|
Initial value problems in discrete fractional calculus
Author(s):
Ferhan
M.
Atici;
Paul
W.
Eloe
Journal:
Proc. Amer. Math. Soc.
137
(2009),
981-989.
MSC (2000):
Primary 39A12, 34A25, 26A33
Posted:
September 10, 2008
MathSciNet review:
2457438
Retrieve article in:
PDF
Abstract |
References |
Similar articles |
Additional information
Abstract:
This paper is devoted to the study of discrete fractional calculus; the particular goal is to define and solve well-defined discrete fractional difference equations. For this purpose we first carefully develop the commutativity properties of the fractional sum and the fractional difference operators. Then a  -th (  ) order fractional difference equation is defined. A nonlinear problem with an initial condition is solved and the corresponding linear problem with constant coefficients is solved as an example. Further, the half-order linear problem with constant coefficients is solved with a method of undetermined coefficients and with a transform method.
References:
-
- 1.
- F. M. ATICI AND P. W. ELOE, A transform method in discrete fractional calculus, International Journal of Difference Equations, Vol. 2, 2 (2007), pp. 165-176.
- 2.
- F. M. ATICI AND P. W. ELOE, Fractional q-calculus on a time scale, Journal of Nonlinear Mathematical Physics, Vol. 14, 3 (2007), pp. 333-344. MR 2350094
- 3.
- M. BOHNER AND G. GUSEINOV, The convolution on time scales, Abstract and Applied Analysis, 2007, Art. ID 58373, 24 pp. MR 2320804 (2008f:44001)
- 4.
- M. BOHNER AND A. PETERSON, Dynamic Equations on Time Scales, Birkhäuser, Boston, 2001. MR 1843232 (2002c:34002)
- 5.
- M. BOHNER AND A. PETERSON, Laplace transform and
 -transform: Unification and extension, Methods and Applications of Analysis, Vol. 9, 1 (2002), pp. 151-157. MR 1948468 (2003m:34027) - 6.
- W. G. KELLEY AND A. C. PETERSON, Difference Equations: An Introduction with Applications, Harcourt/Academic Press, San Diego, CA, 2001. MR 1765695 (2001i:39001)
- 7.
- K. S. MILLER AND B. ROSS, Fractional Difference Calculus, Proceedings of the International Symposium on Univalent Functions, Fractional Calculus and Their Applications, Nihon University, Koriyama, Japan, May 1988, pp. 139-152; Ellis Horwood Ser. Math. Appl., Horwood, Chichester, 1989. MR 1199147 (93h:26010)
- 8.
- K. S. MILLER AND B. ROSS, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, Inc., New York, 1993. MR 1219954 (94e:26013)
- 9.
- I. PODLUBNY, Fractional Differential Equations, Academic Press, San Diego, CA, 1999. MR 1658022 (99m:26009)
- 10.
- S. G. SAMKO, A. A. KILBAS, AND O. I. MARICHEV, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, 1993. MR 1347689 (96d:26012)
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical
Society
with
MSC (2000):
39A12, 34A25, 26A33
Retrieve articles in all Journals with
MSC (2000):
39A12, 34A25, 26A33
Additional Information:
Ferhan
M.
Atici
Affiliation:
Department of Mathematics, Western Kentucky University, Bowling Green, Kentucky 42101-3576
Email:
ferhan.atici@wku.edu
Paul
W.
Eloe
Affiliation:
Department of Mathematics, University of Dayton, Dayton, Ohio 45469-2316
Email:
Paul.Eloe@notes.udayton.edu
DOI:
10.1090/S0002-9939-08-09626-3
PII:
S 0002-9939(08)09626-3
Keywords:
Discrete fractional calculus
Received by editor(s):
February 25, 2008
Posted:
September 10, 2008
Communicated by:
Jane M. Hawkins
Copyright of article:
Copyright
2008,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
