On the Rigorous Correspondence Between Operator Fractional Powers and Fractional Derivatives via the Sonine Kernel

Abstract

Traditional operational calculus, while intuitive and effective in addressing problems in physical fractal spaces, often lacks the rigorous mathematical foundation needed for fractional operations, sometimes resulting in inconsistent outcomes. To address these challenges, we have developed a universal framework for defining the fractional calculus operators using the generalized fractional calculus with the Sonine kernel. In this framework, we prove that the α-th power of a differential operator corresponds precisely to the α-th fractional derivative, ensuring both accuracy and consistency. The relationship between the fractional power operators and fractional calculus is not arbitrary, it must be determined by the specific operator form and the initial conditions. Furthermore, we provide operator representations of commonly used fractional derivatives and illustrate their applications with examples of fractional power operators in physical fractal spaces. A superposition principle is also introduced to simplify fractional differential equations with non-integer exponents by transforming them into zero-initial-condition problems. This framework offers new insights into the commutative properties of fractional calculus operators and their relevance in the study of fractal structures.

1. Introduction

Operational calculus was first introduced by Heaviside [1] as a symbolic technique for solving differential equations in circuits and the electromagnetic theory. By representing the time derivative with the symbol $\mathit{p}$, Heaviside applied algebraic operations to differential equations. Despite lacking formal rigor, this approach provided practical solutions that continued to attract attention. Flegg [2] highlighted the educational significance of this method, while Mildenberger [3] analyzed its foundational challenges. Lützen [4] traced efforts to formalize the approach, and Bengochea and López [5] demonstrated its applications in Bessel functions, expanding its utility. Although criticized for occasional inaccuracies, Heaviside’s method laid the groundwork for ongoing advancements in operational calculus.

In 1959, Mikusiński [6] revolutionized operational calculus by establishing a new algebraic framework that introduced convolution integrals to represent operator actions, enabling more systematic operations. This laid the foundation for further advancements by researchers like Berg [7] and Glaeske et al. [8], who clarified the algebraic structures of the operators. However, challenges remain in providing a unified theory for representing the operators and their physical meanings, especially for the infinite operator domains and fractal structures. In response to these challenges, Yu and Yin [9] combined operational calculus with integral transforms, developing the operator kernel method. A key requirement of this method is that the inverse Laplace transform (ILT) of the unknown operator $\mathit{T}$ with respect to the differential operator $\mathit{p}$ must exist:

$$\underset{p\to \infty }{\lim }\mathit{T}\left(\mathit{p}\right)=0.$$

(1)

Although the differential operator $\mathit{p}$ itself does not satisfy the conditions of Equation (1), Mikusiński [6] addressed this by defining it through the following expression:

$$\mathit{p}f(t)={\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}f(t)+\mathit{p}f(0).$$

(2)

The mathematical representation and physical meaning of the differential operator $\mathit{p}$ have already been clearly established, eliminating the need for further calculations using the operator kernel method. The first power of the differential operator $\mathit{p}$ corresponds to the first derivative of a function. Therefore, we propose the conjecture that the $\mathsf{\alpha }$-th power of the differential operator $\mathit{p}$ should be related to the $\mathsf{\alpha }$-th derivative of the function, where $\alpha >0$. However, over the past century, numerous and complex definitions of fractional derivatives have been developed, including those by Riemann–Liouville (RL) [10], Caputo [11,12], Caputo and Fabrizio (CF) [13], and Atangana and Baleanu (AB) [14,15]. Given the large number and complexity of these definitions, it is impractical for researchers to explore each one in detail to determine the most suitable form.

Therefore, this formal conjecture raises two fundamental questions: first, with the non-uniqueness of fractional derivative definitions, which definition should be chosen? Second, given that $\underset{\mathit{p}\to \mathrm{\infty }}{\lim }{\mathit{p}}^{\alpha }\ne 0$, it neither satisfies Equation (1) nor has a defined form. How, then, should the interaction between the operator and the function be determined? This study addresses these two fundamental questions and ultimately proves the conjecture to be entirely correct: the choice of fractional calculus corresponding to the fractional operators is unique, and this class of operators defines their action on functions through fractional calculus.

In classical fractional calculus, differentiation and integration do not satisfy the commutative property. However, in operational calculus, the operator domain is required to be commutative, which simplifies the calculations. Several studies have explored the commutative properties of the operators in fractional calculus. For instance, Luchko et al. [16,17] and Al-Kandari et al. [18] focused on the Caputo and RL derivatives, while Rani et al. [19] and Hanna et al. [20] examined the Prabhakar and Erdélyi–Kober fractional derivatives. However, these methods address only specific cases and lack generalizability. Luchko’s works [21,22,23] on general fractional derivatives with Sonine kernels, particularly his approach of separating differentiation and convolution, inspired this study. While we adopted the method for the operator representation of the fractional derivatives, our key distinction lies in solving the convolution part. Instead of focusing on specific operators, we employed a more general operator kernel function method, which offers a broader framework for analyzing operator actions on functions.

The structure of this study is outlined as follows: Section 2 establishes a universal expression for defining fractional calculus operators based on the generalized fractional calculus with the Sonine kernel; Section 3 applies the operator kernel method to provide the unique operator representations for general fractional derivatives, including the Caputo, RL, Prabhakar, CF, and AB derivatives; Section 4 demonstrates the application of fractional calculus operator representations through examples of fractional power operators in physical fractal spaces, followed by the introduction of the operator method and the superposition principle for fractional differential equations, with a discussion on these applications.

2. Preliminaries

2.1. Definitions of Fractional Integration and Differentiation

In 1869, Laurent expanded upon the foundational work of Riemann, Sonine, and Letnikov, thereby formalizing the RL fractional integral, which is now widely recognized. The definition is as follows [10]:

$${}_a{}^{RL}I_t^{\alpha }f(t)={\displaystyle \frac{1}{\Gamma (\alpha )}}{\displaystyle {\int }_a^t{(t-\tau )}^{\alpha -1}}f(\tau )\mathrm{d}\tau$$

(3)

where $\mathrm{R}\mathrm{e}\left(\alpha \right)>0$. Unless otherwise specified, the lower limit of integration is assumed to be $0$, and the upper limit is $t$. The notation for the RL fractional integral is abbreviated as ${\mathbf{I}}^{\alpha }$. Similarly, the notation for the fractional derivative is denoted as ${\mathbf{D}}^{\alpha }$. Thus, the RL fractional derivative is defined as follows [10]:

$${}^{RL}D^{\alpha }f(t)={\displaystyle \frac{1}{\Gamma (n-\alpha )}}{\displaystyle \frac{{\mathrm{d}}^n}{\mathrm{d}{t}^n}}{\displaystyle {\int }_0^t{(t-\tau )}^{n-\alpha -1}}f(\tau )d\tau$$

(4)

where $n=\lceil \alpha \rceil$ represents the smallest integer greater than $\alpha$.

Although the RL fractional derivative is theoretically rigorous, it lacks the computational simplicity required in engineering applications, particularly as its result on constants is non-zero. In light of this, many alternative definitions have been proposed from the perspective of practical applications, including the Weyl derivative, the Marchaud derivative, and the Erdélyi–Kober derivative, among others. Notably, in 1966, the Italian geophysicist Caputo introduced the Caputo fractional derivative, which has garnered significant attention [11,12]:

$${}^{C}D^{\alpha }f(t)={\displaystyle \frac{1}{\Gamma (n-\alpha )}}{\displaystyle {\int }_0^t{(t-\tau )}^{n-\alpha -1}}{f}^{(n)}(\tau )d\tau .$$

(5)

In terms of the operational form, the primary difference between the RL derivative and the Caputo derivative lies in the interchange of the order between fractional integration and integer-order differentiation.

