Basic Theory for Differential Equations with Unified Reimann-Liouville and Hadamard Type Fractional Derivatives

Karpuz, Basak; Ozkan, Umut M.; Yalcin, Tugba; Yildiz, Mustafa K.

Research Article

Basic Theory for Differential Equations with Unified Reimann-Liouville and Hadamard Type Fractional Derivatives

Basak Karpuz¹*
Umut M. Ozkan²
Tugba Yalcin²
Mustafa K. Yildiz²

Dokuz Eylul University
Afyon Kocatepe University

* Corresponding author

Abstract

In this paper, we extend the definition of the fractional integral and derivative introduced in [Appl. Math. Comput. 218 (2011)] by Katugampola, which exhibits nice properties only for numbers whose real parts lie in [0,1]. We prove some interesting properties of the fractional integrals and derivatives. Based on these properties, the following concepts for the new type fractional differential equations are explored: Existence and uniqueness of solutions; Solutions of autonomous fractional differential equations; Dependence on the initial conditions; Green's function; Variation of parameters formula.

How to Cite

B. Karpuz, U. M. Ozkan, T. Yalcin, M. K. Yildiz, Basic Theory for Differential Equations with Unified Reimann-Liouville and Hadamard Type Fractional Derivatives, Int. J. Anal. Appl. 13 (2017), 216–230.

BibTeX

@article{Karpuz2017,
  author  = {Karpuz, Basak and Ozkan, Umut M. and Yalcin, Tugba and Yildiz, Mustafa K.},
  title   = {Basic Theory for Differential Equations with Unified Reimann-Liouville and Hadamard Type Fractional Derivatives},
  journal = {Int. J. Anal. Appl.},
  volume  = {13},
  year    = {2017},
  pages   = {216--230},
  doi     = {}
}

References

[1] P.L. Butzer, A.A. Kilbas and J.J. Trujillo, Compositions of Hadamard-type fractional integration operators and the semigroup property, J. Math. Anal. Appl. 269 (2002), no. 2, 387-400.
[2] P.L. Butzer, A.A. Kilbas and J.J. Trujillo, Fractional calculus in the Mellin setting and Hadamard-type fractional integrals, J. Math. Anal. Appl. 269 (2002), no. 1, 1-27.
[3] K. Diethelm, The Analysis of Fractional Differential Equations, Springer-Verlag, Berlin, 2010.
[4] J. Hadamard, Essai sur l ´etude des fonctions donnees par leur developpement de Taylor, Journal de Mathematiques Pures et Appliquees 4 (1892), no. 8, 101-186.
[5] U.N. Katugampola, New approach to a generalized fractional integral, Appl. Math. Comput. 218 (2011), no. 3, 860-865.
[6] U.N. Katugampola, A new approach to generalized fractional derivatives, Bull. Math. Anal. Appl. 4 (2014), no. 6, 1-15.
[7] A.A. Kilbas, Hadamard-type fractional calculus, J. Korean Math. Soc. 38 (2001), no. 6, 1191-1204.
[8] A.A. Kilbas and J.J. Trujillo, Hadamard-type integrals as G-transforms, Integral Transforms Spec. Funct. 14 (2003), no. 5, 413-427.
[9] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations Elsevier Science B.V., Amsterdam, 2006.
[10] K.B. Oldham and J. Spanier, The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order, Academic Press, New York, 1974.
[11] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
[12] S. Pooseh, P. Almeida and D.F.M. Torres, 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.
[13] S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.