In this paper, the operational matrix of Euler functions for fractional derivative of order β in the Caputo sense is derived. Via this matrix, we develop an efficient collocation method for solving nonlinear fractional Volterra integro-differential equations. Illustrative examples are given to demonstrate the validity and applicability of the proposed method, and the comparisons are made with the existing results.

References

References
1.
Diethelm
,
K.
,
2010
,
The Analysis of Fractional Differential Equations
,
Springer-Verlag
,
Berlin, Germany
.10.1007/978-3-642-14574-2
2.
Carpinteri
,
A.
,
Chiaia
,
B.
, and
Cornetti
,
P.
,
2001
, “
Static-Kinematic Duality and the Principle of Virtual Work in the Mechanics of Fractal Media
,”
Comput. Methods Appl. Mech. Eng.
,
191
(
1–2
), pp.
3
19
.10.1016/S0045-7825(01)00241-9
3.
Podlubny
,
I.
,
1999
,
Fractional Differential Equations. Mathematics in Science and Engineering
,
Academic Press
,
New York
, p.
198
.
4.
Oldham
,
K. B.
, and
Spanier
,
J.
,
1974
,
The Fractional Calculus, Integrations and Differentiations of Arbitrary Order
,
Academic Press
,
New York
.
5.
Rossikhin
,
Y. A.
, and
Shitikova
,
M. V.
,
1997
, “
Applications of Fractional Calculus to Dynamic Problems of Linear and Nonlinear Hereditary Mechanics of Solids
,”
ASME Appl. Mech. Rev.
,
50
(
1
), pp.
15
67
.10.1115/1.3101682
6.
Baillie
,
R. T.
,
1996
, “
Long Memory Processes and Fractional Integration in Econometrics
,”
J. Econometrics
,
73
(
1
), pp.
5
59
.10.1016/0304-4076(95)01732-1
7.
Kilbas
,
A. A.
,
Srivastava
,
H. M.
, and
Trujillo
,
J. J.
,
2006
,
Theory and Applications of Fractional Differential Equations
,
Elsevier
,
San Diego, CA
.
8.
Oldham
,
K. B.
, and
Spanier
,
J.
,
1974
,
Fractional Calculus: Theory and Applications, Differentiation and Integration to Arbitrary Order
,
Academic Press
,
New York/London, UK
.
9.
Avudainayagam
,
A.
, and
Vani
,
C.
,
2000
, “
Wavelet Galerkin Method for Integro-Differential Equations
,”
Appl. Numer. Math.
,
32
(
3
), pp.
247
254
.10.1016/S0168-9274(99)00026-4
10.
Maleknejad
,
K.
, and
Tavassoli Kajani
,
M.
,
2004
, “
Solving Linear Integro-Differential Equation System by Galerkin Methods With Hybrid Functions
,”
Appl. Math. Comput.
,
159
(
3
), pp.
603
612
.10.1016/j.amc.2003.10.046
11.
Zhu
,
L.
, and
Fan
,
Q.
,
2012
, “
Solving Fractional Nonlinear Fredholm Integro-Differential Equations by the Second Kind Chebyshev Wavelet
,”
Commun. Nonlinear Sci. Numer. Simul.
,
17
(
6
), pp.
2333
2341
.10.1016/j.cnsns.2011.10.014
12.
Huang
,
L.
,
Li
,
X.
,
Zhaoa
,
Y.
, and
Duana
,
X.
,
2011
, “
Approximate Solution of Fractional Integro-Differential Equations by Taylor Expansion Method
,”
Comput. Math. Appl.
,
62
(
3
), pp.
1127
1134
.10.1016/j.camwa.2011.03.037
13.
Saeedi
,
H.
, and
Moghadam
,
M. M.
,
2011
, “
Numerical Solution of Nonlinear Volterra Integro-Differential Equations of Arbitrary Order by CAS Wavelets
,”
Commun. Nonlinear Sci. Numer. Simul.
,
16
(
3
), pp.
1216
1226
.10.1016/j.cnsns.2010.07.017
14.
Khader
,
M. M.
,
2011
, “
Numerical Solution of Nonlinear Multi-Order Fractional Differential Equations by Implementation of the Operational Matrix of Fractional Derivative
,”
Stud. Nonlinear Sci.
,
2
(
1
), pp.
5
12
.
15.
Balaji
,
S.
, “
Legendre Wavelet Operational Matrix Method for Solution of Fractional Order Riccati Differential Equation
,”
J. Egypt. Math. Soc.
, (in press).
16.
El-Wakil
,
S. A.
,
Elhanbaly
,
A.
, and
Abdou
,
M. A.
,
2006
, “
Adomian Decomposition Method for Solving Fractional Nonlinear Differential Equations
,”
Appl. Math. Comput.
,
182
(
1
), pp.
