This paper compares the performance of Legendre wavelets (LWs) with integer and noninteger orders for solving fractional nonlinear Fredholm integro-differential equations (FNFIDEs). The generalized fractional-order Legendre wavelets (FLWs) are formulated and the operational matrix of fractional derivative in the Caputo sense is obtained. Based on the FLWs, the operational matrix and the Tau method an efficient algorithm is developed for FNFIDEs. The FLWs basis leads to more efficient and accurate solutions of the FNFIDE than the integer-order Legendre wavelets. Numerical examples confirm the superior accuracy of the proposed method.

References

References
1.
Oldham
,
K. B.
, and
Spanier
,
J.
,
1974
,
The Fractional Calculus
,
Academic Press
,
New York
.
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
,
Elsevier
,
San Diego, CA
.
4.
Bagley
,
R. L.
, and
Torvik
,
P. J.
,
1983
, “
A Theoretical Basis for the Application of Fractional Calculus to Viscoelasticity
,”
J. Rheol.
,
27
(
3
), pp.
201
210
.
5.
Baillie
,
R. T.
,
1996
, “
Long Memory Processes and Fractional Integration in Econometrics
,”
J. Econometrics
,
73
(
1
), pp.
5
59
.
6.
He
,
J.
,
1998
, “
Nonlinear Oscillation With Fractional Derivative and Its Applications
,”
International Conference Vibrating Engineering
, Leuven, Belgium, Sept. 16–18, pp.
288
291
.
7.
He
,
J.
,
1999
, “
Some Applications of Nonlinear Fractional Differential Equations and Their Approximations
,”
Bull. Sci. Technol.
,
15
(
2
), pp.
86
90
.
8.
Mainardi
,
F.
,
1997
, “
Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics
,”
Fractals and Fractional Calculus in Continuum Mechanics
,
A.
Carpinteri
and
F.
Mainardi
, eds.,
Springer-Verlag
,
Wien, Austria
, pp.
291
348
.
9.
Panda
,
R.
, and
Dash
,
M.
,
2006
, “
Fractional Generalized Splines and Signal Processing
,”
Signal Process
,
86
(
9
), pp.
2340
2350
.
10.
Bohannan
,
G.
,
2008
, “
Analog Fractional Order Controller in Temperature and Motor Control Applications
,”
J. Vib. Control
,
14
(
9–10
), pp.
1487
1498
.
11.
Kumar
,
S.
,
Kumar
,
D.
,
Abbasbandy
,
S.
, and
Rashidi
,
M. M.
,
2014
, “
Analytical Solution of Fractional Navier-Stokes Equation by Using Modified Laplace Decomposition Method
,”
Ain Shams Eng. J.
,
5
(
2
), pp.
569
574
.
12.
Li
,
Y. Y.
,
Zhao
,
Y.
,
Xie
,
G. N.
,
Baleanu
,
D.
,
Yang
,
X. J.
, and
Zhao
,
K.
,
2014
, “
Local Fractional Poisson and Laplace Equations With Applications to Electrostatics in Fractal Domain
,”
Adv. Math. Phys.
,
2014
, p.
590574
.
13.
Arikoglu
,
A.
, and
Ozkol
,
I.
,
2009
, “
Solution of Fractional Integro-Differential Equations by Using Fractional Differential Transform Method
,”
Chaos Solitons Fractals
,
40
(
2
), pp.
521
529
.
14.
Meerschaert
,
M.
, and
Tadjeran
,
C.
,
2006
, “
Finite Difference Approximations for Two-Sided Space-Fractional Partial Differential Equations
,”
Appl. Numer. Math.
,
56
(
1
), pp.
80
90
.
15.
Suarez
,
L.
, and
Shokooh
,
A.
,
1997
, “
An Eigenvector Expansion Method for the Solution of Motion Containing Fractional Derivatives
,”
ASME J. Appl. Mech.
,
64
(
3
), pp.
629
635
.
16.
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
.
17.
Li
,
Y.
, and
Sun
,
N.
,
2011
, “
Numerical Solution of Fractional Differential Equations Using the Generalized Block Pulse Operational Matrix
,”
Comput. Math. Appl.
,
62
(
3
), pp.
1046
1054
.
18.
