- About this Journal
- Abstracting and Indexing
- Aims and Scope
- Annual Issues
- Article Processing Charges
- Articles in Press
- Author Guidelines
- Bibliographic Information
- Citations to this Journal
- Contact Information
- Editorial Board
- Editorial Workflow
- Free eTOC Alerts
- Publication Ethics
- Reviewers Acknowledgment
- Submit a Manuscript
- Subscription Information
- Table of Contents
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 406757, 13 pages
On the Definitions of Nabla Fractional Operators
1Department of Mathematics, Çankaya University, 06530 Ankara, Turkey
2Department of Mathematics, Western Kentucky University, Bowling Green, KY 42101, USA
Received 12 April 2012; Accepted 4 September 2012
Academic Editor: Dumitru Bǎleanu
Copyright © 2012 Thabet Abdeljawad and Ferhan M. Atici. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- H. L. Gray and N. F. Zhang, “On a new definition of the fractional difference,” Mathematics of Computation, vol. 50, no. 182, pp. 513–529, 1988.
- F. M. Atıcı and P. W. Eloe, “Discrete fractional calculus with the nabla operator,” Electronic Journal of Qualitative Theory of Differential Equations, no. 3, pp. 1–12, 2009.
- K. S. Miller and B. Ross, “Fractional difference calculus,” in Proceedings of the International Symposium on Univalent Functions, Fractional Calculus and Their Applications, pp. 139–152, Nihon University, Koriyama, Japan, 1989.
- T. Abdeljawad, “On Riemann and Caputo fractional differences,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1602–1611, 2011.
- T. Abdeljawad and D. Baleanu, “Fractional differences and integration by parts,” Journal of Computational Analysis and Applications, vol. 13, no. 3, pp. 574–582, 2011.
- F. Jarad, T. Abdeljawad, D. Baleanu, and K. Biçen, “On the stability of some discrete fractional nonautonomous systems,” Abstract and Applied Analysis, vol. 2012, Article ID 476581, 9 pages, 2012.
- J. Fahd, T. Abdeljawad, E. Gündoğdu, and D. Baleanu, “On the Mittag-Leffler stability of q-fractional nonlinear dynamical systems,” Proceedings of the Romanian Academy, vol. 12, no. 4, pp. 309–314, 2011.
- T. Abdeljawad and D. Baleanu, “Caputo q-fractional initial value problems and a q-analogue Mittag-Leffler function,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 12, pp. 4682–4688, 2011.
- F. M. Atici and P. W. Eloe, “A transform method in discrete fractional calculus,” International Journal of Difference Equations, vol. 2, no. 2, pp. 165–176, 2007.
- F. M. Atici and P. W. Eloe, “Initial value problems in discrete fractional calculus,” Proceedings of the American Mathematical Society, vol. 137, no. 3, pp. 981–989, 2009.
- F. M. Atıcı and S. Şengül, “Modeling with fractional difference equations,” Journal of Mathematical Analysis and Applications, vol. 369, no. 1, pp. 1–9, 2010.
- F. M. Atici and P. W. Eloe, “Linear systems of fractional nabla difference equations,” The Rocky Mountain Journal of Mathematics, vol. 41, no. 2, pp. 353–370, 2011.
- F. M. Atıcı and P. W. Eloe, “Gronwall’s inequality on discrete fractional calculus,” Computer and Mathematics with Applications. In press.
- R. A. C. Ferreira and D. F. M. Torres, “Fractional h-difference equations arising from the calculus of variations,” Applicable Analysis and Discrete Mathematics, vol. 5, no. 1, pp. 110–121, 2011.
- C. S. Goodrich, “Existence of a positive solution to a class of fractional differential equations,” Applied Mathematics Letters, vol. 23, no. 9, pp. 1050–1055, 2010.
- Nuno R. O. Bastos, Rui A. C. Ferreira, and Delfim F. M. Torres, “Discrete-time fractional variational problems,” Signal Processing, vol. 91, no. 3, pp. 513–524, 2011.
- G. A. Anastassiou, “Nabla discrete fractional calculus and nabla inequalities,” Mathematical and Computer Modelling, vol. 51, no. 5-6, pp. 562–571, 2010.
- K. Ahrendt, L. Castle, M. Holm, and K. Yochman, “Laplace transforms for the nabla -difference operator and a fractional variation of parameters formula,” Communications in Applied Analysis. In press.
- J. Hein, Z. McCarthy, N. Gaswick, B. McKain, and K. Speer, “Laplace transforms for the nabla-difference operator,” Panamerican Mathematical Journal, vol. 21, no. 3, pp. 79–97, 2011.
- K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993.
- I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999.
- S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, Switzerland, 1993.
- A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Application of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, 2006.