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Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 406910, 12 pages
On Delta and Nabla Caputo Fractional Differences and Dual Identities
1Department of Mathematics and Physical Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia
2Department of Mathematics, Çankaya University, 06530 Ankara, Turkey
Received 13 January 2013; Accepted 14 June 2013
Academic Editor: Shurong Sun
Copyright © 2013 Thabet Abdeljawad. 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.
- I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, 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 Science, Yverdon, Switzerland, 1993.
- A. A. Kilbas, M. H. Srivastava, and J. J. Trujillo, Theory and Application of Fractional Differential Equations, vol. 204 of North Holland Mathematics Studies, Elsevier Science, 2006.
- O. P. Agrawal and D. Baleanu, “A Hamiltonian formulation and a direct numerical scheme for fractional optimal control problems,” Journal of Vibration and Control, vol. 13, no. 9-10, pp. 1269–1281, 2007.
- E. Scalas, “Mixtures of compound Poisson processes as models of tick-by-tick financial data,” Chaos, Solitons & Fractals, vol. 34, no. 1, pp. 33–40, 2007.
- D. Baleanu and J. J. Trujillo, “On exact solutions of a class of fractional Euler-Lagrange equations,” Nonlinear Dynamics, vol. 52, no. 4, pp. 331–335, 2008.
- 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.
- 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.
- T. Abdeljawad, “On Riemann and Caputo fractional differences,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1602–1611, 2011.
- F. M. Atıcı 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. Atıcı 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 P. W. Eloe, “Discrete fractional calculus with the nabla operator,” Electronic Journal of Qualitative Theory of Differential Equations, no. 3, pp. 1–12, 2009.
- F. M. Atıcı and S. Şengül, “Modelling with fractional difference equations,” Journal of Mathematical Analysis and Applications, vol. 369, pp. 1–9, 2010.
- N. R. O. Bastos, R. A. C. Ferreira, and D. F. M. Torres, “Discrete-time fractional variational problems,” Signal Processing, vol. 91, no. 3, pp. 513–524, 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.
- G. A. Anastassiou, “Principles of delta fractional calculus on time scales and inequalities,” Mathematical and Computer Modelling, vol. 52, no. 3-4, pp. 556–566, 2010.
- G. A. Anastassiou, “Nabla discrete fractional calculus and nabla inequalities,” Mathematical and Computer Modelling, vol. 51, no. 5-6, pp. 562–571, 2010.
- G. A. Anastassiou, “Foundations of nabla fractional calculus on time scales and inequalities,” Computers & Mathematics with Applications, vol. 59, no. 12, pp. 3750–3762, 2010.
- T. Abdeljawad and F. M. Atıcı, “On the definitions of nabla fractional operators,” Abstract and Applied Analysis, vol. 2012, Article ID 406757, 13 pages, 2012.
- T. Abdeljawad, “Dual identities in fractional difference calculus within Riemann,” Advances in Difference Equations, vol. 2013, article 36, 2013.
- M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, Mass, USA, 2003.
- G. Boros and V. Moll, Irresistible Integrals; Symbolics, Analysis and Experiments in the Evaluation of Integrals, Cambridge University Press, Cambridge, UK, 2004.
- R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics: Foundation for Computer Science, Addison-Wesley, Reading, Mass, USA, 2nd edition, 1994.
- J. Spanier and K. B. Oldham, “The Pochhammer Polynomials (,” in An Atlas of Functions, pp. 149–156, Hemisphere, Washington, DC, USA, 1987.
- F. M. Atıcı and P. W. Eloe, “Gronwall's inequality on discrete fractional calculus,” Computers & Mathematics with Applications, vol. 64, no. 10, pp. 3193–3200, 2012.
- T. Abdeljawad, D. Baleanu, and F. Jarad, “Existence and uniqueness theorem for a class of delay differential equations with left and right Caputo fractional derivatives,” Journal of Mathematical Physics, vol. 49, no. 8, article 083507, 11 pages, 2008.
- T. Abdeljawad, F. Jarad, and D. Baleanu, “On the existence and the uniqueness theorem for fractional differential equations with bounded delay within Caputo derivatives,” Science in China Series A, vol. 51, no. 10, pp. 1775–1786, 2008.