Abstract

The Laplace transform method is one of the powerful tools in studying the fractional differential equations (FDEs). In this paper, it is shown that the Heaviside expansion method for integer order differential equations is also applicable to the Laplace transforms of multiterm Caputo FDEs of zero initial conditions if the orders of Caputo derivatives are integer multiples of a common real number. The particular solution of a linear multiterm Caputo FDE is obtained by its Laplace transform and the Heaviside expansion method. A Caputo FDE of nonzero initial conditions is transformed to an Caputo FDE of zero initial conditions by an appropriate change of variables. In the latter, the terms originated from the initial conditions appear as nonhomogeneous terms. Thus, the solution to the Caputo FDE of nonzero initial conditions is obtained as the particular solutions to the equivalent Caputo FDE of zero initial conditions. The solutions of a linear multiterm Caputo FDEs of nonzero initial conditions are expressed through the two parameter Mittag–Leffler functions.

References

1.
Podlubny
,
I.
,
1999
,
Fractional Differential Equations
,
Academic Press
,
New York
.
2.
Mainardi
,
F.
,
2010
,
Fractional Calculus and Waves in Linear Viscoelasticity
,
Imperial College Press
,
London
.
3.
Ortigueira
,
M. D.
,
2011
,
Fractional Calculus for Scientists and Engineers
,
Springer
,
Dordrecht, The Netherlands
.
4.
J.
,
Klafter
,
S. C.
,
Lim
, and
R.
,
Metzler
, eds.,
2012
,
Fractional Dynamics: Recent Advances
,
World Scientific
, Hackensack,
NJ
.
5.
Atanackovć
,
T. M.
,
Pilipović
,
S.
,
Stanković
,
B.
, and
Zorica
,
D.
,
2014
,
Fractional Calculus With Applications in Mechanics
,
Wiley
,
London
.
6.
Gorenflo
,
R.
,
Kilbas
,
A. A.
,
Mainardi
,
F.
, and
Rogoson
,
S. V.
,
2014
,
Mittag-Leffler Function, Related Topics and Applications
,
Springer-Verlag
,
Berlin
.
7.
Yu
,
Y.
,
Perdikaris
,
P.
, and
Karniadakis
,
G. E.
,
2016
, “
Fractional Modeling of Viscoelasticity in 3D Cerebral Arteries and Aneurysms
,”
J. Comput. Phys.
,
323
(
2
), pp.
219
242
.10.1016/j.jcp.2016.06.038
8.
Baleanu
,
D.
,
Diethelm
,
K.
,
Scalas
,
E.
, and
Trujillo
,
J. J.
,
2017
,
Fractional Calculus
, 2nd ed.,
World Scientific
,
London
.
9.
Orsingher
,
E.
, and
Beghin
,
L.
,
2004
, “
Time-Fractional Telegraph Equations and Telegraph Processes With Brownian Time
,”
Probab. Theory Relat. Fields
,
128
(
1
), pp.
141
160
.10.1007/s00440-003-0309-8
10.
Camagro
,
R. F.
,
Chiaccho
,
A. O.
, and
de Oliveira
,
E. C.
,
2008
, “
Differentiation to Fractional Orders and the Fractional Telegraph Equation
,”
J. Math. Phys.
,
49
, p.
033505
.10.1063/1.2890375
11.
Chen
,
J.
,
Liu
,
F.
, and
Anh
,
V.
,
2008
, “
Analytical Solution for the Time-Fractional Telegraph Equation by the Method of Separating Variables
,”
J. Math. Anal. Appl.
,
338
(
2
), pp.
1364
1377
.10.1016/j.jmaa.2007.06.023
12.
Vergara
,
V.
,
2014
, “
Asymptotic Behaviour of the Time-Fractional Telegraph Equation
,”
J. Appl. Prob.
,
51
(
3
), pp.
890
893
.10.1239/jap/1409932682
13.
Boyadjiev
,
L.
, and
Luchko
,
T.
,
2017
, “
The Neutral-Fractional Telegraph Equation
,”
Math. Modell. Nat. Phenom.
,
12
(
6
), pp.
51
67
.10.1051/mmnp/2017064
14.
Mamchuev
,
M. O.
,
2017
, “
Solutions of the Main Boundary Value Problems for the Time-Fractional Telegraph Equation by the Green Function Method
,”
Fractional Calculus Appl. Anal.
,
20
(
1
), pp.
190
211
.10.1515/fca-2017-0010
15.
Tenenbaum
,
M.
, and
Pollard
,
H.
,
1963
,
Ordinary Differential Equations
,
Dover
,
New York
.
16.
Oldham
,
K. H.
, and
Spanier
,
J.
,
1974
,
The Fractional Calculus
,
Dover
,
Mineola, NY
.
17.
Miller
,
K. S.
, and
Ross
,
B.
,
1993
,
