Fractional calculus is a rapidly going area from both experimental and theoretical points of view. As a result new methods and techniques should be developed in order to deal with new types of fractional differential equations. In this paper the operational matrix of fractional derivative together with the τ method are used to solve the linear systems of fractional differential equations. The results of this method are shown by solving three illustrative examples. By comparing the obtained results with the analytic solutions and with the ones provided by three standard methods for solving the fractional differential equations we conclude that our method gave comparable results.

References

References
1.
Samko
,
S. G.
,
Kilbas
,
A. A.
, and
Marichev
,
O. I.
,
1993
, “
Fractional Integrals and Derivatives
,”
Theory and Applications
,
Gordon and Breach
,
Langhorne, PA
.
2.
Podlubny
,
I.
,
1999
,
Fractional Differential Equation
,
Academic Press
,
San Diego
.
3.
Kilbas
,
A. A.
,
Srivastava
,
H. M.
, and
Trujillo
,
J. J.
,
2006
,
Theory and Applications of Fractional Differential Equations
,
Elsevier Science
,
Amsterdam
.
4.
Miller
,
K. S.
, and
Ross
,
B.
,
1993
,
An Introduction to the Fractional Calculus and Differential Equations
,
John Wiley
,
New York
.
5.
West
,
B. J.
,
Bologna
,
M.
, and
Grogolini
,
P.
,
2003
,
Physics of Fractal Operators
,
Springer
,
New York
.
6.
Magin
,
R. L.
,
2006
,
Fractional Calculus in Bioengineering
,
Begell House Publisher, Inc.
Connecticut
.
7.
Baleanu
,
D.
,
Diethelm
,
K.
,
Scalas
,
E.
, and
Trujillo
,
J. J.
,
2012
,
Fractional Calculus Models and Numerical Methods
(Series on Complexity, Nonlinearity and Chaos),
World Scientific
,
New Jersey
.
8.
Machado
,
J. A. T.
,
2003
, “
A Probabilistic Interpretation of the Fractional-Order Differentiation
,”
Frac. Calc. Appl. Anal.
,
6
(
1
), pp.
73
80
.
9.
Raspini
,
A.
,
2001
, “
Simple Solutions of the Fractional Dirac Equation of Order 23
,”
Phys. Scr.
,
64
(
1
), pp.
20
22
.10.1238/Physica.Regular.064a00020
10.
Riewe
,
F.
,
1996
, “
Nonconservative Lagrangian and Hamiltonian Mechanics
”.
Phys. Rev. E.
,
53
(
2
), pp.
1890
1899
.10.1103/PhysRevE.53.1890
11.
Riewe
,
F.
,
1997
, “
Mechanics With Fractional Derivatives
,”
Phys. Rev. E.
,
55
(
3
), pp.
3581
3592
.10.1103/PhysRevE.55.3581
12.
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
13.
Klimek
,
K.
,
2001
, “
Fractional Sequential Mechanics-Models With Symmetric Fractional Derivatives
,”
Czech. J. Phys.
,
51
(
12
), pp.
1348
1354
.10.1023/A:1013378221617
14.
Baleanu
,
D.
, and
Muslih
,
S.
,
2005
, “
Lagrangian Formulation of Classical Fields Within Riemann–Liouville Fractional Derivatives
,”
Phys. Scr.
,
72
(
2
), pp.
119
121
.10.1238/Physica.Regular.072a00119
15.
Baleanu
,
D.
, and
Muslih
,
S. I.
,
2005
, “
Formulation of Hamiltonian Equations for Fractional Variational Problems
,”
Czech. J. Phys.
,
55
(
6
), pp.
633
642
.10.1007/s10582-005-0067-1
16.
Baleanu
,
D.
, and
Avkar
,
T.
,
2004
, “
Lagrangians With Linear Velocities Within Riemann–Liouville Fractional Derivatives
,”
Nuovo Cimento B.
,
119
(
1
), pp.
73
79
.
17.
Trujillo
,
J. J.
,
Rivero
,
M.
, and
Bonilla
,
B.
,
1999
, “
On a Riemann–Liouville Generalized Taylor's Formula
,”
J. Math. Anal. Appl.
,
231
(
1
), pp.