2.2. Sonine Kernel Theory

The RL fractional integral operator is, in a generalized sense, a special case of the linear Volterra-type convolution operator [24]:

$${\mathcal{V}}_{m(t)}f(t)={\displaystyle {\int }_0^{t}m}(t-\tau )f(\tau )\mathrm{d}\tau =m(t)\ast f(t)$$

(6)

where ${\mathcal{V}}_{m\left(t\right)}$ represents a convolution operator with the function $m\left(t\right)$ as its kernel. When the convolution kernel $m\left(t\right)$ is chosen as the Gel’fand–Shilov distribution, it becomes:

$$m(t)=h_{\alpha }(t)={\displaystyle \frac{t^{\alpha -1}}{\Gamma (\alpha )}}.$$

(7)

The resulting linear operator, expressed in the form of a convolution as shown in Equation (6), corresponds to the RL fractional integral expression in Equation (3).

Considering that differentiation and integration are inverse operations, applying arbitrary-order integration to a function and then superimposing integer-order differentiation leads to the classical definition of a fractional derivative. The non-uniqueness of the operator sequence results in the emergence of both the RL and Caputo definitions:

$${}^{RL}D^{\alpha }f(t)=D^n{\mathcal{V}}_{k(t)}f(t),$$

(8)

$${}^{C}D^{\alpha }f(t)={\mathcal{V}}_{k(t)}{D}^{n}f(t)$$

(9)

where ${\mathbf{D}}^n={\displaystyle \frac{{\mathrm{d}}^n}{\mathrm{d}{t}^n}}$ represents the $n$-th order integer derivative. When the dual differential kernel function is chosen as $k\left(t\right)=h_{n-\alpha }\left(t\right)$, Equations (8) and (9), respectively, reduce to the classical RL fractional derivative and the Caputo fractional derivative.

Different definitions of fractional calculus exist [25,26], each with a specific applicability and limitations. This paper focuses on the operational calculus approach to fractional operators. Although the approach proposed in this paper may not yet encompass all the possible cases, it holds potential for future extension, expansion, and application.

This study adopts the method of defining generalized fractional calculus through the use of the Sonine dual kernel functions [27,28]. Sonine’s work [29] builds on Abel’s theory [30] to address generalized integral equations:

$$f(t)={\displaystyle \frac{1}{\Gamma (1-\alpha )}}{\displaystyle {\int }_0^t\frac{{\varphi }^{\prime }(\tau )d\tau }{{(t-\tau )}^{\alpha }}}.$$

(10)

Abel’s solution to Equation (10) corresponds exactly to what is now widely recognized as the RL integral:

$$\varphi (t)={\displaystyle \frac{1}{\Gamma (\alpha )}}{\displaystyle {\int }_0^t{(t-\tau )}^{\alpha -1}}f(\tau )\mathrm{d}\tau \triangleq I^{\alpha }f(t),t>0.$$

(11)

In the process of solving Equation (11), Abel introduced the following convolution relationship:

$$h_{\alpha }(t)\ast h_{1-\alpha }(t)=1(t),t>0,0<\alpha <1$$

(12)

where $h_{\alpha }\left(t\right)$ is given by Equation (7), and $1\left(t\right)$ represents a constant function following Mikusiński’s notation [6].

In 1884, Sonine [29] realized that Equation (12) was the key to solving the Abel integral equation and subsequently generalized it:

$$m(t)\ast k(t)=1(t),t>0.$$

(13)

The kernel functions $m\left(t\right),k\left(t\right)$ that satisfy Equation (13) are referred to as the Sonine dual kernel functions [31], denoted by the set $\mathcal{S}$.

Table 1. Sonine integral and differential kernel functions [32,33] *.

$\mathbf{Integral}\mathbf{Kernel}\mathbf{Function}\mathit{m}\left(\mathit{t}\right)$	$\mathbf{Differential}\mathbf{Kernel}\mathbf{Function}\mathit{k}\left(\mathit{t}\right)$
$h_{\alpha }\left(t\right)={\displaystyle \frac{t^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}}$	$h_{1-\alpha }\left(t\right)={\displaystyle \frac{t^{-\alpha }}{\mathsf{\Gamma }\left(1-\alpha \right)}}$
$h_{\alpha ,\lambda }\left(t\right)={\displaystyle \frac{t^{\mathsf{\alpha }-1}}{\mathsf{\Gamma }\left(\alpha \right)}}{e}^{-\mathsf{\lambda }t}$	$h_{1-\alpha ,\lambda }\left(t\right)+{\displaystyle \frac{{\mathsf{\lambda }}^{\alpha }}{\mathsf{\Gamma }\left(1-\alpha \right)}}\mathsf{\gamma }\left(1-\alpha ,\lambda t\right)$
${\left(\sqrt{t}\right)}^{\alpha -1}{J}_{\alpha -1}\left(2\sqrt{t}\right)$	${\left(\sqrt{t}\right)}^{-\alpha }{I}_{-\alpha }\left(2\sqrt{t}\right)$
${\displaystyle \frac{\cos \left(2\sqrt{t}\right)}{\sqrt{\mathsf{\pi }t}}}$	${\displaystyle \frac{\cosh \left(2\sqrt{t}\right)}{\sqrt{\mathsf{\pi }t}}}$
$t^{\alpha -1}\mathsf{\Phi }\left(\beta ,\alpha ;-\lambda t\right)$	${\displaystyle \frac{\sin \left(\mathsf{\pi }\alpha \right)}{\mathsf{\pi }}}{t}^{-\alpha }\mathsf{\Phi }\left(-\beta ,1-\alpha ;-\lambda t\right)$
$1+{\displaystyle \frac{\lambda }{\mathsf{\Gamma }\left(\alpha \right)\sqrt{t}}}$	${\displaystyle \frac{1}{\sqrt{\mathsf{\pi }t}}}-\lambda e^{{\mathsf{\lambda }}^{2}t}\mathrm{erfc}\left(\lambda \sqrt{t}\right)$
$1-{\displaystyle \frac{\lambda }{\mathsf{\Gamma }\left(\alpha \right)}}{t}^{\alpha -1}$	${\displaystyle \frac{1}{\sqrt{\mathsf{\pi }t}}}-\lambda t^{-\alpha }{E}_{1-\alpha ,1-\alpha }\left[\lambda t^{1-\alpha }\right]$
$h_{1-\beta +\alpha }\left(t\right)+h_{1-\beta }\left(t\right)$	$t^{\beta -1}{E}_{\alpha ,\beta }\left[-t^{\alpha }\right]$

* Note: The following symbols and functions are used in the table: $\mathsf{\gamma }\left(1-\mathsf{\alpha },\mathsf{\lambda }t\right)$ denotes the incomplete gamma function; $J_{\alpha -1}\left(2\sqrt{t}\right)$ represents the Bessel function of the first kind; $I_{-\alpha }\left(2\sqrt{t}\right)$ is the modified Bessel function of the first kind; $\mathsf{\Phi }\left(\beta ,\alpha ;-\lambda t\right)$ is the Kummer confluent hypergeometric function; $\mathrm{erfc}\left(\lambda \sqrt{t}\right)$ refers to the complementary error function; and $E_{\alpha ,\beta }\left(z\right)$ is the Mittag–Leffler (ML) function, a generalization of the exponential function used in fractional calculus.

[32,33] presents some common examples of Sonine dual kernel functions.

