Table of Contents Author Guidelines Submit a Manuscript
Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 406910, 12 pages
http://dx.doi.org/10.1155/2013/406910
Research Article

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.

Citations to this Article [22 citations]

The following is the list of published articles that have cited the current article.

  • Dorota Mozyrska, and Ewa Girejko, “On Solutions to Fractional Discrete Systems with Sequential h-Differences,” Abstract and Applied Analysis, 2013. View at Publisher · View at Google Scholar
  • Ioannis K. Dassios, and Dumitru I. Baleanu, “Duality of singular linear systems of fractional nabla difference equations,” Applied Mathematical Modelling, 2014. View at Publisher · View at Google Scholar
  • Mohamad Rafi Segi Rahmat, “Integral Transform Methods for Solving Fractional Dynamic Equations on Time Scales,” Abstract and Applied Analysis, vol. 2014, pp. 1–10, 2014. View at Publisher · View at Google Scholar
  • Weidong Lv, “Solvability for a Discrete Fractional Three-Point Boundary Value Problem at Resonance,” Abstract and Applied Analysis, vol. 2014, pp. 1–7, 2014. View at Publisher · View at Google Scholar
  • Guo-Cheng Wu, Dumitru Baleanu, Zhen-Guo Deng, and Sheng-Da Zeng, “Lattice fractional diffusion equation in terms of a Riesz–Caputo difference,” Physica A: Statistical Mechanics and its Applications, 2015. View at Publisher · View at Google Scholar
  • Guo-Cheng Wu, Dumitru Baleanu, He-Ping Xie, and Sheng-Da Zeng, “Lattice fractional diffusion equation of random order,” Mathematical Methods in the Applied Sciences, 2015. View at Publisher · View at Google Scholar
  • Weidong Lv, “Existence and Uniqueness of Solutions for a Discrete Fractional Mixed Type Sum-Difference Equation Boundary Value Problem,” Discrete Dynamics in Nature and Society, vol. 2015, pp. 1–10, 2015. View at Publisher · View at Google Scholar
  • Saowaluk Chasreechai, Chanakarn Kiataramkul, and Thanin Sitthiwirattham, “On Nonlinear Fractional Sum-Difference Equations via Fractional Sum Boundary Conditions Involving Different Orders,” Mathematical Problems in Engineering, vol. 2015, pp. 1–8, 2015. View at Publisher · View at Google Scholar
  • Thabet Abdeljawad, and Dumitru Baleanu, “Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels,” Advances in Difference Equations, vol. 2016, no. 1, 2016. View at Publisher · View at Google Scholar
  • Thabet Abdeljawad, and Delfim F.M. Torres, “Symmetric duality for left and right Riemann–Liouville and Caputo fractional differences,” Arab Journal of Mathematical Sciences, 2016. View at Publisher · View at Google Scholar
  • Jiraporn Reunsumrit, and Thanin Sitthiwirattham, “On positive solutions to fractional sum boundary value problems for nonlinear fractional difference equations,” Mathematical Methods in the Applied Sciences, vol. 39, no. 10, pp. 2737–2751, 2016. View at Publisher · View at Google Scholar
  • Wei Nian Li, and Weihong Sheng, “Sufficient conditions for oscillation of a nonlinear fractional nabla difference system,” SpringerPlus, vol. 5, no. 1, 2016. View at Publisher · View at Google Scholar
  • Lynn Erbe, Christopher S Goodrich, Baoguo Jia, and Allan Peterson, “Survey of the qualitative properties of fractional difference operators: monotonicity, convexity, and asymptotic behavior of solutions,” Advances in Difference Equations, vol. 2016, no. 1, 2016. View at Publisher · View at Google Scholar
  • Weihong Sheng, and Wei Nian Li, “Forced oscillation for solutions of boundary value problems of fractional partial difference equations,” Advances in Difference Equations, vol. 2016, no. 1, 2016. View at Publisher · View at Google Scholar
  • Dorota Mozyrska, and Małgorzata Wyrwas, “Stability by linear approximation and the relation between the stability of difference and differential fractional systems,” Mathematical Methods in the Applied Sciences, 2016. View at Publisher · View at Google Scholar
  • M. Ganji, and F. Gharari, “Bayesian Estimation in Delta and Nabla Discrete Fractional Weibull Distributions,” Journal of Probability and Statistics, vol. 2016, pp. 1–8, 2016. View at Publisher · View at Google Scholar
  • Christopher S. Goodrich, “The relationship between sequential fractional differences and convexity,” Applicable Analysis and Discrete Mathematics, vol. 10, no. 2, pp. 345–365, 2016. View at Publisher · View at Google Scholar
  • Christopher S. Goodrich, “Monotonicity and non-monotonicity results for sequential fractional delta differences of mixed order,” Positivity, 2017. View at Publisher · View at Google Scholar
  • M. Ganji, and F. Gharari, “The Discrete Delta and Nabla Mittag-Leffler Distributions,” Communications in Statistics - Theory and Methods, pp. 0–0, 2017. View at Publisher · View at Google Scholar
  • Rajendra Dahal, and Christopher S. Goodrich, “An almost sharp monotonicity result for discrete sequential fractional delta differences,” Journal of Difference Equations and Applications, pp. 1–14, 2017. View at Publisher · View at Google Scholar
  • Thabet Abdeljawad, and Qasem M. Al-Mdallal, “Discrete Mittag-Leffler kernel type fractional difference initial value problems and Gronwall’s inequality,” Journal of Computational and Applied Mathematics, 2017. View at Publisher · View at Google Scholar
  • Thabet Abdeljawad, Qasem M. Al-Mdallal, and Mohamed A. Hajji, “Arbitrary Order Fractional Difference Operators with Discrete Exponential Kernels and Applications,” Discrete Dynamics in Nature and Society, vol. 2017, pp. 1–8, 2017. View at Publisher · View at Google Scholar