A Discretization of the Hadamard fractional derivative

Year 2016, Volume: 4 Issue: 1, 31 - 39, 15.04.2016

Ricardo Almeida Nuno R. O. Bastos

https://doi.org/10.36753/mathenot.421356

Cited By: 4

Abstract

We present a new discretization for the Hadamard fractional derivative, that simplifies

the computations. We then apply the method to solve a fractional differential equation

and a fractional variational problem with dependence on the Hadamard fractional

derivative.

Keywords

fractional calculus, discretization methods

References

• [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.