A naturally emerging bivariate Mittag-Leffler function and associated fractional-calculus operators
- PDF / 1,075,163 Bytes
- 27 Pages / 439.37 x 666.142 pts Page_size
- 5 Downloads / 174 Views
A naturally emerging bivariate Mittag-Leffler function and associated fractional-calculus operators Arran Fernandez1
· Cemaliye Kürt1 · Mehmet Ali Özarslan1
Received: 24 November 2019 / Revised: 4 June 2020 / Accepted: 10 June 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract We define an analogue of the classical Mittag-Leffler function which is applied to two variables, and establish its basic properties. Using a corresponding single-variable function with fractional powers, we define an associated fractional integral operator which has many interesting properties. The motivation for these definitions is twofold: firstly, their link with some fundamental fractional differential equations involving two independent fractional orders, and secondly, the fact that they emerge naturally from certain applications in bioengineering. Keywords Mittag-Leffler functions · Fractional integrals · Fractional derivatives · Fractional differential equations · Bivariate Mittag-Leffler functions
1 Introduction The classical Mittag-Leffler function, defined as E α (x) =
∞ n=0
xn , Γ (nα + 1)
Re(α) > 0,
(1)
was proposed (Mittag-Leffler 1903) by the Swedish mathematician Gösta Mittag-Leffler in 1903. It has been extended and generalised in various ways (Gorenflo et al. 2016; Kilbas et al. ρ 2006), with functions denoted by E α,β (x) and E α,β (x) and E α,β,γ (x), the “two-parameter” and “three-parameter” Mittag-Leffler functions, being defined by power series similar to the
Communicated by Roberto Garrappa.
B
Arran Fernandez [email protected] Cemaliye Kürt [email protected] Mehmet Ali Özarslan [email protected]
1
Department of Mathematics, Faculty of Arts and Sciences, Eastern Mediterranean University, via Mersin 10, Famagusta, Northern Cyprus, Turkey 0123456789().: V,-vol
123
200
Page 2 of 27
A. Fernandez et al.
one for E α (x) with modified coefficients to take account of the extra parameters: E α,β (x) = ρ
E α,β (x) =
∞ n=0 ∞ n=0
xn , Γ (nα + β) (ρ)n x n , n!Γ (nα + β)
Re(α) > 0; Re(α) > 0.
Recently, a different type of generalisation has been proposed: the so-called “bivariate” and “multivariate” Mittag-Leffler functions, which are defined not by a power series in a single variable x, but by double power series in two variables x and y, or even multiple power series in an arbitrary number of variables. Multivariate Mittag-Leffler functions were defined in Luchko (1999), Saxena et al. (2011) and further studied in for example (Luchko and Gorenflo 1999; Andualem et al. 2019; Suthar et al. 2019), and various types of bivariate Mittag-Leffler functions have been defined in for example (Garg et al. 2013; Lavault 2018; Özarslan and Kürt 2019). It is interesting to note that more than one type of bivariate function is emerging as an equivalent of the Mittag-Leffler function: so far, we have seen E
∞ ∞ (δ1 )κ1 m (δ2 )κ2 n xm yn δ 1 , κ1 ; δ 2 , κ2 , x, y = γ1 , α1 , β1 ; γ2 , α2 ; γ3 , β2 Γ (α1 m + β1 n + γ1 ) Γ (α2 m + γ2 ) Γ (
Data Loading...
