Note on fractional Mellin transform and applications

In this article, we define the fractional Mellin transform by using Riemann–Liouville fractional integral operator and Caputo fractional derivative of order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \ge 0$$\end{document}α≥0 and study some of their properties. Further, some properties are extended to fractional way for Mellin transform.


Basic definitions of fractional calculus
Fractional calculus is a generalization of the classical calculation and it has been used successfully in various fields of science and engineering. In fact, there are new opportunities in mathematics and theoretical physics appear, when order differential operator or operator becomes an integral arbitrary parameter. The fractional calculus is a powerful tool for the physical description systems that have long-term memory and long term spatial interactions see Podlubny (1999), Miller and Ross (1993), Hilfer (2000), Kilbas et al. (2006) and Samko et al. (1993).
There are different types of fractional derivatives in the current literature. One of the new fractional derivatives that was recently proposed is called Caputo-Fabrizio derivative see Atangana (2016), Caputo and Fabrizio (2015) and Losada and Nieto (2015). However in our study, Riemann-Liouville and Caputo derivatives have been used.
The use of integral transforms to deal with fractional derivatives traces back to Riemann and Liouville (Oldham and Spanier 1974;Widder 1971). Further, in Dattoli et al. (2003) the authors have shown that combined use of integral transforms and special polynomials provides a powerful tool to deal with fractional derivatives and integrals.
In this section, we give the definitions of Riemann-Liouville and Caputo fractional operators along the main properties as follows: Definition 3 The Riemann-Liouville fractional derivative operator of order α of a function f(x) is defined as: Definition 4 (Podlubny 1999) The Riemann-Liouville fractional integral operator of order α ≥ 0 of a function f ∈ C µ , µ ≥ −1 is defined as:

Some properties of Riemann-Liouville fractional operator
If α, β are two positive real number, then: Definition 5 (Caputo 1969 Definition 6 (Fractional Cauchy's integral formula) (Jumarie 2010) Assume that f : U → C, z → f (z) is a fractional analytic function of order α = 1 N , N ≥ 1, N integer. For every a ∈ U consider the disk D ⊂ U with the boundary defined by the circle γ of which the radius is r. Then f (z) is actually infinitely αth differentiable, with As a special case when n = 1, the fractional derivative can be written in the form:

Main results
In this part, some properties of Mellin transform of fractional operator have shown.

Proof
(1) The result is obtained by applying Mellin transform to both sides of the first property (1) in Theorem 3 (2) We apply Mellin transform on the part (2) (2) M f 3 2 (x); s = ∞ 0 x s−1 f 3 2 (x)dx, by using fractional integration by parts and fractional derivative of power function, we obtain Continuing by the induction, then the results in Theorem 5 can be extended to further fractional derivatives as the following theorem: Theorem 6 Let f be Mellin transformable function on R + , and f is a fractional derivative function for all n − 1 < α < n, n ∈ N, then: Remark 4 By using the same technique in above theorem, Mellin transform of fractional integral can be yielded as the following formula: Theorem 7 Let f be Mellin transformable defined on R + , then By the same way as in Theorem 5, the next result follows: Example 1 Solve the problem: By applying the Mellin transform to both side and on using the Theorem 7 we have By solving the equation and applying the inverse Mellin transform by using complex inversion integral in order to cover the f(x) explicitly as the solution Theorem 8 Let f ∈ X (a,b) and holomorphic on the strip St(a, b). In addition f is Mellin transformable function, then where s ∈ St(a, b), and 0 ≤ α ≤ 1.
First of all, for u > 0, let us consider Secondly, we apply fractional Cauchy's integral formula when n = 1 for fractional derivatives By another application of fractional Cauchy's integral formula, we obtain Therefore, the proof of Theorem 8 is fulfilled.
Example 2 Let f (x) = e −x we apply Theorem 8 then we have So,