Extension of the fractional derivative operator of the Riemann-Liouville

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Abstract. By using the generalized beta function, we extend the fractional derivative operator of the Riemann-Liouville and discusses its properties. Moreover, we ...
Available online at www.isr-publications.com/jnsa J. Nonlinear Sci. Appl., 10 (2017), 2914–2924 Research Article

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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;