Abstract
Recently, the conformable derivative and its properties have been introduced. In this work we have investigated in more detail some new properties of this derivative and we have proved some useful related theorems. Also, some new definitions have been introduced.
References
[1] R. Khalil , M. Al Horani, A. Yousef and M. Sababheh “A new definition of fractional derivative” Journal of Computational and Applied Mathematics ,(2014) 65–70. 10.1016/j.cam.2014.01.002Search in Google Scholar
[2] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006. Search in Google Scholar
[3] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, NY, USA, 1993. Search in Google Scholar
[4] S. G. Samko, A. A. Kilbas, and O. I. Maritchev, Integrals and Derivatives of the Fractional Order and Some of Their Applications, in Russian, Nauka i Tekhnika, Minsk, Belarus, 1987. Search in Google Scholar
[5] M. Caputo, “Linear models of dissipation whose Q is almost frequency independent, part II,” Geophysical Journal International, vol. 13, no. 5, pp. 529–539, 1967. 10.1111/j.1365-246X.1967.tb02303.xSearch in Google Scholar
[6] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006. Search in Google Scholar
[7] Abdon Atangana and Aydin Secer, “A Note on Fractional Order Derivatives and Table of Fractional Derivatives of Some Special Functions,” Abstract and Applied Analysis, vol. 2013, Article ID 279681, 8 pages, 2013. doi:10.1155/2013/279681. 10.1155/2013/279681Search in Google Scholar
[8] I. Podlubny, “Geometric and physical interpretation of fractional integration and fractional differentiation,” Fractional Calculus and Applied Analysis, vol. 5, no. 4, pp. 367–386, 2002. Search in Google Scholar
[9] G. Jumarie, “Modified Riemann-Liouville derivative and fractional Taylor series of non-differentiable functions further results,” Computers & Mathematics with Applications, vol. 51, no. 9-10, pp. 1367–1376, 2006. Search in Google Scholar
[10] M. Davison and C. Essex, “Fractional differential equations and initial value problems,” The Mathematical Scientist, vol. 23, no. 2, pp. 108–116, 1998. Search in Google Scholar
[11] Abdon Atangana and Adem Kilicman, “Analytical Solutions of the Space-Time Fractional Derivative of Advection Dispersion Equation,” Mathematical Problems in Engineering, vol. 2013, Article ID 853127, 9 pages, 2013. doi:10.1155/2013/853127. 10.1155/2013/853127Search in Google Scholar
[12] Abdon Atangana and P. D. Vermeulen, “Analytical Solutions of a Space-Time Fractional Derivative of Groundwater Flow Equation,” Abstract and Applied Analysis, vol. 2014, Article ID 381753, 11 pages, 2014. doi:10.1155/2014/381753. 10.1155/2014/381753Search in Google Scholar
[13] Abdon Atangana, “On the Singular Perturbations for Fractional Differential Equation,” The Scientific World Journal, vol. 2014, Article ID 752371, 9 pages, 2014. doi:10.1155/2014/752371 10.1155/2014/752371Search in Google Scholar PubMed PubMed Central
[14] Abdon Atangana and Innocent Rusagara, “On the Agaciro Equation via the Scope of Green Function,” Mathematical Problems in Engineering, vol. 2014, Article ID 201796, 8 pages, 2014. doi:10.1155/2014/201796. 10.1155/2014/201796Search in Google Scholar
[15] Y. Luchko and R. Gorenflo, The Initial Value Problem for Some Fractional Differential Equations with the Caputo Derivative, Preprint Series A08-98, Fachbereich Mathematik und Informatik, Freic Universität, Berlin, Germany, 1998. Search in Google Scholar
[16] D. Baleanu, K. Diethelm, E. Scalas, J.J. Trujillo, Fractional calculus models and numerical methods, Series on Complexity, Nonlinearity and Chaos, World Scientific, Boston, 2012. 10.1142/8180Search in Google Scholar
[17] Thabet Abdeljawad. On the conformable fractional calculus.Journal of computational and Applied Mathematics, 279, pp. 57-66, 2015. 10.1016/j.cam.2014.10.016Search in Google Scholar
[18] U. N. Katugampola, New approach to a generalized fractional integral, Appl. Math. Comput., vol.218, no.3 pp. 860-865, 2011. 10.1016/j.amc.2011.03.062Search in Google Scholar
[19] D. R. Anderson and R. I. Avery, Fractional-order boundary value problem with Sturm-Liouville boundary conditions, Electronic Journal of Differential Equations, vol.2015 , no. 29, pp.1-10, 2015. Search in Google Scholar
[20] U. N. Katugampola, A new fractional derivative with classical properties, arXiv:1410.6535v2. Search in Google Scholar
[21] U. N. Katugampola, New approach to generalized fractional derivatives, B. Math. Anal. App., vol.6 no.4 pp. 1-15, 2014. Search in Google Scholar
[22] Batarfi, H., Losada, J., Nieto, J.J., Shammakh, W., Three-point boundary value problems for conformable fractional differential equations. Journal of Function Spaces 2015 (2015), 706383, 6 pages 10.1155/2015/706383Search in Google Scholar
[23] Benkhettou, N., Hassani, S., Torres, D.F.M., A conformable fractional calculus on arbitrary time scales,Journal of King Saud University - Science, In Press, doi:10.1016/j.jksus.2015.05.003 10.1016/j.jksus.2015.05.003Search in Google Scholar
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