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

Abstract

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. The present result may be useful for applying the Mikusinski operational calculus to the study of fractional calculus in mathematics and to the theory of filters of fractional order in engineering.

1. Introduction

Fractional calculus gains increase interests in processing biomedical signals; see, for example, [1–16]. The fractional integral of the Riemann-Liouville type is widely used in the field; see, for example, [17–22].

Denote by 𝐶(0,∞) the set of piecewise continuous functions on (0, ∞). Let 𝑣>0 and 𝑓(𝑡)∈𝐶(0,∞). Assume that 𝑓(𝑡) is integrable on any finite subinterval of [0, ∞). For 𝑡>0, denote by ₀𝐷_𝑡^−𝑣 the fractional Riemann-Liouville integral operator of order 𝑣 [19]. Then, the fractional Riemann-Liouville integral of order 𝑣 of 𝑓(𝑡) is given by ₀𝐷_𝑡^−𝑣𝑓(𝑡)=1Γ(𝑣)^𝑡₀(𝑡−𝑢)^𝑣−1𝑓(𝑢)𝑑𝑢, (1) where Γ(⋅) is the gamma function. As early as 1919, O’Shaughnessy and Post studied the problem indexed by 433 [23]. The desired solution to Problem 433 is the solution to the differential equation of order 1/2 expressed by 𝑑^1/2𝑓(𝑡)𝑑𝑡^1/2−𝑓(𝑡)𝑡=0. (2) The above needs the differential of order 1/2. They gave the following solution to (2) based on the fractional Riemann-Liouville integral [24]: 𝑓(𝑡)=𝐶𝑡^−1/2−1exp𝑡, (3) where 𝐶 is a constant.

This short paper aims at exhibiting that ₀𝐷_𝑡^−𝑣 is equivalent to the Mikusinski operator 𝑙^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, [25–28], one may immediately infer that the algebra properties of ₀𝐷_𝑡^−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.

2. Derivation

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

The Mikusinski operators are described by convolution [28–30]. Let 𝑎(𝑡) and 𝑏(𝑡) belong to 𝐶(0,∞). Following the usage of Mikusinski’s, we rewrite 𝑎(𝑡) and 𝑏(𝑡) by 𝑎={𝑎(𝑡)},𝑏={𝑏(𝑡)}. (4) The convolution described by Mikusinski is then given by 𝑎𝑏={𝑎(𝑡)}{𝑏(𝑡)}=^𝑡₀𝑎(𝑡−𝜏)𝑏(𝜏)𝑑𝜏. (5) The deconvolution, therefore, is expressed by 𝑎𝑏={𝑎(𝑡)}{𝑏(𝑡.)} (6)

Define 𝑙={1} such that {1}{𝑎(𝑡)}=^𝑡₀𝑎(𝜏)𝑑𝜏. (7)

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 𝐶(0,∞) is a commutative ring is obvious.

Let 𝑎={1} in (7). Then, 𝑙²={1}{1}=^𝑡₀=𝑡𝑑𝜏1. (8) In the general case of 𝑛=1,…, one has 𝑙^𝑛=𝑡^𝑛−1()!,𝑛−1 (9) where 0!=1. The above 𝑙^𝑛 may be termed as a Mikusinski operator.

When one exerts 𝑙^𝑛 on 𝑓(𝑡)∈𝐶(0,∞), that is, 𝑙^𝑛{𝑓(𝑡)}, the following Cauchy formula results: 𝑙^𝑛𝑓(𝑡)=^𝑡₀(𝑡−𝜏)^𝑛−1()!𝑓(𝜏).𝑛−1𝑑𝜏 (10) Considering the generalization of 𝑙^𝑛 in (9) for 𝑣>0 yields another Mikusinski operator given by 𝑙^𝑣=𝑡^𝑣−1()!=𝑡𝑣−1Γ(𝑣^𝑣−1). (11) Further, by taking into account 𝑙^𝑣{𝑓(𝑡)}, we have 𝑙^𝑣𝑓(𝑡)=^𝑡₀(𝑡−𝜏)^𝑣−1Γ(𝑣)𝑓(𝜏)𝑑𝜏. (12) Releasing the usage of Mikusinski in {⋅} for the purpose of his operational calculus, we have 𝑙^𝑣𝑓(𝑡)=^𝑡₀(𝑡−𝜏)^𝑣−1Γ(𝑣)𝑓(𝜏)𝑑𝜏. (13) This completes the derivation because (13) is the definition of the fractional Riemann-Liouville integral of order 𝑣.

3. Discussions

From (1) and (13), one sees that the fractional Riemann-Liouville integral operator of order 𝑣, that is, ₀𝐷_𝑡^−𝑣, is equivalent to the Mikusinski operator 𝑙^𝑣 though the originality of Mikusinski’s by introducing 𝑙^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, 𝑙^𝑣 may yet be an alternative of ₀𝐷_𝑡^−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 𝑓(𝑡)∈𝐶(0,∞) is not necessary in (1). In fact, (1) exists if𝑓(𝑡) is a generalized function [32]. In addition, 𝑓(𝑡) may be a random function such as the Brownian motion [33]. Due to the consistence of ₀𝐷_𝑡^−𝑣 with 𝑙^𝑣, one sees that a generalized function𝑓(𝑡) 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 [34–36], may be interesting.

4. 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.

Acknowledgments

This work was supported in part by the 973 plan under the project Grant no. 2011CB302800 and by the National Natural Science Foundation of China under the project Grant nos. 61272402, 61070214, and 60873264.