Initial value problems in discrete fractional calculus

Authors:
Ferhan M. Atici and Paul W. Eloe

Journal:
Proc. Amer. Math. Soc. **137** (2009), 981-989

MSC (2000):
Primary 39A12, 34A25, 26A33

DOI:
https://doi.org/10.1090/S0002-9939-08-09626-3

Published electronically:
September 10, 2008

MathSciNet review:
2457438

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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.

**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)**

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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:
https://doi.org/10.1090/S0002-9939-08-09626-3

Keywords:
Discrete fractional calculus

Received by editor(s):
February 25, 2008

Published electronically:
September 10, 2008

Communicated by:
Jane M. Hawkins

Article copyright:
© Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.