Kochubei [34,35] conducted an in-depth study of all the “Volterra-type linear operators” in generalized fractional calculus. Drawing on this foundation, he introduced a formal definition for the RL-type fractional derivatives:

$$\begin{array}{ll}\hfill D_{k(t)}f(t)& ={\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{\displaystyle {\int }_0^{t}k}(t-\tau )f(\tau )d\tau -k(t)u(0^{+})\hfill \\ & ={\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}(k\ast f)-k(t)u(0^{+}).\hfill \end{array}$$

(14)

And the corresponding dual expression for fractional integrals:

$$\begin{array}{ll}{I}_{m(t)}f(t)& ={\displaystyle {\int }_0^{t}m(t-\tau )f(\tau )\mathrm{d}\tau }\hfill \\ & =m(t)\ast f(t).\end{array}$$

(15)

This paper follows Kochubei’s approach, utilizing Equation (15) to define fractional integrals. In cases where $m\left(t\right)$ contains only a single parameter besides the variable $t$, this parameter can be interpreted as the order of the fractional integral, as seen with α in Equation (7). When the kernel function is clearly defined, the subscript is omitted, and the fractional integral is denoted as:

$$\begin{array}{ll}\hfill I^{\alpha }f(t)& =I_{m(\alpha ,t)}f(t)\hfill \\ & ={\displaystyle {\int }_0^{t}m(\alpha ,t-\tau )f(\tau )\mathrm{d}\tau }\\ & =m(\alpha ,t)\ast f(t).\end{array}$$

(16)

The RL-type fractional derivative is defined as:

$$\begin{array}{ll}\hfill {}^{RL}D^{\alpha }f(t)& =D^n{\mathcal{V}}_{k(\alpha ,t)}f(t)\hfill \\ & ={\displaystyle \frac{{\mathrm{d}}^n}{\mathrm{d}{t}^n}}{\displaystyle {\int }_0^{t}k}(\alpha ,t-\tau )f(\tau )\mathrm{d}\tau \\ & ={\displaystyle \frac{{\mathrm{d}}^n}{\mathrm{d}{t}^n}}\left[k(\alpha ,t)\ast f(t)\right].\end{array}$$

(17)

The Caputo-type fractional derivative is defined as:

$$\begin{array}{ll}\hfill {}^{C}D^{\alpha }f(t)& =V_{k(\alpha ,t)}{D}^{n}f(t)\hfill \\ & ={\displaystyle {\int }_0^{t}k}(\alpha ,t-\tau )f^{(n)}(\tau )\mathrm{d}\tau \\ & =k(\alpha ,t)\ast f^{(n)}(t)\end{array}$$

(18)

where $n=\lceil \alpha \rceil$, and the integral kernel $m\left(t\right)$ and differential kernel $k\left(t\right)$ satisfy the Sonine relationship given by Equation (13). From this process, it becomes evident that in the classical fractional calculus theory, the operations of integration and differentiation cannot be interchanged arbitrarily, as changing their order fundamentally alters the definition of the fractional derivative. This non-commutative property introduces significant challenges in computation. Therefore, this study aims to relate the classical fractional representation to the operator forms, thereby imparting a commutative property to the operations.

2.3. Operational Calculus

Following the notation established by Mikusiński [6] and Yu and Yin [9], this paper adopts the convention of representing functions using Latin or Greek letters along with variables, such as $f\left(t\right)$ and $g\left(t\right)$. When presenting the expression of a function, variables may be omitted, for instance, $e^{-t}$. The operator symbols are denoted by bold Latin or Greek letters, such as $\mathit{f},\mathit{g},\mathit{\delta },\mathrm{and}\mathit{p}$. Operators that involve parameters are represented as $\mathit{f}(\lambda ,\mu )$.

The operator domain $\mathcal{O}$ is generated using the function ring $\mathcal{K}$, where each element of $\mathcal{K}$ corresponds to an element in $\mathcal{O}$ [6]

$$\mathit{f}\simeq f(t)\mathit{f}\in \mathcal{O},f(t)\in \mathcal{K}.$$

(19)

The equivalence relation ‘ $\simeq$’ shows that the element $\mathit{f}$ in the operator domain $\mathcal{O}$ corresponds to $f\left(t\right)$ in the function ring $\mathcal{K}$.

Definition 1.

Operator addition corresponds to the addition of their functions:

$$\mathit{f}+\mathit{g}\simeq f(t)+g(t),\mathit{f},\mathit{g}\in \mathcal{O},f(t),g(t)\in \mathcal{K}.$$

(20)

Definition 2.

Operator multiplication is defined as the convolution of their functions:

$$\mathit{f}\cdot \mathit{g}\simeq f(t)\ast g(t)={\displaystyle {\int }_0^{t}f(t-\tau )g(\tau )\mathrm{d}\tau },\mathit{f},\mathit{g}\in \mathcal{O},f(t),g(t)\in \mathcal{K}.$$

(21)

It can be proven that the triplet $<f,+,\cdot >$ satisfies the field properties. To avoid confusion, the multiplication symbol ‘ $\cdot$’ for the operator domain is omitted, i.e., $\mathit{f}\cdot \mathit{g}=\mathit{f}\mathit{g}$. The equivalence class of the operators is represented as:

$$({\mathit{f}}_1,{\mathit{g}}_1)~({\mathit{f}}_2,{\mathit{g}}_2)\iff {\displaystyle \frac{{\mathit{f}}_1}{{\mathit{g}}_1}}={\displaystyle \frac{{\mathit{f}}_2}{{\mathit{g}}_2}}\iff {\mathit{f}}_1(t)\ast {\mathit{g}}_2(t)={\mathit{f}}_2(t)\ast {\mathit{g}}_1(t).$$

(22)

The operator $\mathit{l}$ is defined as $1\left(t\right)\equiv 1$. It is important to distinguish between a constant function in the functional sense and a number in the algebraic sense: a number cannot be convolved with a function. According to the definition, the action of $\mathit{l}$ on a function $f\left(t\right)\in \mathcal{K}$ is expressed as:

$$lf={\displaystyle {\int }_0^{t}f(\tau )\mathrm{d}\tau },\mathit{f},l\in \mathcal{O},f(t)\in \mathcal{K}.$$

(23)

In view of Equation (23), the operator $\mathit{l}$, defined as $\mathit{l}\triangleq 1\left(t\right)$, is called the integral operator.

Remark 1.

According to Equation (23),  $\mathit{p}\mathit{l}f=\mathit{I}f=f$ , where $\mathit{I}$ represents the identity operator, also referred to as the unit operator  $\mathbf{1}$ , thus the operator $\mathit{p}$ is often known as a differential operator or a derivative operator. In the theory developed here, the differential operator $\mathit{p}$ is not entirely equivalent to the derivative operator ${\displaystyle \frac{d}{dt}}$, but rather $\mathit{p}f\left(t\right)={\displaystyle \frac{d}{dt}}f\left(t\right)+\mathit{p}f\left(0\right)$. There is no element corresponding to $\mathit{p}$ for $C^n$-smooth functions [16,17].

Remark 2.

According to Equation (21), the convolution of two constant functions  $C_1\left(t\right)\equiv C_1$ and  $C_2\left(t\right)\equiv C_2$ is not a constant value, i.e.,  $C_1\left(t\right)\ast C_2\left(t\right)=C_1{C}_{2}t$. The numerical operator is denoted in bold as  $\mathit{C}=\mathit{p}C\left(t\right)$. It is important to distinguish between the constant  $C$, the constant function  $C\left(t\right)$, and the numerical operator  $\mathit{C}$, where  $\mathit{C}=\mathit{p}C\left(t\right)$ and  $C\left(t\right)\equiv C$.

Remark 3.

When the numerical operator acts on another operator, we have  $\mathit{C}f=\mathit{p}C\left(t\right)\ast f\left(t\right)=Cf\left(t\right)$. Because of this property, in cases where it causes no confusion, it can be written as  $\mathit{C}f=Cf=Cf\left(t\right)$. Therefore,  $\mathit{I}=\mathbf{1}=1$ and  $\mathit{p}=\mathit{I}/\mathit{l}=1/l$.

