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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 406757, 13 pages
http://dx.doi.org/10.1155/2012/406757
Research Article

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.

Linked References

  1. 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. View at Publisher · View at Google Scholar
  2. 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.
  3. 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.
  4. T. Abdeljawad, “On Riemann and Caputo fractional differences,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1602–1611, 2011. View at Publisher · View at Google Scholar
  5. 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.
  6. 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. View at Publisher · View at Google Scholar
  7. 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.
  8. 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. View at Publisher · View at Google Scholar
  9. 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.
  10. 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. View at Publisher · View at Google Scholar
  11. 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. View at Publisher · View at Google Scholar
  12. 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. View at Publisher · View at Google Scholar
  13. F. M. Atıcı and P. W. Eloe, “Gronwall’s inequality on discrete fractional calculus,” Computer and Mathematics with Applications. In press.
  14. 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. View at Publisher · View at Google Scholar
  15. 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. View at Publisher · View at Google Scholar
  16. 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.
  17. G. A. Anastassiou, “Nabla discrete fractional calculus and nabla inequalities,” Mathematical and Computer Modelling, vol. 51, no. 5-6, pp. 562–571, 2010. View at Publisher · View at Google Scholar
  18. 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.
  19. 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.
  20. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993.
  21. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999.
  22. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, Switzerland, 1993.
  23. 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.