This paper is devoted to investigating the relation between Hadamard-type fractional derivatives and finite part integrals in Hadamard sense; that is to say, the Hadamard-type fractional derivative of a given function can be expressed by the finite part integral of a strongly singular integral, which actually does not exist. Besides, our results also cover some fundamental properties on absolutely continuous functions, and the logarithmic series expansion formulas at the right end point of interval for functions in certain absolutely continuous spaces.

References

References
1.
Kilbas
,
A. A.
,
Srivastava
,
H. M.
, and
Trujillo
,
J. J.
,
2006
,
Theory and Applications of Fractional Differential Equations
,
Elsevier Science
,
Amsterdam, The Netherlands
.
2.
Li
,
C. P.
, and
Ma
,
L.
,
2016
, “
Lyapunov-Schmidt Reduction for Fractional Differential Systems
,”
ASME J. Comput. Nonlinear Dyn.
,
11
(
5
), p.
051022
.
3.
Li
,
C. P.
, and
Zeng
,
F. H.
,
2015
,
Numerical Methods for Fractional Calculus
,
Chapman and Hall/CRC
,
Boca Raton, FL
.
4.
Machado
,
J. A. T.
,
Silva
,
M. F.
,
Barbosa
,
R. S.
,
Jesus
,
I. S.
,
Reis
,
C. M.
,
Marcos
,
M. G.
, and
Galhano
,
A. F.
,
2010
, “
Some Applications of Fractional Calculus in Engineering
,”
Math. Probl. Eng.
,
2010
, p.
639801
.
5.
Ma
,
L.
, and
Li
,
C. P.
,
2016
, “
Center Manifold of Fractional Dynamical System
,”
ASME J. Comput. Nonlinear Dyn.
,
11
(
2
), p.
021010
.
6.
Li
,
C. P.
,
Yi
,
Q.
, and
Kurths
,
J.
,
2017
, “
Fractional Convection
,”
ASME J. Comput. Nonlinear Dyn.
, epub.
7.
Pinto
,
C. M. A.
, and
Machado
,
J. A. T.
,
2013
, “
Fractional Model for Malaria Transmission Under Control Strategies
,”
Comput. Math. Appl.
,
66
(
5
), pp.
908
916
.
8.
Pinto
,
C. M. A.
,
2017
, “
Persistence of Low Levels of Plasma Viremia and of the Latent Reservoir in Patients Under ART: A Fractional-Order Approach
,”
Commun. Nonlinear Sci. Numer. Simul.
,
43
, pp.
251
260
.
9.
Pinto
,
C. M. A.
, and
Carvalho
,
A. R. M.
,
2017
, “
The Role of Synaptic Transmission in a HIV Model With Memory
,”
Appl. Math. Comput.
,
292
, pp.
76
95
.
10.
Jarad
,
F.
,
Abdeljawad
,
T.
, and
Baleanu
,
D.
,
2017
, “
On the Generalized Fractional Derivatives and Their Caputo Modification
,”
J. Nonlinear Sci. Appl.
,
10
(
5
), pp.
2607
2619
.
11.
Baleanu
,
D.
,
Wu
,
G. C.
, and
Zeng
,
S. D.
,
2017
, “
Chaos Analysis and Asymptotic Stability of Generalized Caputo Fractional Differential Equations
,”
Chaos Solitons Fractals
,
102
, pp.
99
105
.
12.
Kilbas
,
A. A.
,
2001
, “
Hadamard-Type Fractional Calculus
,”
J. Korean Math. Soc.
,
38
(
6
), pp.
1191
1204
.http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.475.7888&rep=rep1&type=pdf
13.
Hadamard
,
J.
,
1892
, “
Essai Sur l’étude des Fonctions Données Par Leur Développment de Taylor
,”
J. Math. Pures Appl.
,
8
(4), pp.
101
186
.
14.
Gong
,
Z. Q.
,
Qian
,
D. L.
,
Li
,
C. P.
, and
Guo
,
P.
,
2012
, “
On the Hadamard Type Fractional Differential System
,”
Fractional Dynamics and Control
,
D.
Baleanu
, J. Machado, and A. Luo, eds.,
Springer
,
New York
, pp.