3. Representation of Commutable Operators in Fractional Calculus

3.1. Riemann–Liouville Fractional Integral

Based on the inverse relationship between integral operators and differential operators:

$$\mathit{p}\mathit{l}=\mathit{I},\mathit{p}={\displaystyle \frac{\mathit{I}}{\mathit{l}}},\mathit{l}={\displaystyle \frac{\mathit{I}}{\mathit{p}}}.$$

(24)

When the order of the integral operator in Equation (24) becomes arbitrary, we obtain the arbitrary-order integral operator ${\mathit{l}}^{\alpha }$:

$${\mathit{p}}^{\alpha }{\mathit{l}}^{\alpha }=\mathit{I},{\mathit{p}}^{\alpha }={\displaystyle \frac{\mathit{I}}{{\mathit{l}}^{\alpha }}},{\mathit{l}}^{\alpha }={\displaystyle \frac{\mathit{I}}{{\mathit{p}}^{\alpha }}}.$$

(25)

This paper only discusses the case where $\alpha$ is a real number.

According to the operator kernel function method provided by Yu and Yin [9], the ILT of the operator ${\mathit{l}}^{\alpha }$ with respect to the differential operator $\mathit{p}$ yields the kernel function of the integral operator:

$${\mathit{l}}^{\alpha }={\displaystyle \frac{1}{2\pi i}}{\displaystyle {\int }_c{e}^{pt}\frac{1}{{\mathit{p}}^{\alpha }}\mathrm{d}\mathit{p}}={\displaystyle \frac{t^{\alpha -1}}{\Gamma (\alpha )}}.$$

(26)

By comparing Equations (26) and (7), it can be observed that the kernel function of the integral operator is precisely the integral kernel of the RL fractional integral. This demonstrates that the RL integral operator is entirely equivalent to the integral operator of the operational calculus, namely:

$$I^{\alpha }={\mathit{l}}^{\alpha },(\alpha >0).$$

(27)

Equation (27) indicates that, during calculations, the RL fractional integral ${\mathbf{I}}^{\alpha }$ can be interchangeably replaced with the integral operator ${\mathit{l}}^{\alpha }$ from operational calculus, and vice versa.

3.2. Caputo Fractional Derivative

3.2.1. Integral–Differential Property Method

According to the operational relationship between integration and differentiation, differentiation is the left inverse of integration. However, the reverse does not hold; performing differentiation first and then integration introduces a correction term:

$$I^{\alpha }\left[{}^{c}D^{\alpha }f(t)\right]=f(t)-{\displaystyle \sum _{i=0}^{n-1}{f}^{(i)}}(0+){\displaystyle \frac{x^i}{i!}}.$$

(28)

The integral–differential property method is derived by transforming this relationship. Luchko [16] used this method to obtain the operator form of the Caputo fractional derivative.

Applying the $\alpha$-order differential operator ${\mathit{p}}^{\alpha }$ to both sides of Equation (28), we obtain:

$${}^{C}D^{\alpha }f(t)={\mathit{p}}^{\alpha }\cdot f(t)-{\mathit{p}}^{\mathit{\alpha }}\cdot f_{\alpha },f_{\alpha }(t)\triangleq {\displaystyle \sum _{i=0}^{n-1}{f}^{(i)}}(0+){\displaystyle \frac{t^i}{i!}}.$$

(29)

The fractional differential operator ${\mathit{p}}^{\alpha }$ and the integral operator ${\mathit{l}}^{\alpha }$ satisfy the inverse relationship within the operator domain, as given by Equation (25). Substituting Equations (26) and (7) into Equation (29), we obtain:

$${}^{C}D^{\alpha }f(t)={\mathit{p}}^{\alpha }\cdot f(t)-{\displaystyle \sum _{i=0}^{n-1}{\mathit{p}}^{\alpha -i-1}}{f}^{(i)}(0+).$$

(30)

Equation (30) is the operator representation of the Caputo fractional derivative. It is evident that the order of the Caputo fractional derivative is reflected as the exponent of the differential operator $\mathit{p}$. To align the fractional power of the operator with the fractional derivative, a correction involving initial value terms is necessary.

3.2.2. Integral Transform Method

Yu and Yin [9] revealed the similarity between operator algebra and integral transforms. The Laplace transform (LT) of the Caputo fractional derivative is given by:

$$\mathcal{L}\left[{}^{C}D^{\alpha }f(t)\right](s)=s^{\alpha }F(s)-{\displaystyle \sum _{i=0}^{n-1}{s}^{\alpha -i-1}}{f}^{(i)}(0).$$

(31)

Upon observation, it is evident that Equations (29) and (31) are completely identical in their form. The LT converts the original function in the time domain into its image function in the frequency domain, thus the symbol $s$ in Equation (31) represents a complex variable in the frequency domain. In contrast, the symbol $\mathit{p}$ in Equation (29) denotes a differential operator, which is a concept within the operator domain. Interestingly, despite originating from different premises, both lead to the same formal expression. Essentially, as proven by Yu and Yin [9], one of the logical foundations of operational calculus algebra is integral transforms.

3.2.3. Operational Calculus Method

Apart from the first two methods, the operator expression of the fractional derivative can also be derived through the definition in operational calculus. According to the definition of the generalized fractional derivative, the Caputo fractional derivative is expressed as:

$$\begin{array}{ll}\hfill {}^{C}D^{\alpha }f(t)& =h_{1-\alpha }(t)\ast \left[{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}f(t)\right]\hfill \\ & ={\mathit{l}}^{1-\alpha }\left[\mathit{p}f(t)-\mathit{p}f(0)\right]\\ & ={\mathit{p}}^{\alpha }f(t)-{\mathit{p}}^{\alpha }f(0).\end{array}$$

(32)

Extending Equation (32) to a more general case:

$$\begin{array}{ll}\hfill {}^{C}D^{\alpha }f(t)& =h_{n-\alpha }(t)\ast \left[{\displaystyle \frac{{\mathrm{d}}^n}{\mathrm{d}{t}^n}}f(t)\right]\hfill \\ & ={\mathit{l}}^{n-\alpha }\left[{\mathit{p}}^{n}f(t)-{\displaystyle \sum _{i=0}^{n-1}{\mathit{p}}^{n-i-1}}{f}^{(i)}(0)\right]\\ & ={\mathit{p}}^{\alpha }f(t)-{\displaystyle \sum _{i=0}^{n-1}{\mathit{p}}^{\alpha -i-1}}{f}^{(i)}(0).\end{array}$$

(33)

Unsurprisingly, although Equations (29), (31), and (33) use three different calculation methods, the results are identical. In terms of the calculation process, the intermediate steps in the operational calculus method are simpler and more intuitive.

3.3. Riemann–Liouville Fractional Derivative

3.3.1. Integral–Differential Property Method

The RL fractional differentiation and integration are also non-commutative. In Section 3.1, we presented the operator representation of the RL fractional integral. In this section, we extend the analysis by constructing the operator representation for the RL fractional derivative. This formulation is grounded in the composition principles of fractional integration and differentiation, as discussed in [10]:

$${\mathbf{I}}^{\alpha }\left[{}^{RL}D^{\alpha }f(t)\right]=f(t)-{\displaystyle \sum _{i=1}^n{\left[{}^{RL}D^{\alpha -i}f(t)\right]}_{t=0}}{\displaystyle \frac{t^{\alpha -i}}{\Gamma (\alpha -i+1)}}.$$

(34)

Applying the differential operator ${\mathit{p}}^{\alpha }$ to both sides:

$${}^{RL}D^{\alpha }f(t)={\mathit{p}}^{\alpha }f(t)-{\mathit{p}}^{\alpha }{{\displaystyle \sum _{i=1}^n\left[{}^{RL}D^{\alpha -i}f(t)\right]}}_{t=0}{\displaystyle \frac{t^{\alpha -i}}{\Gamma (\alpha -i+1)}}.$$

(35)

Equation (35) represents the commutative operator form of the RL fractional derivative.

