On the Fractional Derivative of Dirac Delta Function and Its Application

The Dirac delta function and its integer-order derivative are widely used to solve integer-order differential/integral equation and integer-order system in related fields. On the other hand, the fractional-order system gets more and more attention. This paper investigates the fractional derivative of the Dirac delta function and its Laplace transform to explore the solution for fractionalorder system. The paper presents the Riemann-Liouville and the Caputo fractional derivative of the Dirac delta function, and their analytic expression. The Laplace transform of the fractional derivative of the Dirac delta function is given later. The proposed fractional derivative of the Dirac delta function and its Laplace transform are effectively used to solve fractional-order integral equation and fractional-order system, the correctness of each solution is also verified.


Introduction
The Dirac delta function δðtÞ was introduced by the theoretical physicist Paul Dirac to develop tools for the quantum field theory [1]. Till now, it is widely used to describe the impulse phenomenon in physics, mathematics, and engineering fields [2][3][4][5]. In control theory and signal process, the Dirac delta function is a basic typical input or testing signal to study the system's response [6,7]. The integerorder derivative of the Dirac delta function δðtÞ, i.e., ith order derivative δ ðiÞ ðtÞði ∈ ℕÞ is also well-studied and finds great significance in applications of related fields [8,9]. The Laplace transform of the Dirac delta function δðtÞ and its integer-order derivative δ ðiÞ ðtÞ are L½δðtÞ = s and L½δ ðiÞ ðtÞ = s i , ði ∈ ℕÞ, respectively [10]. They are effectively used to solve various kinds of integer-order differential/integral equation and integer-order system.
On the other hand, the fractional-order system which is characterized with "process memory" and "historical hered-ity" gets more and more attention [11][12][13][14][15]. Many different definitions of integrals and fractional derivatives, such as the ψ-Hilfer integral and the ψ-Hilfer fractional derivative, are proposed and find great significance in applications [16][17][18]. Just as solving integer-order integral/differential equation and system, it will be significant to investigate the fractional derivative of The Dirac delta function δðtÞ, i.e., 0 D α t δðtÞ and its Laplace transform L½ 0 D α t δðtÞ for exploring the solution for fractional-order integral/differential equation or fractional-order system.
Research on the fractional derivative of the Dirac delta function 0 D α t δðtÞ is just beginning so far. From the point of view of viscoelasticity, the literature [19] showed that the memory function of complex materials is the fractional derivative of the Dirac delta function, and the Laplace transform of the fractional derivative of the Dirac delta function is derived. However, the literature [19] only discussed the Riemann-Liouville fractional derivative, and the fractional derivative order discussed in [19] is limited in ℝ + .
In this paper, we explore the Riemann-Liouville fractional derivative and the Caputo fractional derivative of the Dirac delta function under the widely used sense, and the order of the fractional derivative is generalized to ℝ. Comparing with [19], our research method is simpler while results are more general.
This paper is organized as follows: Section 2 Basic Preparation and Section 3 investigates the fractional derivative of Dirac delta function, including the Riemann-Liouville fractional derivative of δðtÞ, i.e., R 0 D α t δðtÞ and the Caputo fractional derivative of δðtÞ, i.e., C 0 D α t δðtÞ. Section 4 studies the Laplace transform of the fractional derivative of Dirac delta function; Section 5 presents application with examples to verify the correctness of the theorems proposed in the paper. Section 6 briefly summarizes the main conclusions of the paper.

Basic Preparation
The fractional derivative of function f , i.e., a D α t f is the basic tool used in fractional calculus and related applications. The basis for its definition is an arbitrary-order integral a I p t f , namely definition 1 given below [11]. Supposing L½ f ðtÞ = FðsÞ, the Laplace transform of R 0 D α t f is as follows: If the function f has continuous derivatives up to the m-order, then its Caputo fractional derivative C 0 D α t f is as follows: The Laplace transform of C 0 D α t f only involves the initial value of the integer-order derivative of the function f ðtÞ, which brings great convenience to its application in engineering.
The Dirac delta function δðtÞ and its ith-order derivative δ ðiÞ ðtÞði ∈ ℕÞ belong to a generalized function. δðtÞ is generally defined as follows [1]: Main useful properties of δðtÞ include [8,10]: 2 Advances in Mathematical Physics

The Fractional Derivative of Dirac Delta Function
Corresponding to the above two definitions of fractional derivative, the Riemann-Liouville fractional derivative of δðtÞ, i.e., R 0 D α t δðtÞ and the Caputo fractional derivative of δðtÞ, i.e., C 0 D α t δðtÞ could be defined as in Definition 2 and Definition 3. In fact, we can give the analytic expression for them further. Theorem 1 gives the analytic formula of R 0 D α t δðtÞ.

The Laplace Transform of the Fractional Derivative of Dirac Delta Function
The Laplace transform is of great importance in solving differential/integral equation and system analysis. Theorem 3 gives the Laplace transform of 0 D α t δðtÞ.
Proof. We will prove it by classifying the different values of α.

Remark 4.
It is well-known that the role of the Laplace transform in differential/integral equation and system analysis is significant. In classical cases, objects involved in the Laplace transform are often the rational proper fraction of s [21]. For example, to the transfer function essentially discussed in control theory, the case that the power of the polynomial of s in the numerator is higher than the power of the polynomial of s in the denominator is nearly ignored. Therefore, Theorem 4 allows us to implement Laplace transform to general rational fraction. This provides a new way for the solution of differential equations and the analysis of control systems, especially of the singular systems.

Application with Examples
Example 1 (Solving fractional Volterra integral equation of the first kind [22]). Investigate the following integral equation: Obviously, the left hand of (10) is the p-order integral of the function yðτÞ, implementing the Laplace transform on the both side of the equation results in: YðsÞ/s p = kΓðm + 1Þ/ s m+1 . Consequently, The solution of (10) could be derived by implementing the inverse Laplace transform on (11) with Theorem 4: We verify the correctness of (12) according to p − m − 1 > 0 and p − m − 1 < 0 respectively.
Substituting it into the left hand of (10) yields:

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Verification: substituting (15) into the left hand of (10) yields: Example 2 (Computation of convolution of power functions). Compute the convolution of power functions: Implementing the Laplace transform on the both side of (17), we get Implementing the inverse Laplace transform on the both side of (18) with Theorem 4, we have Obviously, the correctness of (19) is confirmed. Note that this kind of convolution is not easy to compute for the vast major value of α and β by traditional way, such as integration by substitution.
Example 3 (Solution for fractional singular system [23]). Find the solution for the next fractional singular system: Implementing the Laplace transform on the both side of (22) yields: After transposition and sorting out: Verification: system (22) is equal to The correctness of x 2 ðtÞ = −uðtÞ is obvious from (29-2). Taking x 2 ðtÞ = −uðtÞ, x 20 = −uð0Þ into account, substituting x 1 ðtÞ in (28) into the right side of (29-1) brings us  Advances in Mathematical Physics