255
265
.10.1006/jmaa.1998.6224
18.
Machado
,
J. A. T.
, and
Galhano
,
M. S. A.
,
2008
, “
Statistical Fractional Dynamics
,”
ASME J. Comput. Nonlin. Dynam.
,
3
(
2
), pp.
021201-1
021201-5
.10.1115/1.2833481
19.
Momani
,
S.
, and
Odibat
,
Z.
,
2007
, “
Numerical Approach to Differential Equations of Fractional Order
,”
J. Comput. Appl. Math.
,
207
(
1
), pp.
96
110
.10.1016/j.cam.2006.07.015
20.
Abdulaziz
,
O.
,
Hashim
,
I.
, and
Momani
,
S.
,
2008
, “
Solving Systems of Fractional Differential Equations by Homotopy Perturbation Method
,”
Phys. Lett. A
,
372
(
4
), pp.
451
459
.10.1016/j.physleta.2007.07.059
21.
Sweilam
,
N. H.
,
Khader
,
M. M.
, and
Al-Bar
,
R. F.
,
2007
, “
Numerical Studies for a Multi-Order Fractional Differential Equation Method
,”
Phys. Lett. A
,
371
(
1
), pp.
26
33
.10.1016/j.physleta.2007.06.016
22.
Saadatmandi
,
A.
, and
Dehghan
,
M.
,
2010
, “
A New Operational Matrix for Solving Fractional-Order Differential Equations
,”
Comput. Math. Appl.
,
59
(
3
), pp.
1326
1336
.10.1016/j.camwa.2009.07.006
23.
Saadatmandi
,
A.
, and
Dehghan
,
M.
,
2011
, “
A Tau Approach for Solution of the Space Fractional Diffusion Equation
,”
Comput. Math. Appl.
,
62
(
3
), pp.
1135
1142
.10.1016/j.camwa.2011.04.014
24.
Saadatmandi
,
A.
, and
Dehghan
,
M.
,
2011
, “
A Legendre Collocation Method for Fractional Integro-Differential Equations
,”
J. Vib. Control
,
17
(
13
), pp.
2050
2058
.10.1177/1077546310395977
25.
Saadatmandi
,
A.
,
Dehghan
,
M.
, and
Azizi
,
M. R.
,
2012
, “
The Sinc–Legendre Collocation Method for a Class of Fractional Convection-Diffusion Equations With Variable Coefficients
,”
Commun. Nonlinear Sci. Numer. Simul.
,
17
(
11
), pp.
4125
4136
.10.1016/j.cnsns.2012.03.003
26.
Esmaeili
,
S.
,
Shamsi
,
S.
, and
Luchko
,
Y.
,
2011
, “
Numerical Solution of Fractional Differential Equations With a Collocation Method Based on Muntz Polynomials
,”
Comput. Math. Appl.
,
62
(
3
), pp.
918
929
.10.1016/j.camwa.2011.04.023
27.
Odibat
,
Z.
,
2011
, “
On Legendre Polynomials Approximation With the VIM or HAM for Numerical Treatment of Nonlinear Fractional Differential Equations
,”
J. Comput. Appl. Math.
,
235
(
9
), pp.
2956
2968
.10.1016/j.cam.2010.12.013
28.
Lanczos
,
C.
,
1956
,
Applied Analysis
,
Prentice-Hall
,
Englewood Cliffs, New York
.
29.
Canuto
,
C.
,
Quarteroni
,
A.
,
Hussaini
,
M. Y.
, and
Zang
,
T. A.
,
1988
,
Spectral Methods in Fluid Dynamic
,
Prentice-Hall
,
Englewood Cliffs, New York
.
30.
Odibat
,
Z. M.
,
2010
, “
Analytic Study on Linear Systems of Fractional Differential Equations
,”
Comput. Math. Appl.
,
59
(
3
), pp.
1171
1183
.10.1016/j.camwa.2009.06.035
31.
Gottlieb
,
D.
,
Hussaini
,
M. Y.
, and
Orszg
,
S.
,
1984
,
Theory and Applications of Spectral Methods in Spectral Methods for Partial Differential Equations
,
(Society for Industrial and Applied Mathematics)
,
Philadelphia, PA
.
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