By comparing the operator representation of the Caputo fractional derivative in Equation (30) with that in Equation (35), it is observed that both share the operator part ${\mathit{p}}^{\alpha }$, but differ in the constant term related to the initial value correction.

3.3.2. Integral Transform Method

Taking the LT of the RL-type fractional derivative, we obtain:

$$\mathcal{L}\left[{}^{RL}D^{\alpha }f(t)\right](s)=s^{\alpha }F(s)-{\displaystyle \sum _{i=0}^{n-1}{s}^i}{\left[{}^{RL}D^{\alpha -i-1}f(t)\right]}_{t=0}.$$

(36)

Similar to the meaning of Equation (31), the LT maps the function to the complex variable space, and Equation (36) is formally similar to the operator representation of the RL fractional derivative in Equation (35).

Considering the special case where $f\left(t\right)=C\left(t\right)\equiv C$, substituting into Equation (35) or Equation (36), we obtain:

$${}^{RL}D^{\alpha }C(t)={\mathit{p}}^{\alpha }C(t)-{\mathit{p}}^{\alpha }{{\displaystyle \sum _{i=0}^{n-1}\left[{}^{RL}D^{\alpha -i-1}C(t)\right]}}_{t=0}{\displaystyle \frac{t^{\alpha -i-1}}{\Gamma (\alpha -i)}}.$$

(37)

where $C\left(t\right)$ represents a constant function with value $C$, to distinguish it from a numerical constant.

When $0<\alpha <1$, ${\left[{}_{}{}^{RL}{\mathbf{D}}^{\alpha -i-1}C\left(t\right)\right]}_{t=0}=0$. Additionally, taking the RL fractional derivative of a constant function results in ${}_{}{}^{RL}{\mathbf{D}}^{\alpha } C\left(t\right)=C{t}^{-\alpha }/\mathsf{\Gamma }\left(1-\alpha \right)$ Thus, we obtain:

$${\displaystyle \frac{C{t}^{-\alpha }}{\Gamma (1-\alpha )}}={}^{RL}D^{\alpha }C(t)={\mathit{p}}^{\alpha }C.$$

(38)

Equation (38) demonstrates that the fractional power of the differential operator ${\mathit{p}}^{\alpha }$ acting on a constant function results in a negative exponential power function. The order of the operator reflects the order of differentiation, which implies a connection between the derivative’s order and the exponent of the resulting function.

3.3.3. Operational Calculus Method

This subsection establishes the operator representation of the RL fractional derivative using the operational calculus method. According to Sonine’s generalized fractional derivative theory, the RL fractional derivative is expressed as:

$${}^{RL}D^{\alpha }f(t)={\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left[{\displaystyle \frac{t^{-\alpha }}{\Gamma (1-\alpha )}}\ast f(t)\right]={\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{\displaystyle {\int }_0^t\frac{{(t-\tau )}^{-\alpha }}{\Gamma (1-\alpha )}f(\tau )\mathrm{d}\tau }.$$

(39)

The RL fractional derivative can be expressed as a convolution between the kernel function $h_{n-\alpha }\left(t\right)$ and the target function, followed by differentiation. By substituting the relationship between $h_{n-\alpha }\left(t\right)$ and the differentiation operator, along with the function derivative $\mathit{p}x\left(t\right)={\displaystyle \frac{df}{dt}}x\left(t\right)+\mathit{p}f\left(0\right)$, we attain the expanded form of the fractional derivative. This leads to:

$$\begin{array}{ll}\hfill {}^{RL}D_t^{\alpha }f(t)& ={\displaystyle \frac{{\mathrm{d}}^n}{\mathrm{d}{t}^n}}\left[h_{n-\alpha }(t)\ast f(t)\right]\hfill \\ & ={\mathit{p}}^{\alpha }f(t)-{\displaystyle \sum _{i=0}^{n-1}{\mathit{p}}^{n-1-i}{\left[{\mathit{l}}^{n-\alpha }f(t)\right]}^{(i)}(0)}\\ & ={\mathit{p}}^{\alpha }f(t)-{{\displaystyle \sum _{i=0}^{n-1}{\mathit{p}}^{n-1-i}\left[{}^{RL}D_t^{\alpha -n+i}f(t)\right]}}_{t=0}\\ & \stackrel{(n-1-i)\to i}{=}{\mathit{p}}^{\alpha }f(t)-{\displaystyle \sum _{i=0}^{n-1}{\mathit{p}}^i{\left[{}^{RL}D_t^{\alpha -i-1}f(t)\right]}_{t=0}}.\end{array}$$

(40)

Ultimately, Equations (39), (36), and (40) yield the same result.

3.4. Prabhakar Fractional Derivative

Rani et al. [19] and Giusti et al. [24] studied the representation of the Prabhakar-type fractional derivative. The Prabhakar-type fractional derivative involves a three-parameter generalized ML function, which is defined as:

$$E_{\alpha .\beta }^{\gamma }(z)={\displaystyle \sum _{k=0}^{\infty }\frac{{(\gamma )}_k{z}^k}{k!\Gamma (\alpha k+\beta )}},\mathrm{Re}(\alpha )>0$$

(41)

where ${\left(\gamma \right)}_k$ denotes the Pochhammer symbol, also referred to as the rising factorial in mathematics. It is calculated as:

$${(\gamma )}_k={\displaystyle \frac{\Gamma (\gamma +k)}{\Gamma (\gamma )}}.$$

(42)

According to Equation (41), the Prabhakar integral kernel is defined as:

$$e_{\alpha ,\beta }^{\gamma }(\lambda ;t)\triangleq t^{\beta -1}{E}_{\alpha ,\beta }^{\gamma }(\lambda t^{\alpha }).$$

(43)

Thus, the Prabhakar fractional integral can be expressed through convolution as:

$$\begin{array}{ll}\hfill E_{\alpha ,\beta ,\lambda }^{\gamma }f(t)& =e_{\alpha ,\beta }^{\gamma }(\lambda ;t)\ast f(t)\hfill \\ & ={\displaystyle {\int }_0^t{e}_{\alpha ,\beta }^{\gamma }(\lambda ;t-\tau )f(\tau )\mathrm{d}\tau }.\end{array}$$

(44)

By using Equation (43) and the Sonine duality relationship from Equation (13), the dual kernel representation of the Prabhakar-type fractional integral and derivative can be calculated as:

$$k(t)=e_{\alpha ,n-\beta }^{-\gamma }(\lambda ;t),$$

(45)

$$m(t)=e_{\alpha ,\beta }^{\gamma }(\lambda ;t).$$

(46)

Using the definition of the Sonine kernel function, the Caputo–Prabhakar-type fractional derivative is expressed as:

$${}^{C-P}D_{\alpha ,\beta ,\lambda }^{\gamma }f(t)=E_{\alpha ,n-\beta ,\lambda }^{-\gamma }{D}^{n}f(t),n=⌈\beta ⌉.$$

(47)

To express Equation (47) in the operator form, it is necessary to determine the operator form of the integral $E_{\alpha ,n-\beta ,\lambda }^{-\gamma }$. This step can be calculated using the operator kernel function method provided by Yu and Yin [9], considering the case $0<\beta <1$, where $n=1$. In this case, the operator form of the integral is:

$$\begin{array}{ll}\hfill E_{\alpha ,1-\beta ,\lambda }^{-\gamma }f(t)& =e_{\alpha ,1-\beta }^{-\gamma }(\lambda ;t)\ast f(t)\hfill \\ & =\mathcal{L}\left[e_{\alpha ,1-\beta }^{-\gamma }(\lambda ;t)\right](\mathit{p})\cdot f(t)\\ & ={\mathit{p}}^{\beta -1}{(1-\lambda {\mathit{p}}^{-\alpha })}^{\gamma }f(t).\end{array}$$

