The multiterm fractional variable-order differential equation has a massive application in physics and engineering problems. Therefore, a numerical method is presented to solve a class of variable order fractional differential equations (FDEs) based on an operational matrix of shifted Chebyshev polynomials of the fourth kind. Utilizing the constructed operational matrix, the fundamental problem is reduced to an algebraic system of equations which can be solved numerically. The error estimate of the proposed method is studied. Finally, the accuracy, applicability, and validity of the suggested method are illustrated through several examples.

References

1.
Baleanu
,
D.
,
Golmankhaneh
,
A. K.
,
Golmankhaneh
,
A. K.
, and
Nigmatullin
,
R. R.
,
2010
, “
Newtonian Law With Memory
,”
Nonlinear Dyn.
,
60
(1–2), pp.
81
86
.
2.
Engheta
,
N.
,
1996
, “
On Fractional Calculus and Fractional Multipoles in Electromagnetism
,”
IEEE Trans. Antennas Propag.
,
44
(
4
), pp.
554
566
.
3.
Golmankhaneh
,
A. K.
,
Golmankhaneh
,
A. K.
, and
Baleanu
,
D.
,
2011
, “
On Nonlinear Fractional Klein-Gordon Equation
,”
Signal Process.
,
91
(
3
), pp.
446
451
.
4.
Magin
,
R. L.
,
2010
, “
Fractional Calculus Models of Complex Dynamics in Biological Tissues
,”
Comput. Math. Appl.
,
59
(
5
), pp.
1586
1593
.
5.
Nagy
,
A. M.
, and
Sweilam
,
N. H.
,
2014
, “
An Efficient Method for Solving Fractional Hodgkin-Huxley Model
,”
Phys. Lett. A
,
378
(30–31), pp.
1980
1984
.
6.
Sweilam
,
N. H.
,
Nagy
,
A. M.
, and
El-Sayed
,
A. A.
,
2015
, “
Second Kind Shifted Chebyshev Polynomials for Solving Space Fractional Order Diffusion Equation
,”
Chaos, Solitons Fractals
,
73
, pp.
141
147
.
7.
Sweilam
,
N. H.
,
Nagy
,
A. M.
, and
El-Sayed
,
A. A.
,
2016
, “
On the Numerical Solution of Space Fractional Order Diffusion Equation Via Shifted Chebyshev Polynomials of the Third Kind
,”
J. King Saud Univ. Sci.
,
28
(
1
), pp.
41
47
.
8.
Samko
,
S. G.
, and
Ross
,
B.
,
1993
, “
Integration and Differentiation to a Variable Fractional Order
,”
Integr. Transfer Spec. Funct.
,
1
(
4
), pp.
277
300
.
9.
Bhrawy
,
A. H.
, and
Zaky
,
M. A.
,
2016
, “
Numerical Algorithm for the Variable-Order Caputo Fractional Functional Differential Equation
,”
Nonlinear Dyn.
,
85
(
3
), pp.
1815
1823
.
10.
Coimbra
,
C. F. M.
,
2003
, “
Mechanics With Variable-Order Differential Operators
,”
Ann. Phys.
,
12
(
11
), pp.
692
703
.
11.
Soon
,
C. M.
,
Coimbra
,
C. F. M.
, and
Kobayashi
,
M. H.
,
2005
, “
The Variable Viscoelasticity Oscillator
,”
Ann. Phys.
,
14
(
6
), pp.
378
389
.
12.
Bhrawy
,
A. H.
, and
Alshomrani
,
M.
,
2012
, “
A Shifted Legendre Spectral Method for Fractional-Order Multi-Point Boundary Value Problems
,”
Adv. Differ. Equations
,
8
, pp.
1
19
.
13.
Sweilam
,
N. H.
,
Nagy
,
A. M.
, and
El-Sayed
,
A. A.
,
2016
, “
Solving Time-Fractional Order Telegraph Equation Via Sinc-Legendre Collocation Method
,”
Mediterr. J. Math.
,
13
(
6
), pp.
5119
5133
.
14.
Babolian
,
E.
, and
Hosseini
,
M. M.
,
2002
, “
A Modified Spectral Method for Numerical Solution of Ordinary Differential Equations With Non-Analytic Solution
,”
Appl. Math. Comput.
,
132
(2–3), pp.
341
351
.
15.
El-Mesiry
,
A.
,
El-Sayed
,
A.
, and
El-Saka
,
H.
,
2005
, “
Numerical Methods for Multi-Term Fractional (Arbitrary) Orders Differential Equations
,”
Appl. Math. Comput.
,
160
(
3
), pp.
683
699
.
16.
Esmaeili
,
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.
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
.
18.
Ford
,
N. J.
, and
Connolly
,
J. A.
,
2009
, “
Systems-Based Decomposition Schemes for the Approximate Solution of Multi-Term Fractional Differential Equations
,”
Comput. Appl. Math.
,
229
(
2
), pp.
382
391
.
19.
Chen
,
Y. M.
,
Wei
,
Y. Q.
,
Liu
,
D. Y.
, and
Yu
,
H.
,
2015
, “
Numerical Solution for a Class of Nonlinear Variable Order Fractional Differential Equations With Legendre Wavelets
,”
Appl. Math. Lett.
,
46
, pp.
83
88
.
20.
Doha
,
E. H.
,
Bhrawy
,
A. H.
, and
Ezz-Eldien
,
S. S.
,
2011
, “
Efficient Chebyshev Spectral Methods for Solving Multi-Term Fractional Orders Differential Equations
,”
Appl. Math. Model.