159
171
.
15.
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
.
16.
Jarad
,
F.
,
Abdeljawad
,
T.
, and
Baleanu
,
D.
,
2012
, “
Caputo-Type Modification of the Hadamard Fractional Derivatives
,”
Adv. Differ. Equations
,
2012
(
1
), pp.
1
8
.
17.
Ma
,
L.
, and
Li
,
C. P.
,
2017
, “
On Hadamard Fractional Calculus
,”
Fractals
,
25
(
3
), p.
1750033
.
18.
Hadamard
,
J.
,
1923
,
Lectures on Cauchy's Problem in Partial Differential Equations
,
Yale University Press
,
New Haven, CT
.
19.
Ioakimidis
,
N. I.
,
1982
, “
Application of Finite-Part Integrals to the Singular Integral Equations of Crack Problems in Plane and Three-Dimensional Elasticity
,”
Acta Mech.
,
45
(
1
), pp.
31
47
.
20.
Elliott
,
D.
,
1993
, “
An Asymptotic Analysis of Two Algorithms for Certain Hadamard Finite-Part Integrals
,”
IMA J. Numer. Anal.
,
13
(
3
), pp.
3273
3286
.
21.
Samko
,
S. G.
,
Kilbas
,
A. A.
, and
Marichev
,
O. I.
,
1993
,
Fractional Integral and Derivatives: Theory and Applications
,
Gordon and Breach Science Publishers
,
Philadelphia, PA
.
22.
Monegato
,
G.
,
2009
, “
Definitions, Properties and Applications of Finite-Part Integrals
,”
J. Comput. Appl. Math.
,
229
(
2
), pp.
425
439
.
23.
Kutt
,
H. R.
,
1975
, “
The Numerical Evaluation of Principal Value Integrals by Finite-Part Integration
,”
Numer. Math.
,
24
(
3
), pp.
205
210
.
24.
Paget
,
D. F.
,
1981
, “
The Numerical Evaluation of Hadamard Finite-Part Integrals
,”
Numer. Math.
,
36
(
4
), pp.
447
453
.
25.
George
,
T.
, and
George
,
D.
,
1990
, “
Gauss Quadrature Rules for Finite Part Integrals
,”
Int. J. Numer. Methods Eng.
,
30
(
1
), pp.
13
26
.
26.
Diethelm
,
K.
,
1997
, “
Generalized Compound Quadrature Formulae for Finite-Part Integrals
,”
IMA J. Numer. Anal.
,
17
(
3
), pp.
479
493
.
27.
Sun
,
W. W.
, and
Wu
,
J. M.
,
2008
, “
Interpolatory Quadrature Rules for Hadamard Finite-Part Integrals and Their Superconvergence
,”
IMA J. Numer. Anal.
,
28
(
3
), pp.
580
597
.
28.
Wu
,
J. M.
,
Dai
,
Z. H.
, and
Zhang
,
X. P.
,
2010
, “
The Superconvergence of the Composite Midpoint Rule for the Finite-Part Integral
,”
J. Comput. Appl. Math.
,
233
(
8
), pp.
1954
1968
.
29.
Zeng
,
F. H.
,
Mao
,
Z. P.
, and
Karniadakis
,
G. E.
,
2017
, “
A Generalized Spectral Collocation Method With Tunable Accuracy for Fractional Differential Equations With End-Point Singularities
,”
SIAM J. Sci. Comput.
,
39
(
1
), pp.
A360
A383
.
30.
Butzer
,
P. L.
,
Kilbas
,
A. A.
, and
Trujillo
,
J. J.
,
2002
, “
Fractional Calculus in the Mellin Setting and Hadamard-Type Fractional Integrals
,”
J. Math. Anal. Appl.
,
269
(
1
), pp.
1
27
.
31.
Varberg
,
D. E.
,
1965
, “
On Absolutely Continuous Functions
,”
Am. Math. Mon.
,
72
(
8
), pp.
831
841
.
32.
Cannarsa
,
P.
, and
D'Aprile
,
T.
,
2015
,
Introduction to Measure Theory and Functional Analysis
,
Springer International Publishing
,
Cham, Switzerland
.
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