It has been known that the initialization of fractional operators requires time-varying functions, a complicating factor. This paper simplifies the process of initialization of fractional differential equations by deriving Laplace transforms for the initialized fractional integral and derivative that generalize those for the integer-order operators. The new transforms unify the initialization of systems of fractional and ordinary differential equations. The paper provides background on past work in the area and determines the Laplace transforms for the initialized fractional integral and fractional derivatives of any (real) order. An application provides insight and demonstrates the theory.

1.
Oldham
,
K. B.
, and
Spanier
,
J.
, 1974,
The Fractional Calculus
,
Mathematics in Science and Engineering, Academic
,
New York
.
2.
Davis
,
H. T.
, 1936,
The Theory of Linear Operators
,
Principia
,
Bloomington, IN
.
3.
Lorenzo
,
C. F.
, and
Hartley
,
T. T.
, 1998, “
Initialization, Conceptualization, and Application in the Generalized Fractional Calculus
,” NASA TP-1998–208415.
4.
Lorenzo
,
C. F.
, and
Hartley
,
T. T.
, 2000, “
Initialized Fractional Calculus
,”
Int. J. Appl. Math Comput. Sci.
0867-857X,
3
(
3
), pp.
249
266
.
5.
Lorenzo
,
C. F.
, and
Hartley
,
T. T.
, 2001, “
Initialization in Fractional-Order Systems
,”
European Control Conference
,
Porto, Portugal
, Sept. 4–7.
6.
Ortigueira
,
M. D.
, 2003, “
On the Initial Conditions in Continuous-Time Fractional Linear Systems
,”
Signal Process.
0165-1684,
83
(
11
), pp.
2301
3218
.
7.
Hartley
,
T. T.
, and
Lorenzo
,
C. F.
, 2002, “
Dynamics and Control of Initialized Fractional-Order Systems
,”
Nonlinear Dyn.
0924-090X,
29
(
1–4
), pp.
201
233
.
8.
Lorenzo
,
C. F.
, and
Hartley
,
T. T.
, 2007, “
Initialization of Fractional Differential Equations: Background and Theory
,”
Proceedings of IDETC/CIE
, DETEC2007-34810.
9.
Lorenzo
,
C. F.
, and
Hartley
,
T. T.
, 2007, “
Initialization of Fractional Differential Equations: Theory and Application
,”
Proceedings of IDETC/CIE
, DETEC2007-34814.
10.
Erdelyi
,
A.
,
Oberhettinger
,
F.
, and
Tricomi
,
F. G.
, 1954,
Tables of Integral Transforms
, Bateman Manuscript Project,
McGraw-Hill
, p.
137
.
11.
Spanier
,
J.
, and
Oldham
,
K.
, 1987,
An Atlas of Functions
,
Hemisphere
,
New York
,
Springer-Verlag
,
London
.
12.
Apostol
,
T. M.
, 1957,
Mathematical Analysis
,
Addison-Wesley
,
Reading, MA
, LCCC 578-8707.
13.
Wylie
,
C. R.
, 1975,
Advanced Engineering Mathematics
,
McGraw-Hill
,
New York
.
14.
Lorenzo
,
C. F.
, and
Hartley
,
T. T.
, 1999, “
Generalized Functions for the Fractional Calculus
,” NASA TP-1999–209424.
This content is only available via PDF.
You do not currently have access to this content.