A Discretization of the Hadamard fractional derivative
We present a new discretization for the Hadamard fractional derivative, that simplifiesthe computations. We then apply the method to solve a fractional differential equationand a fractional variational problem with dependence on the Hadamard fractionalderivative.
___
- [1] Butzer, P.L., Kilbas, A. A. and Trujillo, J. J., Mellin transform analysis and integration by parts
for Hadamard-type fractional integrals, J. Math. Anal. Appl. 270 (2002), no. 1, 1-15.
- [2] Butzer, P.L., Kilbas, A. A. and Trujillo, J. J., Fractional calculus in the Mellin setting and
Hadamard-type fractional integrals, J. Math. Anal. Appl. 269 (2002), no. 1, 1-27.
- [3] Butzer, P.L., Kilbas, A. A. and Trujillo, J. J., Stirling functions of the second kind in the setting
of difference and fractional calculus, Numer. Funct. Anal. Optim. 24 (2003), no. 7-8, 673-711.
- [4] Hadamard, J., Essai sur l’etude des fonctions donnees par leur developpment de Taylor, J.
Pure Appl. Math. 4 (1892), no. 8, 101-186.
- [5] Jarad, F., Abdeljawad, T. and Baleanu, D., Caputo-type modification of the Hadamard fractional
derivatives, Advances in Difference Equations August 2012 (2012), 2012–142.
- [6] Kilbas, A.A., Hadamard-type fractional calculus, J. Korean Math. Soc. 38 (2001), no. 6, 1191-
1204.
- [7] Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J., Theory and applications of fractional differential
equations. North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam,
2006.
- [8] Kilbas, A.A. and Titioura, A.A., Nonlinear differential equations with Marchaud-Hadamardtype
fractional derivative in the weighted space of summable functions, Math. Model. Anal. 12
(2007), no. 3, 343-356.
- [9] Pooseh, S., Almeida, R. and Torres, D. F. M., Expansion formulas in terms of integer-order
derivatives for the Hadamard fractional integral and derivative, Numer. Funct. Anal. Optim.
33 (2012), no. 3, 301–319.
- [10] Qassim, M. D., Furati, K. M. and Tatar, N.E., On a Differential Equation Involving Hilfer-
Hadamard Fractional Derivative, Abstract and Applied Analysis vol. 2012 (2012), Article ID
391062, 17 pages, doi:10.1155/2012/391062.
- [11] Qian, D., Gong, Z. and Li, C., A generalized Gronwall inequality and its application to
fractional differential equations with Hadamard derivatives, 3rd Conference on Nonlinear
Science and Complexity (NSC10), Cankaya University, Ankara, Turkey, 28–31 July, 2010.