Tripathi
,
M. P.
,
Baranwal
,
V. K.
,
Pandey
,
R. K.
, and
Singh
,
O. P.
,
2013
, “
A New Numerical Algorithm to Solve Fractional Differential Equations Based on Operational Matrix of Generalized Hat Functions
,”
Commun. Nonlinear Sci. Numer. Simul.
,
18
(
6
), pp.
1327
1340
.
19.
Momani
,
S.
, and
Al-Khaled
,
K.
,
2005
, “
Numerical Solutions for Systems of Fractional Differential Equations by the Decomposition Method
,”
Appl. Math. Comput.
,
162
(
3
), pp.
1351
1365
.
20.
Quan
,
X. J.
,
Han
,
H. L.
, and
Wang
,
J.
,
2014
, “
The Adomian Decomposition Method for Sloving Nonlinear Volterra Integral Equations of Fractional Order
,”
J. Jiangxi Normal Univ. (Natural Sci. Ed.)
,
5
, p.
18
.
21.
Odibat
,
Z.
, and
Momani
,
S.
,
2006
, “
Application of Variational Iteration Method to Nonlinear Differential Equations of Fractional Order
,”
Int. J. Nonlinear Sci. Numer. Simul.
,
7
(1), pp.
27
34
.
22.
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
.
23.
Hashim
,
I.
,
Abdulaziz
,
O.
, and
Momani
,
S.
,
2009
, “
Homotopy Analysis Method for Fractional IVPs
,”
Commun. Nonlinear Sci. Numer. Simul.
,
14
(
3
), pp.
674
684
.
24.
Baleanu
,
D.
,
Darzi
,
R.
, and
Agheli
,
B.
,
2018
, “
A Reliable Mixed Method for Singular Integro-Differential Equations of Non-Integer Order
,”
Math. Modell. Natural Phenom.
,
13
(
1
). p. 4.
25.
Yin
,
Y.
,
Yanping
,
C.
, and
Yunqing
,
H.
,
2014
, “
Convergence Analysis of the Jacobi Spectral-Collocation Method for Fractional Integro-Differential Equations
,”
Acta Math. Sci.
,
34
(
3
), pp.
673
690
.
26.
Mokhtary
,
P.
,
2015
, “
Reconstruction of Exponentially Rate of Convergence to Legendre Collocation Solution of a Class of Fractional Integro-Differential Equations
,”
J. Comput. Appl. Math.
,
279
, pp.
145
158
.
27.
Bhrawy
,
A. H.
, and
Zaky
,
M. A.
,
2015
, “
A Shifted Fractional-Order Jacobi Orthogonal Functions: An Application for System of Fractional Differential Equations
,”
Appl. Math. Modell.
,
40
(2), pp. 832–845.
28.
Bhrawy
,
B. A. H.
,
Alhamed
,
Y. A.
, and
Baleanu
,
D.
,
2014
, “
New Specral Techniques for Systems of Fractional Differential Equations Using Fractional-Order Generalized Laguerre Orthogonal Functions
,”
Fractional Calculus Appl. Anal.
,
17
(4), pp.
1138
1157
.
29.
Kazem
,
S.
,
Abbasbandy
,
S.
, and
Kumar
,
S.
,
2013
, “
Fractional-Order Legendre Functions for Solving Fractional-Order Differential Equations
,”
Appl. Math. Modell.
,
37
(
7
), pp.
5498
5510
.
30.
Doha
,
E. H.
,
Bhrawy
,
A. H.
,
Baleanu
,
D.
,
Ezz-Eldien
,
S. S.
, and
Hafez
,
R. M.
,
2015
, “
An Efficient Numerical Scheme Based on the Shifted Orthonormal Jacobi Polynomials for Solving Fractional Optimal Control Problems
,”
Adv. Differ. Equations
,
2015
(
1
), pp.
1
17
.
31.
Saeedi
,
H.
, and
Mohseni Moghadam
,
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
.
32.
Saeedi
,
H.
,
Moghadam
,
M. M.
,
Mollahasani
,
M.
, and
Chuev
,
G. N.
,
2011
, “
A CAS Wavelet Method for Solving Nonlinear Fredholm Integro-Differential Equations of Fractional Order
,”
Commun. Nonlinear. Sci. Numer. Simul.
,
16
(
3
), pp.
1154
1163
.
33.
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
.
34.
Zhu
,
L.