(48)

Substituting Equation (48) into Equation (47), the operator representation of the Caputo–Prabhakar-type fractional derivative is obtained as:

$$\begin{array}{ll}\hfill {}^{C-P}D_{\alpha ,\beta ,\lambda }^{\gamma }f(t)& ={\mathit{p}}^{\beta -1}{(1-\lambda {\mathit{p}}^{-\alpha })}^{\gamma }\left[\mathit{p}f(t)-\mathit{p}f(0+)\right]\hfill \\ & ={\mathit{p}}^{\beta }{(1-\lambda {\mathit{p}}^{-\alpha })}^{\gamma }f(t)-{\mathit{p}}^{\beta }{(1-\lambda {\mathit{p}}^{-\alpha })}^{\gamma }f(0).\end{array}$$

(49)

Similarly, the operator representation of the RL–Prabhakar-type fractional derivative can be obtained as:

$${}^{RL-P}D_{\alpha ,\beta ,\lambda }^{\gamma }f(t)={\mathit{p}}^{\beta }{(1-\lambda {\mathit{p}}^{-\alpha })}^{\gamma }f(t)-{\left[{\mathit{p}}^{\beta }{(1-\lambda {\mathit{p}}^{-\alpha })}^{\gamma }f(t)\right]}_{t=0}.$$

(50)

3.5. Caputo–Fabrizio Fractional Derivative

This section discusses the operator representation of the CF fractional derivative [13]. The question of whether the CF derivative, which is based on a non-singular kernel, can be classified as a ‘fractional derivative’ will not be examined in detail here. This topic has been thoroughly explored in the literature [25,26,36]. Despite the debate surrounding the naming of fractional derivatives, it is undeniable that the CF derivative has demonstrated significant utility in addressing practical problems [37,38]. The CF fractional derivative is defined as follows [13]:

$${}^{CF}D^{\alpha }f(t)={\displaystyle \frac{M(\alpha )}{1-\alpha }}{\displaystyle {\int }_0^t\exp \left[-\frac{\alpha }{1-\alpha }(t-\tau )\right]f^{\prime }(t)\mathrm{d}\tau }$$

(51)

where $M\left(\alpha \right)$ is the normalization function, and it satisfies $M\left(0\right)=M\left(1\right)=1$.

In fact, the CF derivative is a special case of the Prabhakar fractional derivative:

$$\begin{array}{ll}\hfill {}^{CF}D^{\alpha }f(t)& ={\displaystyle \frac{M(\alpha )}{1-\alpha }}\left[{}^{C-P}D_{1,0,-\alpha /(1-\alpha )}^{-1}f(t)\right]\hfill \\ & ={\displaystyle \frac{M(\alpha )}{1-\alpha }}{E}_{1,1,-\alpha /(1-\alpha )}^1{f}^{\prime }(t).\end{array}$$

(52)

Using the result from Equation (49), the operator form of the CF fractional derivative can be directly obtained as:

$${}^{CF}D^{\alpha }f(t)=M(\alpha ){\displaystyle \frac{\mathit{p}f(t)-\mathit{p}f(0)}{(1-\alpha )\mathit{p}+\alpha }}.$$

(53)

3.6. Atangana–Baleanu Fractional Derivative

Similar to the CF fractional derivative, this section discusses another special case of the Prabhakar-type fractional derivative—the ABC derivative (AB operator in the Caputo sense). The definition of the ABC derivative is given as follows [15]:

$${}^{ABC}D^{\alpha }f(t)={\displaystyle \frac{B(\alpha )}{1-\alpha }}{\displaystyle {\int }_0^t{E}_{\alpha }\left[-\frac{\alpha }{1-\alpha }{(t-\tau )}^{\alpha }\right]f^{\prime }(t)\mathrm{d}\tau }$$

(54)

where $B\left(\alpha \right)$ is the normalization function, and it satisfies $B\left(0\right)=B\left(1\right)=1$. This normalization function $B\left(\alpha \right)$ ensures that the ABC derivative recovers classical results for integer values of α, making it a generalization of the standard fractional derivatives.

The ABC derivative can be viewed as a generalization of the CF derivative, where the exponential function is extended to a single-parameter ML function [39]. It can also be derived from the Prabhakar-type fractional derivative:

$$\begin{array}{ll}\hfill {}^{ABC}D^{\alpha }f(t)& ={\displaystyle \frac{B(\alpha )}{1-\alpha }}\left[{}^{C-P}D_{\alpha ,0,-\alpha /(1-\alpha )}^{-1}f(t)\right]\hfill \\ & ={\displaystyle \frac{B(\alpha )}{1-\alpha }}{E}_{\alpha ,1,-\alpha /(1-\alpha )}^1{f}^{\prime }(t).\end{array}$$

(55)

Using the result from Equation (49), the operator form of the ABC fractional derivative can be directly obtained as:

$${}^{ABC}D^{\alpha }f(t)=B(\alpha ){\displaystyle \frac{\mathit{p}f(t)-\mathit{p}f(0)}{(1-\alpha )\mathit{p}+\alpha {\mathit{p}}^{1-\alpha }}}.$$

(56)

By observing Equations (53) and (56), it can be noted that they exhibit a formal similarity. This similarity arises from the analogous convolution kernel functions in their derivative definitions.

3.7. Generalized Fractional Derivative with the Sonine Kernel Definition

For the fractional calculus defined by the Sonine kernel, the operator representation of the commutative–derivative operator can be similarly derived using the method outlined in the previous section. Each derivative, defined by a differential kernel function, is categorized into two types: the RL-type and the Caputo-type.

Table 2. Operator representation of the Caputo-type and RL-type fractional derivatives defined by the Sonine differential kernels.

