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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 546062, 11 pages
A Generalized q-Mittag-Leffler Function by q-Captuo Fractional Linear Equations
Department of Mathematics and Computer Science, Çankaya University, 06530 Ankara, Turkey
Received 23 January 2012; Accepted 20 February 2012
Academic Editor: Juan J. Trujillo
Copyright © 2012 Thabet Abdeljawad et al. 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, Academic Press, San Diego, Calif, USA, 1999.
- S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science, Yverdon, Switzerland, 1993.
- A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204, North-Holland Mathematics Studies, 2006.
- R. L. Magin, Fractional Calculus in Bioengineering, Begell House, Connecticut, Conn, USA, 2006.
- I. J. Jesus and J. A. Tenreiro Machado, “Application of integer and fractional models in electrochemical systems Math,” Mathematical Problems in Engineering, vol. 2012, Article ID 248175, 17 pages, 2012.
- D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus Models and Numerical Methods, World Scientific, New York, NY, USA, 2012.
- T. Ernst, “The history of q-calculus and new method (Licentiate Thesis),” U.U.D.M. Report 2000, http://www.math.uu.se/thomas/Lics.pdf.
- F. Chen, X. Luo, and Y. Zhou, “Existence results for nonlinear fractional difference equation,” Advances in Difference Equations, vol. 2011, Article ID 713201, 12 pages, 2011.
- 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.
- K. A. Aldwoah and A. E. Hamza, “Difference time scales,” International Journal of Mathematics and Statistics, vol. 9, no. S11, pp. 106–125, 2011.
- A. M. C. Brito da Cruz, N. Martins, and D. F. M. Torres, “Higher-order Hahn’s quantum variational calculus,” Nonlinear Analysis, vol. 75, no. 3, pp. 1147–1157, 2012.
- A. B. Malinowska and D. F. M. Torres, “The Hahn quantum variational calculus,” Journal of Optimization Theory and Applications, vol. 147, no. 3, pp. 419–442, 2010.
- N. R. O. Bastos, R. A. C. Ferreira, and D. F. M. Torres, “Necessary optimality conditions for fractional difference problems of the calculus of variations,” Discrete and Continuous Dynamical Systems A, vol. 29, no. 2, pp. 417–437, 2011.
- H. Koçak, T. Öziş, and A. Yıldırım, “Homotopy perturbation method for the nonlinear dispersive K(m,n,1) equations with fractional time derivatives,” International Journal of Numerical Methods for Heat & Fluid Flow, vol. 20, no. 2, pp. 174–185, 2010.
- A. Yildirim and S. A. Sezer, “Analytical solution of linear and non-linear space-time fractional reaction-diffusion equations,” International Journal of Chemical Reactor Engineering, vol. 8, article A110, 2010.
- A. Yildirim and S. T. Mohyud-Din, “Analytical approach to space- and time-fractional burgers equations,” Chinese Physics Letters, vol. 27, no. 9, Article ID 090501, 2010.
- M. A. Balci and A. Yildirim, “Analysis of fractional nonlinear differential equations using the homotopy perturbation method, Zeitschrift fr Naturforschung A,” A Journal of Physical Sciences, vol. 66, no. 2, pp. 87–92, 2011.
- A. Inan and Y. Ahmet, “Ahmet Application of variational iteration method to fractional initial-value problems,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 7, pp. 877–883, 2009.
- W. A. Al-Salam, “q-analogues of Cauchy's formulas,” Proceedings of the American Mathematical Society, vol. 17, pp. 182–184, 1952.
- W. A. Al-Salam, “Some fractional q-integrals and q-derivatives,” Proceedings of the Edinburgh Mathematical Society, vol. 15, pp. 135–140, 1966.
- W. A. Al-Salam and A. Verma, “A fractional Leibniz q-formula,” Pacific Journal of Mathematics, vol. 60, no. 2, pp. 1–9, 1975.
- R. P. Agarwal, “Certain fractional q-integrals and q-derivatives,” vol. 66, pp. 365–370, 1969.
- M. R. Predrag, D. M. Sladana, and S. S. Miomir, “Fractional integrals and derivatives in q-calculus,” Applicable Analysis and Discrete Mathematics, vol. 1, no. 1, pp. 311–323, 2007.
- M. Bohner and A. Peterson, Dynamic Equations on Time Scales, Birkhäuser, Boston, Mass, USA, 2001.
- T. Abdeljawad and B. Dumitru, “Fractional differences and integration by parts,” Journal of Computational Analysis and Applications, vol. 13, no. 3, pp. 574–582, 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. Atici and P. W. Eloe, “Fractional q-calculus on a time scale,” Journal of Nonlinear Mathematical Physics, vol. 14, no. 3, pp. 333–344, 2007.
- 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, 1989.
- 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.
- A. A. Kilbas and M. Saigo, “On solution of integral equation of Abel-Volterra type,” Differential and Integral Equations, vol. 8, no. 5, pp. 993–1011, 1995.
- T. Abdeljawad and B. Dumitru, “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.
- J. Čermák and L. Nechvátal, “On (q,h)-analogue of fractional calculus,” Journal of Nonlinear Mathematical Physics, vol. 17, no. 1, pp. 51–68, 2010.
- T. Abdeljawad, F. Jarad, and D. Baleanu, “Variational optimal-control problems with delayed arguments on time scales,” Advances in Difference Equations, vol. 2009, Article ID 840386, 15 pages, 2009.
- T. Abdeljawad, “A note on the chain rule on time scales,” Journal of Science and Arts, vol. 9, pp. 1–6, 2008.