, and
Fan
,
Q.
,
2013
, “
Numerical Solution of Nonlinear Fractional-Order Volterra Integro-Differential Equations by SCW
,”
Commun. Nonlinear. Sci. Numer. Simul.
,
18
(
5
), pp.
1203
1213
.
35.
Mohammadi
,
F.
,
2014
, “
Numerical Solution of Bagley-Torvik Equation Using Chebyshev Wavelet Operational Matrix of Fractional Derivative
,”
Int. J. Adv. Appl. Math. Mech.
,
2
(
1
), pp.
83
91
.
36.
Shiralashetti
,
S. C.
, and
Deshi
,
A. B.
,
2016
, “
An Efficient Haar Wavelet Collocation Method for the Numerical Solution of Multi-Term Fractional Differential Equations
,”
Nonlinear Dyn.
,
83
(
1–2
), pp.
293
303
.
37.
Wang
,
Y.
, and
Fan
,
Q.
,
2012
, “
The Second Kind Chebyshev Wavelet Method for Solving Fractional Differential Equations
,”
Appl. Math. Comput.
,
218
(
17
), pp.
8592
8601
.
38.
Heydari
,
M. H.
,
Hooshmandasl
,
M. R.
, and
Mohammadi
,
F.
,
2014
, “
Two-Dimensional Legendre Wavelets for Solving Time-Fractional Telegraph Equation
,”
Adv. Appl. Math. Mech.
,
6
(
02
), pp.
247
260
.
39.
Mohammadi
,
F.
, and
Adhami
,
P.
,
2016
, “
Numerical Study of Stochastic Volterra-Fredholm Integral Equations by Using Second Kind Chebyshev Wavelets
,”
Random Operators Stochastic Equations
,
24
(
2
), pp.
129
141
.
40.
Razzaghi
,
M.
, and
Yousefi
,
S.
,
2001
, “
The Legendre Wavelets Operational Matrix of Integration
,”
Int. J. Syst. Sci.
,
32
(
4
), pp.
495
502
.
41.
Mohammadi
,
F.
, and
Hosseini
,
M. M.
,
2011
, “
A Comparative Study of Numerical Methods for Solving Quadratic Riccati Differential Equations
,”
J. Franklin Inst.
,
348
(
2
), pp.
156
164
.
42.
Mohammadi
,
F.
,
2011
, “
A New Legendre Wavelet Operational Matrix of Derivative and Its Applications in Solving the Singular Ordinary Differential Equations
,”
J. Franklin Inst.
,
348
(
8
), pp.
1787
1796
.
43.
Mohammadi
,
F.
,
Hosseini
,
M. M.
, and
Mohyud-Din
,
S. T.
,
2011
, “
Legendre Wavelet Galerkin Method for Solving Ordinary Differential Equations With Non-Analytic Solution
,”
Int. J. Syst. Sci.
,
42
(
4
), pp.
579
585
.
44.
Mohammadi
,
F.
,
2016
, “
A Computational Wavelet Method for Numerical Solution of Stochastic Volterra-Fredholm Integral Equations
,”
Wavelet Linear Algebra
,
3
(
1
), pp.
13
25
.http://wala.vru.ac.ir/article_19924.html
45.
Mohammadi
,
F.
, and
Ciancio
,
A.
,
2017
, “
Wavelet-Based Numerical Method for Solving Fractional Integro-Differential Equation With a Weakly Singular Kernel
,”
Wavelet Linear Algebra
,
4
(
1
), pp.
53
73
.
46.
Xu
,
X.
, and
Xu
,
D.
,
2017
, “
Legendre Wavelets Method for Approximate Solution of Fractional-Order Differential Equations Under Multi-Point Boundary Conditions
,”
Int. J. Comput. Math.
,
95
(5), pp. 998–1014.
47.
Liu
,
N.
, and
Lin
,
E. B.
,
2010
, “
Legendre Wavelet Method for Numerical Solutions of Partial Differential Equations
,”
Numer. Methods Partial Differ. Equations
,
26
(
1
), pp.
81
94
.
48.
Canuto
,
C.
,
Hussaini
,
M.
,
Quarteroni
,
A.
, and
Zang
,
T.
,
1988
,
Spectral Methods in Fluid Dynamics
,
Springer
, Berlin.
You do not currently have access to this content.