313
324
.10.1016/j.amc.2006.02.055
17.
Ertürk
,
V. S.
, and
Momani
,
S.
,
2008
, “
Solving Systems of Fractional Differential Equations Using Differential Transform Method
,”
J. Comput. Appl. Math.
,
215
(
1
), pp.
142
151
.10.1016/j.cam.2007.03.029
18.
Ertürk
,
V. S.
,
Momani
,
S.
, and
Odibat
,
Z.
,
2088
, “
Application of Generalized Differential Transform Method to Multi-Order Fractional Differential Equations
,”
Commun. Nonlinear Sci. Numer. Simul.
,
13
(
8
), pp.
1642
1654
.10.1016/j.cnsns.2007.02.006
19.
Karimi Vanani
,
S.
, and
Aminataei
,
A.
,
2011
, “
Operational Tau Approximation for a General Class of Fractional Integro-Differential Equations
,”
J. Comput. Appl. Math.
,
30
(
3
), pp.
655
674
.
20.
Khader
,
M. M.
, and
Sweilam
,
N. H.
,
2013
, “
On the Approximate Solutions for System of Fractional Integro-Differential Equations Using Chebyshev Pseudo-Spectral Method
,”
Appl. Math. Modell.
,
37
(
24
), pp.
9819
9828
.10.1016/j.apm.2013.06.010
21.
Doha
,
E. H.
,
Bhrawy
,
A. H.
, and
Ezz-Eldien
,
S. S.
,
2012
, “
A New Jacobi Operational Matrix: An Application for Solving Fractional Differential Equations
,”
Appl. Math. Modell.
,
36
(
10
), pp.
4931
4943
.10.1016/j.apm.2011.12.031
22.
Yang
,
Y.
,
Chen
,
Y.
, and
Huang
,
Y.
,
2014
, “
Convergence Analysis of the Jacobi Spectral-Collocation Method for Fractional Integro-Differential Equations
,”
Acta Math. Sci.
,
34
(
3
), pp.
673
690
.10.1016/S0252-9602(14)60039-4
23.
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
.10.1016/j.amc.2012.12.072
24.
Mirzaee
,
F.
, and
Bimesl
,
S.
,
2014
, “
A New Euler Matrix Method for Solving Systems of Linear Volterra Integral Equations With Variable Coefficients
,”
J. Egypt. Math. Soc.
,
22
(
2
), pp.
238
248
.10.1016/j.joems.2013.06.016
25.
Mirzaee
,
F.
, and
Bimesl
,
S.
,
2014
, “
Application of Euler Matrix Method for Solving Linear and a Class of Nonlinear Fredholm Integro-Differential Equations
,”
Mediterr. J. Math.
,
11
(
3
), pp.
999
1018
.10.1007/s00009-014-0391-4
26.
Hille
,
E.
, and
Phillips
,
R. S.
,
1974
,
Functional Analysis and Semi-Groups
,
American Mathematical Society
, Colloquium Publications, p.
31
.
27.
Caputo
,
M.
,
1967
, “
Linear Model of Dissipation Whose Q is Almost Frequency Independent
,”
Geophys. J. R. Astron. Soc.
,
13
(
5
), pp.
529
539
.10.1111/j.1365-246X.1967.tb02303.x
28.
Kim
,
T.
,
2012
, “
Identities Involving Frobenius–Euler Polynomials Arising From Non-Linear Differential Equations
,”
J. Number Theory
,
132
(
12
), pp.
2854
2865
.10.1016/j.jnt.2012.05.033
29.
Chu
,
W.
, and
Wang
,
C. Y.
,
2009
, “
Arithmetic Identities Involving Bernoulli and Euler Numbers
,”
Results Math.
,
55
(
1–2
), pp.
65
77
.10.1007/s00025-009-0378-9
30.
Tohidi
,
E.
,
Bhrawy
,
A. H.
, and
Erfani
,
Kh.
,
2013
, “
A Collocation Method Based on Bernoulli Operational Matrix for Numerical Solution of Generalized Pantograph Equation
,”
Appl. Math. Modell.
,
37
(
6
), pp.
4283
4294
.10.1016/j.apm.2012.09.032
31.
Srivastava
,
H. M.
,
2004
, “
Remarks on Some Relationships Between the Bernoulli and Euler Polynomials
,”
Appl. Math. Lett.
,
17
(
4
), pp.
375
380
.10.1016/S0893-9659(04)90077-8
32.
Mirzaee
,
F.
, and
Bimesl
,
S.
,
2013
, “
A New Approach to Numerical Solution of Second-Order Linear Hyperbolic Partial Differential Equations Arising From Physics and Engineering
,”
Results Phys.
,
3
, pp.
241
247
.10.1016/j.rinp.2013.10.002
33.
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
You do not currently have access to this content.