In this paper, we present a new type of fractional operator, the Caputo–Katugampola derivative. The Caputo and the Caputo–Hadamard fractional derivatives are special cases of this new operator. An existence and uniqueness theorem for a fractional Cauchy-type problem, with dependence on the Caputo–Katugampola derivative, is proved. A decomposition formula for the Caputo–Katugampola derivative is obtained. This formula allows us to provide a simple numerical procedure to solve the fractional differential equation (FDE).

References

References
1.
Kilbas
,
A. A.
,
Srivastava
,
H. M.
, and
Trujillo
,
J. J.
,
2006
,
Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies
,
Elsevier Science B.V.
,
Amsterdam, The Netherlands
.
2.
Podlubny
,
I.
,
1999
,
Fractional Differential Equations Mathematics in Science and Engineering
, Vol.
198
,
Academic Press
,
San Diego, CA
.
3.
Samko
,
S. G.
,
Kilbas
,
A. A.
, and
Marichev
,
O. I.
,
1993
,
Fractional Integrals and Derivatives
[translated from the 1987 Russian original],
Gordon and Breach
,
Yverdon, Switzerland
.
4.
Katugampola
,
U. N.
,
2011
, “
New Approach to a Generalized Fractional Integral
,”
Appl. Math. Comput.
,
218
(
3
), pp.
860
865
.
5.
Katugampola
,
U. N.
,
2014
, “
A New Approach to Generalized Fractional Derivatives
,”
Bull. Math. Anal. Appl.
,
6
(
4
), pp.
1
15
.
6.
Katugampola
,
U. N.
,
2016
, “
Existence and Uniqueness Results for a Class of Generalized Fractional Differential Equations
,” arXiv:1411.5229
7.
Gambo
,
Y. Y.
,
Jarad
,
F.
,
Baleanu
,
D.
, and
Abdeljawad
,
T.
,
2014
, “
On Caputo Modification of the Hadamard Fractional Derivatives
,”
Adv. Differ. Equations
,
2014
(
10
), pp.
1
12
.
8.
Jarad
,
F.
,
Abdeljawad
,
T.
, and
Baleanu
,
D.
,
2012
, “
Caputo-Type Modification of the Hadamard Fractional Derivatives
,”
Adv. Differ. Equations
,
2012
(
142
), pp.
1
8
.
9.
Kilbas
,
A. A.
, and
Marzan
,
S. A.
,
2004
, “
Cauchy Problem for Differential Equation With Caputo Derivative
,”
Fractional Calculus Appl. Anal.
,
7
(
3
), pp.
297
321
.
10.
Atanacković
,
T. M.
, and
Stankovic
,
B.
,
2008
, “
On a Numerical Scheme for Solving Differential Equations of Fractional Order
,”
Mech. Res. Commun.
,
35
(
7
), pp.
429
438
.
11.
Pooseh
,
S.
,
Almeida
,
R.
, and
Torres
,
D. F. M.
,
2013
, “
Numerical Approximations of Fractional Derivatives With Applications
,”
Asian J. Control
,
15
(
3
), pp.
698
712
.
12.
Pooseh
,
S.
,
Almeida
,
R.
, and
Torres
,
D. F. M.
,
2012
, “
Expansion Formulas in Terms of Integer-Order Derivatives for the Hadamard Fractional Integral and Derivative
,”
Numer. Funct. Anal. Optim.
,
33
(
3
), pp.
301
319
.
13.
Ford
,
N. J.
, and
Morgado
,
M. L.
,
2011
, “
Fractional Boundary Value Problems: Analysis and Numerical Methods
,”
Fractional Calculus Appl. Anal.
,
14
(
4
), pp.
554
567
.
14.
Ford
,
N. J.
, and
Morgado
,
L. M.
,
2012
, “
Distributed Order Equations as Boundary Value Problems
,”
Comput. Math. Appl.
,
64
(
10
), pp.
2973
2981
.
15.
Gracia
,
J. L.
, and
Stynes
,
M.
,
2015
, “
Central Difference Approximation of Convection in Caputo Fractional Derivative Two-Point Boundary Value Problems
,”
J. Comput. Appl. Math.
,
273
(
C
), pp.
103
115
.
16.
Sousa
,
E.
,
2014
, “
An Explicit High Order Method for Fractional Advection Diffusion Equations
,”
J. Comput. Phys.
,
278
, pp.
257
274
.
17.
Yan
,
Y.
,
Pal
,
K.
, and
Ford
,
N. J.
,
2014
, “
Higher Order Numerical Methods for Solving Fractional Differential Equations
,”
BIT Numer. Math.
,
54
(
2
), pp.
555
584
.
18.
Yang
,
Q.
,
Liu
,
F.
, and
Turner
,
I.
,
2010
, “
Numerical Methods for Fractional Partial Differential Equations With Riesz Space Fractional Derivatives
,”
Appl. Math. Modell.
,
34
(
1
), pp.
200
218
.
19.
Atanacković
,
T. M.
,
Janevb
,
M.
,
Pilipovicc
,
S.
, and
Zoricab
,
D.
,
2014
, “
Convergence Analysis of a Numerical Scheme for Two Classes of Non-Linear Fractional Differential Equations
,”
Appl. Math. Comput.
,
243
, pp.
611
623
.
20.
Garrappa
,
R.
,
2007
, “
Some Formulas for Sums of Binomial Coefficients and Gamma Functions
,”
Int. Math. Forum
,
2
(13–16), pp.
725
733
.
21.
Tricomi
,
F. G.
, and
Erdélyi
,
A.
,
1951
, “
The Asymptotic Expansion of a Ratio of Gamma Functions
,”
Pac. J. Math.
,
1
(
1
), pp.
133
142
.
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