Differential Kernel	Operator Part	Caputo Type	RL Type
${\displaystyle \frac{t^{-\mu }}{\mathsf{\Gamma }\left(1-\mu \right)}}$	${\mathit{p}}^{\mu }$	$-{\displaystyle \sum _{i=0}^{n-1}}{\mathit{p}}^{\mu -i-1}{f}^{\left(i\right)}\left(0\right)$	$-{\displaystyle \sum _{i=0}^{n-1}}{\mathit{p}}^i{\left[{\mathbf{D}}_t^{\mu -\mathcal{i}-1}f\left(t\right)\right]}_{t=0}$
$t^{-\beta }{E}_{\alpha ,m-\beta }^{-\gamma }\left(\mathsf{\lambda }{t}^{\alpha }\right)$	${\mathit{p}}^{\beta }{\left(1-\lambda {\mathit{p}}^{-\alpha }\right)}^{\gamma }$	$-{\mathit{p}}^{\beta }{\left(1-\lambda p^{-\alpha }\right)}^{\gamma }f\left(0\right)$	$-{\left[{\mathit{p}}^{\beta }{\left(1-\lambda {\mathit{p}}^{-\alpha }\right)}^{\gamma }f\left(t\right)\right]}_{t=0}$
${\displaystyle \frac{M\left(\alpha \right)}{1-\alpha }}{E}_{\mathrm{1,1}}^1\left({\displaystyle \frac{-\alpha }{1-\alpha }}t\right)$	${\displaystyle \frac{M\left(\alpha \right)\mathit{p}}{\left(1-\alpha \right)\mathit{p}+\alpha }}$	$-{\displaystyle \frac{M\left(\alpha \right)f\left(0\right)}{\left(1-\alpha \right)\mathit{p}+\alpha }}$	$-{\left[{\displaystyle \frac{M\left(\alpha \right)f\left(t\right)}{\left(1-\alpha \right)\mathit{p}+\alpha }}\right]}_{t=0}$
${\displaystyle \frac{B\left(\alpha \right)}{1-\alpha }}{E}_{\alpha ,1}^1\left({\displaystyle \frac{-\alpha }{1-\alpha }}{t}^{\alpha }\right)$	${\displaystyle \frac{B\left(\alpha \right)\mathit{p}}{\left(1-\alpha \right)\mathit{p}+\alpha {\mathit{p}}^{1-\alpha }}}$	$-{\displaystyle \frac{B\left(\alpha \right)f\left(0\right)}{\left(1-\alpha \right)\mathit{p}+\alpha {\mathit{p}}^{1-\alpha }}}$	$-{\left[{\displaystyle \frac{B\left(\alpha \right)f\left(t\right)}{\left(1-\alpha \right)\mathit{p}+\alpha {\mathit{p}}^{1-\alpha }}}\right]}_{t=0}$
${\displaystyle \frac{\cosh \left(2\sqrt{t}\right)}{\sqrt{\mathsf{\pi }t}}}$	$e^{{\displaystyle \frac{1}{\mathit{p}}}}{\mathit{p}}^{{\displaystyle \frac{1}{2}}}$	$-f\left(0\right)e^{{\displaystyle \frac{1}{\mathit{p}}}}{\mathit{p}}^{-{\displaystyle \frac{1}{2}}}$	${-\left[e^{{\displaystyle \frac{1}{\mathit{p}}}}{\mathit{p}}^{-{\displaystyle \frac{1}{2}}}f\left(t\right)\right]}_{t=0}$
${\left(\sqrt{t}\right)}^{-\alpha }{I}_{-\alpha }\left(2\sqrt{t}\right)$	$e^{{\displaystyle \frac{1}{\mathit{p}}}}{\mathit{p}}^{\alpha }$	$-f\left(0\right)e^{{\displaystyle \frac{1}{\mathit{p}}}}{\mathit{p}}^{\alpha -1}$	$-{\left[e^{{\displaystyle \frac{1}{\mathit{p}}}}{\mathit{p}}^{\alpha -1}f\left(t\right)\right]}_{t=0}$
${\displaystyle \frac{1}{\sqrt{\mathsf{\pi }t}}}-\lambda e^{{\lambda }^{2}t}\mathrm{erfc}\left(\lambda \sqrt{t}\right)$	${\displaystyle \frac{\mathit{p}}{\sqrt{\mathit{p}+\lambda }}}$	$-{\displaystyle \frac{f\left(0\right)}{\sqrt{\mathit{p}+\lambda }}}$	$-{\left[{\displaystyle \frac{f\left(t\right)}{\sqrt{\mathit{p}+\lambda }}}\right]}_{t=0}$

lists the operator expressions for the fractional calculus operators defined by the Sonine kernel, as discussed in this study. By comparing the fractional derivatives for each kernel definition, it is found that the core expressions of the Caputo-type and RL-type operators are largely the same, with the differences introduced by the correction terms based on the initial conditions. The correction term for the Caputo-type derivative is obtained by multiplying the main operator expression by the initial value of the function, whereas the correction term for the RL-type derivative is derived by applying the main operator directly to the function and then taking the initial value of the whole expression.

This indicates that, when applying the operational calculus method to solve fractional calculus problems, fractional differential operators cannot be treated as equivalent to fractional derivatives. Instead, the selection of the appropriate form of fractional calculus and operator representation should be based on the specific requirements of the problem. Moreover, when using the operator methods to analyze stiffness or compliance operators in physical fractal structures, a direct correspondence between fractional calculus and the resulting operators can be established, providing a clear mathematical framework for analyzing fractal structures.

4. Discussion

4.1. Fractal Operator Representation as a Function

To bridge the theoretical results with real-world applications, this section demonstrates how fractional operators can be used to solve problems in physical fractal systems. Hu et al. [40,41], in their study of complex viscoelastic behavior, utilized the operational calculus method to investigate the fractal tree model shown in Figure 1, abstracting the fractional power operators from the physical fractal space.

The operator form of the stress–strain relationship for the fractal tree is given as:

$$\sigma (t)={\displaystyle \frac{\eta }{E}}{\mathit{p}}^{{\displaystyle \frac{1}{2}}}\epsilon (t).$$

(57)

The theory established by Minkusiński only addressed the case where the order of the differential operator $\mathit{p}$ is an integer, and it could not provide a mathematical representation for operators of the form seen in Equation (57). Nevertheless, by comparing Equation (2) with Equation (57), earlier scholars astutely noticed that the differential operator $\mathit{p}$ is related to the first derivative of a function, suggesting that the operator ${\mathit{p}}^{{\displaystyle \frac{1}{2}}}$ should be connected to some form of fractional derivative. Therefore, Equation (57) was further expressed as:

$$\sigma (t)={\displaystyle \frac{\eta }{E}}{\displaystyle \frac{{\mathrm{d}}^{\frac{1}{2}}}{\mathrm{d}{t}^{\frac{1}{2}}}}\epsilon (t).$$

(58)

As mentioned in the introduction, there are numerous definitions for the fractional derivative $\frac{{\mathrm{d}}^{1/2}}{\mathrm{d}{t}^{1/2}}$. The key question is: which fractional derivative should be chosen to obtain the functional form of ${\mathit{p}}^{{\displaystyle \frac{1}{2}}}$? Earlier scholars could not provide an answer to this question. The research presented in this paper resolves the issue by demonstrating that the fractional derivative corresponding to the operator ${\mathit{p}}^{{\displaystyle \frac{1}{2}}}$ must be either the RL fractional derivative or the Caputo fractional derivative.

By comparing Equation (2) with Equation (57) once again, it becomes evident that in addition to the choice of the fractional derivative, there seems to be a missing term of Equation (57) that needs to be added. Compared with the question of which fractional derivative to choose, which this paper focuses on solving, determining the missing term is a relatively straightforward task. In fact, while establishing the operator representation of fractional calculus, we also addressed the issue of which correction term should be added here. Specifically, in this case, for zero initial strain, i.e., $\epsilon \left(0\right)=0$, ${\mathit{p}}^{{\displaystyle \frac{1}{2}}}$ is expressed as the half-order Caputo fractional derivative.

Thus, the final link in the logical chain is complete: the correspondence between the differential operator and fractional calculus cannot be arbitrarily chosen. The specific method of representation is as follows: first, drawing from the findings shown in Table 2, select the appropriate form of fractional calculus corresponding to the operator (i.e., choose the relevant row in Table 2); second, depending on the types of boundary conditions, select either the Caputo-type or RL-type representation (i.e., choose between the third and fourth columns of Table 2).

Unlike existing studies on the operator representation of fractional calculus [16,17,18,19,20], the method proposed in this paper is both universal and extensible. Its universality lies in the fact that the method is not restricted to a specific definition of fractional calculus but applies to a class of problems represented by the Sonine kernel, and of course, also includes solving fractional power operators abstracted from physical fractal spaces. In this paper, we provide several specific examples to demonstrate the solution process, leaving more definition forms to the reader and future research.

4.2. The Operator Method and Superposition Principle for Fractional Differential Equations

The operator representation of fractional calculus is an effective method for solving fractional differential equations. By transforming differential equations into operator algebraic equations, the solving process can be simplified. This approach is particularly valuable when dealing with fractional differential equations that have non-zero initial conditions. However, it is essential to note that the fractional derivatives cannot simply be replaced by the fractional powers of differential operators. In practical engineering problems, the initial or boundary conditions are often determined by the integer-order derivatives of functions, significantly reducing the complexity of solving such equations.

