Integral Equations Involving Generalized Mittag-Leffler Function
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INTEGRAL EQUATIONS INVOLVING GENERALIZED MITTAG-LEFFLER FUNCTION R. Desai,1 I. A. Salehbhai,2 and A. K. Shukla3,4
UDC 517.5
γ,q (z) We deal with the solution of the integral equation with generalized Mittag-Leffler function E↵,β specifying the kernel with the help of a fractional integral operator. The existence of the solution is justified and necessary conditions for the integral equation admitting a solution are discussed. In addition, the solution of the integral equation is obtained.
1. Introduction and Preliminaries The Mittag-Leffler function was defined by G. Mittag-Leffler [5], by the following formula: E↵ (z) =
1 X
n=0
zn , Γ(↵n + 1)
(1)
where z is a complex variable and ↵ ≥ 0. This function was obtained as a solution of some fractional-order differential equations. The Mittag-Leffler function is a direct generalization of the exponential function, hyperbolic functions, and trigonometric functions, namely, E1 (z) = ez ,
E2 (z 2 ) = cosh z,
and
E2 (−z 2 ) = cos z.
For 0 < ↵ < 1 and |z| < 1, it interpolates between the exponential function ez and the geometric function 1
X 1 = zk . 1−z k=0
Due to its vast involvement in the fields of physics, engineering and applied sciences, numerous authors defined and studied various generalizations of the Mittag-Leffler function, namely, E↵,β (z) was introduced by Wiman [16], γ γ,q (z) was suggested by Prabhakar [6], E↵,β (z) was defined and studied by Shukla and Prajapati [9], etc. E↵,β γ,q (z) for ↵, β, γ 2 C, Re(↵), Re(β), Re(γ) > 0, Shukla and Prajapati [9] defined a generalization of (1) as E↵,β and q 2 (0, 1) [ N in the form γ,q E↵,β (z)
=
1 X
n=0
zn , Γ(↵n + β) n! (γ)qn
1
(2)
K. J. Somaiya College Engineering, University of Mumbai, Mumbai, India; S. V. National Institute of Technology, Surat, India; e-mail: [email protected]. 2 Government Engineering College, Bharuch, India; e-mail: [email protected]. 3 S. V. National Institute of Technology, Surat, India; e-mail: [email protected]. 4 Corresponding author. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 5, pp. 620–627, May, 2020. Original article submitted April 1, 2017. 712
0041-5995/20/7205–0712
© 2020
Springer Science+Business Media, LLC
I NTEGRAL E QUATIONS I NVOLVING G ENERALIZED M ITTAG -L EFFLER F UNCTION
713
where (γ)qn =
Γ(γ + qn) Γ(γ)
is the generalized Pochhammer symbol, which in particular, reduces to q qn
◆ q ✓ Y γ+r−1
r=1
q
n
1,1 for q 2 N. Moreover, it is a generalization of the exponential function as E1,1 (z) = exp(z). In addition, 1,1 (z) = E↵ (z), E↵,1
1,1 E↵,β (z) = E↵,β (z),
and
γ,1 γ E↵,β (z) = E↵,β (z).
Further, Shukla and Prajapati [10] studied some properties of the generalized Mittag-Leffler-type function and the generated integral operator γ,q f (x) E↵,β,w;a+
=
Zx a
γ,q � (x − t)β−1 E↵,β w(x − t)↵ f (t) dt
(3)
for ↵, β, γ, w 2 C, Re(↵), Re(β), Re(γ) > 0, and q 2 (0, 1) [ N. Fractional integral operators play an important role in the solution of several problems in science and engineering. Many fractional
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