An Introduction to the Fractional Calculus and Fractional Differential Equations
,
Wiley
,
New York
.
18.
Kilbas
,
A. A.
,
Srivastava
,
H. M.
, and
Trujillo
,
J. J.
,
2006
,
Theory and Applications of Fractional Differential Equations
,
Elsevier
,
Amsterdam, The Netherlands
.
19.
Diethelm
,
K.
,
2010
,
The Analysis of Fractional Differential Equations
,
Springer
,
Berlin
.
20.
Gorenflo
,
R.
, and
Mainardi
,
F.
,
1997
, “
Fractional Calculus: Integral and Differential Equations of Fractional Order
,”
Fractals and Fractional Calculus in Continuum Mechanics
,
A.
,
Carpinteri
, and
F.
,
Mainardi
, eds.,
Springer
,
Wien, Austria
, pp.
223
276
.
21.
Diethelm
,
K.
, and
Ford
,
N. J.
,
2004
, “
Multi-Order Fractional Differential Equations and Their Numerical Solution
,”
Appl. Math. Comput.
,
154
(
3
), pp.
621
640
.10.1016/S0096-3003(03)00739-2
22.
Li
,
C.
, and
Deng
,
W.
,
2007
, “
Remarks on Fractional Derivatives
,”
Appl. Math. Comput.
,
187
(
2
), pp.
777
784
.10.1016/j.amc.2006.08.163
23.
Prabhakar
,
T. R.
,
1971
, “
A Singular Integral Equation With a Generalized Mittag-Leffler Function in the Kernel
,”
Yokohama Math. J.
,
19
(
1
), pp.
7
15
.
24.
Luchko
,
Y.
, and
Gorenflo
,
R.
,
1999
, “
An Operational Method for Solving Fractional Differential Equations With the Caputo Derivatives
,”
Acta Mathematica Vietnamica
,
24
(
2
), pp.
207
203
.
25.
Li
,
C.
, and
Cai
,
M.
,
2020
,
Theory and Numerical Approximations of Fractional Integrals and Derivatives
,
SIAM
,
Philadelphia, PA
.
26.
Fukunaga
,
M.
,
2019
, “
Free Oscillation Solution for Fractional Differential System
,”
ASME J. Comput. Nonlinear Dyn.
,
14
(
12
), p.
124502
.10.1115/1.4044922
27.
Fukunaga
,
M.
,
2002
, “
On Initial Value Problems in Fractional Differential Equations
,”
Int. J. Appl. Math.
,
9
(
2
), pp.
219
236
.
28.
Fukunaga
,
M.
, and
Shimizu
,
N.
,
2004
, “
Analytical and Numerical Solutions for Fractional Viscoelastic Equations
,”
JSME Int. J., Ser. C
,
47
(
1
), pp.
251
259
.10.1299/jsmec.47.251
29.
Fukunaga
,
M.
, and
Shimizu
,
N.
,
2004
, “
Role of Prehistories in the Initial Value Problems of Fractional Viscoelastic Equations
,”
Nonlinear Dyn.
,
38
(
1–4
), pp.
207
220
.10.1007/s11071-004-3756-6
30.
Fukunaga
,
M.
,
Shimizu
,
N.
, and
Nasuno
,
H.
,
2009
, “
A Nonlinear Fractional Derivative Models of Impulse Motion for Viscoelastic Materials
,”
Phys. Scr.
,
2009
(
T136
), p.
014010
.10.1088/0031-8949/2009/T136/014010
31.
Fukunaga
,
M.
, and
Shimizu
,
N.
,
2014
, “
Comparison of Fractional Derivative Models for Finite Deformation With Experiments of Impulse Response
,”
J. Vib. Control
,
20
(
7
), pp.
1033
1041
.10.1177/1077546313481051
32.
Nasuno
,
H.
,
Shimizu
,
N.
, and
Fukunaga
,
M.
,
2007
, “
Fractional Derivative Consideration on Nonlinear Viscoelastic Static and Dynamical Behavior Under Large Pre-Displacement
,”
Advances in Fractional Calculus, Theoretical Development and Applications in Physics and Engineering
,
J.
Sabatier
,
O. P.
Agrawal
, and
J. A. T.
Machado
, eds.,
Springer
,
Dordrecht, The Netherlands
, pp.
363
376
.
33.
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
,
New York
, pp.
291
348
.
34.
Fukunaga
,
M.
,
2019
, “
Mode Analysis on Onset of Turing Instability in Time-Fractional Reaction-Subdiffusion Equations by Two-Dimensional Numerical Simulations
,”
ASME J. Comput. Nonlinear Dyn.
,
14
(
6
), p.
061005
.10.1115/1.4043149
35.
Diethelm
,
K.
,
Siegmund
,
S.
, and
Tuan
,
H. T.
,
2017
, “
Aymptotic Behavior of Solutions of Linear Multi-Order Fractional Differential Equation Systems
,”
Fractional Calvulus Appl. Anal.
,
20
(
5
), pp.
1165
1195
.10.1515/fca-2017-0062
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