For example, consider a fractional differential equation in the form of the Caputo derivative:

$$a_1{D}^{{\gamma }_1}y(t)+a_2{D}^{{\gamma }_2}y(t)+\cdots +a_m{D}^{{\gamma }_m}y(t)=u(t),n\ge {\gamma }_1>{\gamma }_2\dots >{\gamma }_m.$$

(59)

The initial conditions are:

$$y(0)=c_0,y^{\prime }(0)=c_1,\dots ,y^{(m)}(0)=c_m.$$

(60)

The operator representation of the Caputo fractional derivative from Equation (33) is substituted into Equation (59), yielding:

$$\left(a_1{\mathit{p}}^{{\gamma }_1}+a_2{\mathit{p}}^{{\gamma }_2}+\cdots +a_m{\mathit{p}}^{{\gamma }_m}\right)y(t)=u(t)+u_0(t)$$

(61)

where $u_0\left(t\right)$ is:

$$u_0(t)={\displaystyle \sum _{i=1}^m{\displaystyle \sum _{j=0}^{⌈{\gamma }_i⌉-1}{\mathit{p}}^{{\gamma }_i-j-1}{y}^{(j)}}(0)}.$$

(62)

Simplifying, the operator form solution of the equation is obtained as:

$$y(t)={\displaystyle \frac{u(t)+u_0(t)}{\left(a_1{\mathit{p}}^{{\gamma }_1}+a_2{\mathit{p}}^{{\gamma }_2}+\cdots +a_m{\mathit{p}}^{{\gamma }_m}\right)}}.$$

(63)

Equation (63) formally provides the solution to the problem Equation (59). However, in the case where the exponents ${\mathsf{\gamma }}_i$ in the operator:

$$T(p)={\displaystyle \frac{1}{\left(a_1{\mathit{p}}^{{\gamma }_1}+a_2{\mathit{p}}^{{\gamma }_2}+\cdots +a_m{\mathit{p}}^{{\gamma }_m}\right)}}$$

(64)

are non-integers, it is not possible to directly perform a rational decomposition of this fractional expression. As a result, the operator kernel function method cannot be used to derive a functional solution from the operator form, requiring further development.

To address this, we propose a superposition principle that transforms the problem into one with zero initial conditions, thus bypassing the complexities of solving the initial value component directly. Specifically, we express the solution to the original problem as:

$$\mathit{T}(\mathit{p})\left[y_0(t)+y_1(t)\right]=u(t).$$

(65)

The initial conditions for $y_0\left(t\right)$ and $y_1\left(t\right)$ in the equation are given as:

$$y_0(0)=c_0,{y_0}^{\prime }(0)=c_1,\dots ,{y_0}^{(m)}=c_m,$$

(66)

$$y_1(0)={y_1}^{\prime }(0)=\dots ={y_1}^{(m)}=0.$$

(67)

For the integer-order initial value problem, let $y_0\left(t\right)$ be in polynomial form:

$$y_0(t)={\displaystyle \sum _{i=0}^m\frac{y^{(i)}(0)}{i!}}.$$

(68)

Substituting Equation (68) back into Equation (65), we obtain:

$$y_1(t)={\displaystyle \frac{1}{\mathit{T}(\mathit{p})}}u(t)-{\displaystyle \sum _n\frac{y^{(n)}(0)}{n!}}.$$

(69)

Thus, solving the function $y_1\left(t\right)$ with zero initial conditions is sufficient to obtain the solution to the original problem. In practice, finding $y_1\left(t\right)$ is simpler than directly solving the original equation for $y_0\left(t\right)$. The following example demonstrates the difference between the method outlined in this section and the direct application of the operator kernel function method discussed earlier.

For example, consider the Bagley–Torwik fractional differential equation:

$$A{y}^{″}(t)+B^C{D}^{{\displaystyle \frac{3}{2}}}y(t)+Cy(t)=C(t+1),y(0)=y^{\prime }(0)=1$$

(70)

where A, B, and C are constants.

First, we use the operator algebra method to solve the equation directly. For simplicity, let $A=0,B=C=1$, so the equation becomes:

$${}^{C}D^{{\displaystyle \frac{3}{2}}}y(t)+y(t)=t+1,y(0)=y^{\prime }(0)=1.$$

(71)

Expressed in the operator form:

$$\left({\mathit{p}}^{{\displaystyle \frac{3}{2}}}y(t)-{\mathit{p}}^{{\displaystyle \frac{1}{2}}}-{\mathit{p}}^{-{\displaystyle \frac{1}{2}}}\right)+y(t)=t+1.$$

(72)

After simplification, the operator form solution to the original problem is:

$$y(t)={\displaystyle \frac{t+1}{{\mathit{p}}^{\frac{3}{2}}+1}}+{\displaystyle \frac{{\mathit{p}}^{\frac{1}{2}}+{\mathit{p}}^{-\frac{1}{2}}}{{\mathit{p}}^{\frac{3}{2}}+1}}.$$

(73)

It is evident that using the direct method to solve presents difficulties in converting the result from Equation (73) into a functional form, even under the simplified conditions of $A=0,B=C=1$.

Next, we will solve the equation using the linear decomposition method presented in this section. For the Bagley–Torwik equation, based on the initial conditions in Equation (70), let $y_0=t+1$. Substituting this into Equation (65), we obtain:

$$\mathit{T}\left(\mathit{p})y_1\right(t)=C(t+1)-\mathit{T}\left(\mathit{p}\right)[y(t)+1]$$

(74)

where $\mathit{T}\left(\mathit{p}\right)=A{\mathit{p}}^2+B{\mathit{p}}^{{\displaystyle \frac{3}{2}}}+C$; therefore,

$$\mathit{T}\left(\mathit{p}\right)[y\left(t\right)+1]=C[y\left(t\right)+1].$$

(75)

Equation (74) becomes:

$$y_1(t)={\displaystyle \frac{1}{\mathit{T}(\mathit{p})}}\cdot 0.$$

(76)

The solution to Equation (76) is $y_1\equiv 0$. Therefore, the solution to the Bagley–Torwik equation is $y\left(t\right)=t+1$, independent of the values of A, B, and C.

This example illustrates that the fractional calculus operator theory proposed in this paper not only allows for the forward transformation of the fractional powers of the operators into fractional calculus functions but also enables the reverse application of the operator methods to solve fractional differential equations. However, this approach has limitations. Unlike integer-order operators, fractional-order operators often cannot be directly resolved using the operator kernel function method to obtain convolution kernels. To address this, we utilize the linear decomposition method, which transforms problems that are complicated by initial conditions into zero-initial-condition problems, allowing for a more straightforward application of operator algebra methods and avoiding the complexities of solving the operator kernel function for the initial value components.

5. Conclusions

This study introduces a new framework that utilizes Heaviside’s operational calculus to address the key issues related to fractional power operators in fractional calculus. Fractional integration is represented as the convolution of the integral kernel function $m\left(t\right)$ with the target function, utilizing the operator kernel method to obtain the operator representation of generalized fractional integrals.

For fractional differentiation, we categorized each differential kernel function $k\left(t\right)$ into Caputo-type and RL-type fractional derivatives by combining them with integer-order derivatives. Using the operator kernel method alongside the defining equations for integer-order derivatives and operators, we established the operator representation for generalized fractional derivatives. The primary expressions for both types are identical, differing only in the correction terms based on the initial conditions.

Ultimately, we addressed the core question posed in the introduction: the operator ${\mathit{p}}^{\alpha }$ can only correspond to either the Caputo or RL fractional derivative. Depending on the initial conditions, the operator acting on a function equals the associated fractional derivative plus a correction term related to these values. For the other operators, the appropriate choice can be made using Table 2.

Additionally, we proposed a superposition principle to handle fractional differential equations with non-integer exponents in the operator, transforming the problem into one with zero initial conditions, simplifying the solving process, and avoiding the complexities of the initial value component.

This research provides a unified method for representing fractional power operators, making a significant contribution to the understanding of these operators within fractional calculus and their practical applications, particularly in physical fractal spaces. Although the approach proposed in this chapter may not yet encompass all possible cases, it holds potential for future extension, expansion, and application.