In this paper, a numerical method for solving the fractional Bagley–Torvik equation is given. This method is based on using fractional Taylor vector approximation. The operational matrix of the fractional integration for fractional Taylor vector is given and is utilized to reduce the solution of the Bagley–Torvik equation to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of this technique.

References

1.
Torvik
,
P. J.
, and
Bagley
,
R. L.
,
1984
, “
On the Appearance of the Fractional Derivative in the Behavior of Real Materials
,”
ASME J. Appl. Mech.
,
51
(
2
), pp.
294
298
.
2.
Atanackovic
,
T. M.
, and
Zorica
,
D.
,
2013
, “
On the Bagley—Torvik Equation
,”
ASME J. Appl. Mech.
,
80
(
4
), p.
041013
.
3.
Podlubny
,
I.
,
1999
,
Fractional Differential Equations
,
Academic Press
,
San Diego, CA
.
4.
Diethelm
,
K.
, and
Ford
,
N. J.
,
2002
, “
Numerical Solution of the Bagley—Torvik Equation
,”
BIT Numer. Math.
,
42
(
3
), pp.
490
507
.
5.
Cenesiz
,
Y.
,
Keskin
,
Y.
, and
Kurnaz
,
A.
,
2010
, “
The Solution of the Bagley—Torvik Equation With the Generalized Taylor Collocation Method
,”
J. Franklin Inst.
,
347
(
2
), pp.
452
466
.
6.
Saha Ray
,
S.
,
2012
, “
On Haar Wavelet Operational Matrix of General Order and Its Application for the Numerical Solution of Fractional Bagley Torvik Equation
,”
Appl. Math. Comput.
,
218
(
9
), pp.
5239
5248
.
7.
Yuzbasi
,
S.
,
2013
, “
Numerical Solution of the Bagley—Torvik Equation by the Bessel Collocation Method
,”
Math. Methods Appl. Sci.
,
36
(
3
), pp.
300
312
.
8.
Saha Ray
,
S.
, and
Bera
,
R. K.
,
2005
, “
Analytical Solution of the Bagley Torvik Equation by Adomian Decomposition Method
,”
Appl. Math. Comput.
,
168
(
1
), pp.
398
410
.
9.
Cermak
,
J.
, and
Kisela
,
T.
, 2014, “
Exact and Discretized Stability of the Bagley–Torvik Equation
,”
J. Comput. Appl. Math.
,
269
(
1
), pp.
53
67
.
10.
Celik
,
I.
,
2006
, “
Collocation Method and Residual Correction Using Chebyshev Series
,”
Appl. Math. Comput.
,
174
(
2
), pp.
910
920
.
11.
Abbasbandy
,
S.
, and
Taati
,
A.
,
2009
, “
Numerical Solution of the System of Nonlinear Volterra Integro-Differential Equations With Nonlinear Differential Part by the Operational Tau Method and Error Estimation
,”
J. Comput. Appl. Math.
,
231
(
1
), pp.
106
113
.
12.
Yuzbasi
,
S.
,
2013
, “
Numerical Solutions of Fractional Riccati Type Differential Equations by Means of the Bernstein Polynomials
,”
Appl. Math. Comput.
,
219
(
11
), pp.
6328
6343
.
13.
Kilbas
,
A. A.
,
Srivastava
,
H. M.
, and
Trujillo
,
J. J.
,
2006
,
Theory and Applications of Fractional Differential Equations
,
Elsevier B.V.
,
Amsterdam, The Netherlands
.
14.
Jafari
,
H.
,
Yousefi
,
S. A.
,
Firoozjaee
,
M. A.
,
Momani
,
S.
, and
Khalique
,
C. M.
,
2011
, “
Application of Legendre Wavelets for Solving Fractional Differential Equations
,”
Comput. Math. Appl.
,
62
(
3
), pp.
1038
1045
.
15.
Diethelm
,
K.
,
Ford
,
N.
, and
Freed
,
A. D.
,
2004
, “
Detailed Error Analysis for a Fractional Adams Method
,”
Numer. Algorithms
,
36
(
1
), pp.
31
52
.
16.
Esmaeilli
,
S.
, and
Shamsi
,
M.
,
2011
, “
A Pseudo-Spectral Scheme for the Approximate Solution of a Family of Fractional Differential Equations
,”
Commun. Nonlinear Sci. Numer. Simul.
,
16
(
9
), pp.
3646
3654
.
17.
Rehman
,
M. U.
, and
Ali Khan
,
R.
,
2012
, “
A Numerical Method for Solving Boundary Value Problems for Fractional Differential Equations
,”
Appl. Math. Modell.
,
36
(
3
), pp.
894
907
.
18.
Ma
,
X.
, and
Huang
,
C.
,
2013
, “
Numerical Solution of Fractional Integro-Differential Equations by a Hybrid Collocation Method
,”
Appl. Math. Comput.
,
219
(
12
), pp.
6750
6760
.
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