,
35
(
12
), pp.
5662
5672
.
21.
Doha
,
E. H.
,
Bhrawy
,
A. H.
, and
Ezz-Eldien
,
S. S.
,
2011
, “
A Chebyshev Spectral Method Based on Operational Matrix for Initial and Boundary Value Problems of Fractional Order
,”
Comput. Math. Appl.
,
62
(
5
), pp.
2364
2373
.
22.
Bhrawy
,
A. H.
,
Taha
,
T. M.
, and
Machado
,
J. A. T.
,
2015
, “
A Review of Operational Matrices and Spectral Techniques for Fractional Calculus
,”
Nonlinear Dyn.
,
81
(
3
), pp.
1023
1052
.
23.
Ghoreishi
,
F.
, and
Yazdani
,
S.
,
2011
, “
An Extension of the Spectral TAU Method for Numerical Solution of Multi-Order Fractional Differential Equations With Convergence Analysis
,”
Comput. Math. Appl.
,
61
(
1
), pp.
30
43
.
24.
Vanani
,
S. K.
, and
Aminataei
,
A.
,
2011
, “
Tau Approximate Solution of Fractional Partial Differential Equations
,”
Comput. Math. Appl.
,
62
(
3
), pp.
1075
1083
.
25.
Youssri
,
Y. H.
, and
Abd-Elhameed
,
W. M.
,
2016
, “
Spectral Solutions for Multi-Term Fractional Initial Value Problems Using a New Fibonacci Operational Matrix of Fractional Integration
,”
Prog. Fract. Differ. Appl.
,
2
(
2
), pp.
141
151
.
26.
Zhou
,
F. Y.
, and
Xu
,
X.
,
2016
, “
The Third Kind Chebyshev Wavelets Collocation Method for Solving the Time-Fractional Convection Diffusion Equations With Variable Coefficients
,”
Appl. Math. Comput.
,
280
, pp.
11
29
.
27.
Keshavarz
,
E.
,
Ordokhani
,
Y.
, and
Razzaghi
,
M.
,
2014
, “
Bernoulli Wavelet Operational Matrix of Fractional Order Integration and Its Applications in Solving the Fractional Order Differential Equations
,”
Appl. Math. Model.
,
38
(
24
), pp.
6038
6051
.
28.
Tavares
,
D.
,
Almeida
,
R.
, and
Torres
,
D. F. M.
,
2016
, “
Caputo Derivatives of Fractional Variable Order: Numerical Approximations
,”
Commun. Nonlinear Sci. Numer. Simul.
,
35
, pp.
69
87
.
29.
Atanackovic
,
T. M.
,
Janev
,
M.
,
Pilipovic
,
S.
, and
Zorica
,
D.
,
2013
, “
An Expansion Formula for Fractional Derivatives of Variable Order
,”
Cent. Eur. J. Phys.
,
11
(
10
), pp.
1350
1360
.
30.
Chen
,
Y. M.
,
Liu
,
L. Q.
,
Li
,
B. F.
, and
Sun
,
Y.
,
2014
, “
Numerical Solution for the Variable Order Linear Cable Equation With Bernstein Polynomials
,”
Appl. Math. Comput.
,
238
, pp.
329
341
.
31.
Liu
,
J.
,
Li
,
X.
, and
Wu
,
L.
,
2016
, “
An Operational Matrix of Fractional Differentiation of the Second Kind of Chebyshev Polynomial for Solving Multiterm Variable Order Fractional Differential Equation
,”
Math. Probl. Eng.
,
2016
, pp.
1
10
.
32.
Maleknejad
,
K.
,
Nouri
,
K.
, and
Torkzadeh
,
L.
,
2016
, “
Operational Matrix of Fractional Integration Based on the Shifted Second Kind Chebyshev Polynomials for Solving Fractional Differential Equations
,”
Mediterr. J. Math.
,
13
(
3
), pp.
1377
1390
.
33.
Wang
,
L. F.
,
Ma
,
Y. P.
, and
Yang
,
Y. Q.
,
2014
, “
Legendre Polynomials Method for Solving a Class of Variable Order Fractional Differential Equation
,”
Comput. Model. Eng. Sci.
,
101
(
2
), pp.
97
111
.
34.
Shen
,
S.
,
Liu
,
F.
,
Chen
,
J.
,
Turner
,
I.
, and
Anh
,
V.
,
2012
, “
Numerical Techniques for the Variable Order Time Fractional Diffusion Equation
,”
Ann. Phys.
,
218
(
22
), pp.
10861
10870
.
35.
Sweilam
,
N. H.
, and
AL-Mrawm
,
H. M.
,
2011
, “
On the Numerical Solutions of the Variable Order Fractional Heat Equation
,”
Stud. Nonlinear Sci.
,
2
(1), pp.
31
36
.
36.
Mason
,
J. C.
, and
Handscomb
,
D. C.
,
2003
,
Chebyshev Polynomials
,
Chapman and Hall
,
New York
.
37.
Sweilam
,
N. H.
,
Nagy
,
A. M.
, and
El-Sayed
,
A. A.
,
2016
, “
Numerical Approach for Solving Space Fractional Order Diffusion Equations Using Shifted Chebyshev Polynomials of the Fourth Kind
,”
Turk. J. Math.
,
40
, pp.
1283
1297
.
38.
Veselić
,
K.
,
2011
,
Damped Oscillations of Linear Systems—A Mathematical Introduction
,
Springer
,
Berlin
.
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