Essay on Fractional Riemann-Liouville Integral Operator versus Mikusinski’s

This paper presents the representation of the fractional Riemann-Liouville integral by using the Mikusinski operators. The Mikusinski operators discussed in the paper may yet provide a new view to describe and study the fractional Riemann-Liouville integral operator.Thepresent resultmay be useful for applying theMikusinski operational calculus to the study of fractional calculus in mathematics and to the theory of filters of fractional order in engineering.

The desired solution to Problem 433 is the solution to the differential equation of order 1/2 expressed by The above needs the differential of order 1/2.They gave the following solution to (2) based on the fractional Riemann-Liouville integral [24]: where  is a constant.This short paper aims at exhibiting that 0  −V  is equivalent to the Mikusinski operator  V .The significance of our analysis is as follows.Since the algebra properties of the Mikusinski operators are satisfactorily studied and well known, see, for example, [25][26][27][28], one may immediately infer that the algebra properties of 0  −V  are consistent with the Mikusinski operators.Moreover, the present result suggests that the Mikusinski operators may be used for studying differential equations or filters of fractional order in signal processing.
The remainder of this paper is organized as follows.We will derive (1) from the point of view of the Mikusinski operators in Section 2. Discussions are given in Section 3, which is followed by conclusions.

Derivation
In this section, we will first brief the Mikusinski operators.Then, the derivation of ( 1) is given based on the Mikusinski operators.

Mathematical Problems in Engineering
The Mikusinski operators are described by convolution [28][29][30].Let () and () belong to (0, ∞).Following the usage of Mikusinski's, we rewrite () and () by The convolution described by Mikusinski is then given by The deconvolution, therefore, is expressed by Define  = {1} such that The representations ( 5) and ( 6) may be convenient to study the operations of the convolution and its inverse from a view of algebra.For instance, that (0, ∞) is a commutative ring is obvious.
When one exerts   on () ∈ (0, ∞), that is,   {()}, the following Cauchy formula results: Considering the generalization of   in (9) for V > 0 yields another Mikusinski operator given by Further, by taking into account  V {()}, we have Releasing the usage of Mikusinski in {⋅} for the purpose of his operational calculus, we have This completes the derivation because ( 13) is the definition of the fractional Riemann-Liouville integral of order V.

Discussions
From ( 1) and ( 13), one sees that the fractional Riemann-Liouville integral operator of order V, that is, 0  −V  , is equivalent to the Mikusinski operator  V though the originality of Mikusinski's by introducing  V may be for the purpose of his theory of operational calculus.
On the one hand, we recall that the Mikusinski operational calculus is a useful tool for studying differential equations.On the other hand,  V may yet be an alternative of 0  −V  , so that the Mikusinski operational calculus may be expanded into the field of differential equations and signal processing of fractional order, which attracts interest in biomedical engineering; see, for example, [31].
We note that () ∈ (0, ∞) is not necessary in (1).In fact, (1) exists if() is a generalized function [32].In addition, () may be a random function such as the Brownian motion [33].Due to the consistence of 0  −V  with  V , one sees that a generalized function() in ( 13) may also be allowed.Finally, we should remember that the fractional Riemann-Liouville integral operator may be extended to its corresponding, more precisely, the fractional Riemann-Liouville differential operator if V < 0. This may correspond to the deconvolution in the Mikusinsiki's operational calculus, which we will work on in the future.Finally, we mention that possible applications of the Mikusinsiki's operational calculus to other issues, for example, in [34][35][36], may be interesting.

Conclusions
We have exhibited that the fractional Riemann-Liouville integral operator may be expressed by using the Mikusinsiki operators, giving another outlook of the fractional Riemann-Liouville integral operator.Thus, we have noticed that the Mikusinski operational calculus may yet be a tool for studying differential equations or systems of fractional order.