Available online at www.isr-publications.com/jnsa J. Nonlinear Sci. Appl., 10 (2017), 2914–2924 Research Article
Journal Homepage: www.tjnsa.com - www.isr-publications.com/jnsa
Extension of the fractional derivative operator of the Riemann-Liouville Dumitru Baleanua,b,∗, Praveen Agarwalc,d , Rakesh K. Parmare , Maysaa M. Alqurashif , Soheil Salahshourg a Department b Institute
of Mathematics, Cankaya University, Ankara, Turkey.
of Space Sciences, Magurele-Bucharest, Romania.
c Department
of Mathematics, Anand International College of Engineering, Jaipur-303012, Republic of India.
d Department
of Mathematics, University Putra Malaysia, 43400 UPM, Serdang, Selangor, Malaysia.
e Department
of Mathematics, Govt. College of Engineering and Technology, Bikaner-334004, Rajasthan, India.
f Department
of Mathematics, King Saud University, P. O. Box 22452, Riyadh 11495, Saudi Arabia.
g Department
of Computer Engineering, Mashhad Branch, IAU, Iran.
Communicated by A. Atangana
Abstract By using the generalized beta function, we extend the fractional derivative operator of the Riemann-Liouville and discusses its properties. Moreover, we establish some relations to extended special functions of two and three variables via generating c functions. 2017 All rights reserved. Keywords: Hypergeometric function of two and three variables, fractional derivative operator, generating functions, Mellin transform. 2010 MSC: 33C05, 33C15.
1. Introduction and preliminaries The concept of extension of fractional operators attracts the attention of many researchers (see, [1]). The present investigation was motivated by above mentioned works. We start from the Riemann–Liouville (RL) fractional derivative operator Dµ z (the reader may check [16] and [4, p. 70 et seq.]): Z z 1 (z − t)−µ−1 f (t) dt, < (µ) < 0 , Γ (−µ) 0 Dµ (1.1) m
z {f (z)} := µ−m d {f (z)} (µ) (m D , m − 1 5 < < m ∈ N) . z dzm We recall that
∗ Corresponding
λ Dµ = z z
Γ (λ + 1) zλ−µ , Γ (λ − µ + 1)
< (λ) > −1 ,
author Email addresses:
[email protected] (Dumitru Baleanu),
[email protected] (Praveen Agarwal), (Rakesh K. Parmar),
[email protected] (Maysaa M. Alqurashi),
[email protected] (Soheil Salahshour) doi:10.22436/jnsa.010.06.06 Received 2016-05-02
(1.2)
D. Baleanu, et al., J. Nonlinear Sci. Appl., 10 (2017), 2914–2924
2915
where the path of integration in (1.1) denotes a line in the complex t-plane starting from t = 0 and ending at t = z. Various extensions and generalizations of the RL fractional derivative operator were reported by various researchers (see, e.g., [4, 7, 8, 13, 16, 20]). In particular, in [10] the authors introduced the extended Riemann–Liouville fractional derivative operator Dµ,p z , namely: Zz 1 pz2 −µ−1 (z − t) f (t) dt, < (µ) < 0 , exp − Γ (−µ) 0 (z − t)t Dµ,p (1.3) z {f (z)} = m
d µ−m {f (z)} , m − 1 5 < (µ) < m (m ∈ N) . Dz dzm Clearly, the special case of (1.3), when p = 0, reduces immediately to RL fractional derivative (see, [18, 19]). Making use of the definition (1.3), they derived the following extension of the fractional derivative (1.2) λ Bp (λ + 1, −µ; p) λ−µ Dµ,p z , z = z Γ (−µ)
−1;
