,
D. F. M.
, 2009, “
Calculus of Variations with Fractional Derivatives and Fractional Integrals
,”
Appl. Math. Lett.
,
22
, pp.
1816
1820
.
16.
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
.
17.
Sousa
,
E.
, 2010, “
How to Approximate the Fractional Derivative of Order 1
<α<1,” Proceedings of FDA’10, the 4th IFAC Workshop Fractional Differentiation and its Applications, Badajoz, Spain, October 18–20, 2010.
18.
Odibat
,
Z.
, and
Momani
,
S.
, 2008, “
An Algorithm for the Numerical Solution of Differential Equations of Fractional Order
,”
J. Appl. Math. Inf.
,
26
(
1–2
), pp.
15
27
.
19.
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
.
20.
Momani
,
S.
,
Jaradat
,
O. K.
, and
Ibrahim
,
R.
, 2008, “
Numerical Approximations of a Dynamic System Containing Fractional Derivatives
,”
J. Appl. Sci.
,
8
, pp.
1079
1084
.
21.
Kumar
,
P.
and
Agrawal
,
O. P.
, 2006, “
An Approximate Method for Numerical Solution of Fractional Differential Equations
,”
Signal Process.
,
86
, pp.
2602
2610
.
22.
Diethelm
,
K.
,
Ford
,
N. J.
, and
Freed
,
A. D.
, 2002, “
A Predictor–Corrector Approach for the Numerical Solution of Fractional Differential Equations
,”
Nonlinear Dyn.
,
29
(
1–4
), pp.
3
22
.
23.
Diethelm
,
K.
,
Ford
,
N. J.
,
Freed
,
A. D.
, and
Luchko
,
Y.
, 2005, “
Algorithms for the Fractional Calculus: A Selection of Numerical Methods
,”
Comput. Methods Appl. Mech. Eng.
,
194
, pp.
743
773
.
24.
Klimek
,
M.
, 2009,
On Solutions of Linear Fractional Differential Equations of a Variational Type
,
Czestochowa University of Technology
,
Czestochowa
.
25.
Agrawal
,
O. P.
, 2010, “
A Series Solution Technique for a Class of Fractional Differential Equations
,” Third Conference on Mathematical Methods in Engineering International Symposium, Instituto Politécnico de Coimbra, Coimbra, Portugal, October 21–24, 2010.
You do not currently have access to this content.