The Fractional-Order Effect Induced by Space–Time Order Mismatch in Operator Differential Equations

Abstract

This paper reports an interesting phenomenon in which fractional-order effects can be induced by the mismatch of the differential orders of space and time; that is, fractional-order effects can be induced by space–time symmetry breakage. Classical mathematical equations can be transformed into differential equations of undetermined operators. We confirmed that the presence of fractional-order operator solutions in operator differential equations is contingent upon the mismatch of differential orders of space and time, which can induce both fractional operators in the time domain and fractional operators in the space domain. The introduction of symmetry breakage and operators of space and time offers novel insights into understanding nonlocal phenomena within the space–time continuum.

1. Introduction

This article aims to address the following fundamental questions: What is the origin of fractional-order operators? Which factors determine the order of fractional-order operators? What is the relationship between the space–time symmetry breakage and nonlocal effects?

Historically, fractional-order operators originated from the “brainstorming” of mathematicians. At that time, Leibniz gave a very wise but essentially meaningless reply [1,2]. This exchange between L’Hôpital and Leibniz illustrates that fractional-order operators were initially conceived as abstract mathematical constructs driven by curiosity and imagination. In other words, unlike integer-order operators, fractional-order operators did not originate from direct physical observations or practical applications.

From a mathematical perspective, the origin of fractional orders is merely a matter of logical definition. The only constraint in mathematics is logical consistency, and mathematics shows no interest in the physical reality of fractional orders at all. The existing multiple definitions of fractional calculus are based on the generalization of integer-order calculus through integration or differentiation [2,3] and do not originate from physical reality. Podlubny [4] explored the meaning of fractional calculus from a geometric perspective, while the physical or mechanical interpretation still awaits more in-depth exploration.

However, from a physics and mechanics perspective, elucidating the origin of fractional order is not an optional inquiry but an essential requirement. The rationale is straightforward: the fundamental equations in physics and mechanics are predominantly conservation laws expressed through operators, which are traditionally of integer order. The gradient operator $\nabla$, for instance, is a first-order differential operator; the Laplace operator ${\nabla }^2$ is a second-order differential operator. The biharmonic operator ${\nabla }^2{\nabla }^2$, on the other hand, is a fourth-order differential operator. The above operators are all of integer orders. Not only that, but all of the integer-order operators have both mathematical and physical meanings, and the factors that determine the order of an operator are both mathematical and physical. In other words, classical integer-order operators and their orders possess well-defined mathematical foundations and physical contexts. Nevertheless, in contrast to integer-order operators, fractional-order operators continue to pose several fundamental questions that remain unresolved. For example, despite the widespread application of fractional-order operators across various disciplines [5,6,7], these applications often appear superficial—researchers tend to introduce fractional-order operators phenomenologically without exploring their underlying physical essence. This scenario is unsurprising, given the lack of understanding of where fractional-order operators originate from and what factors determine their order.

Obviously, if the above-mentioned fundamental issues remain unsolved, the application of fractional-order operators will only remain at the phenomenological level, thereby greatly restricting the development of many disciplines. However, the progress made in previous research [8] has given us a chance to break the deadlock.

In our previous work [8], an explanation was provided regarding the origin and order of fractional derivatives—fractal operators in the physical fractal space are of the fractional order. We abstracted the physical fractal space from the biological structure and the movements within living organisms and found that, within a physical fractal space, a fractional-order time derivative operator must necessarily exist. Given its presence in a fractal space, this operator was termed a “fractal operator” [8]. It has been established that the fractional order of the fractal operator is mainly contingent upon two factors: firstly, the topological index of the physical fractal space, and secondly, the integer order of the component operator within this space.

Recently, we found that the origin of fractional derivatives is not singular, and the factors determining the fractional order are diverse. From the perspective of space–time symmetry breakage, a new explanation for the origin of fractional orders can also be provided, and from this, an understanding of the space–time duality of operators can be obtained. Fractional calculus and fractional operators are frequently used to handle explicit space–time problems [9,10,11], but the induction of the fractional orders by the characteristics of space–time is a novel conclusion.

This paper confirms that, in classical mathematical physics equations, fractional operators also exist, encompassing fractional operators of both time and space. The order of these fractional operators is contingent upon a specific form of the space–time symmetry breakage: the discrepancy or mismatch between the differential orders of time and space derivatives, that is, the mismatch of time and space orders, within integer-order differential equations. The mismatch of differential orders in space and time can induce fractional-order effects, that is, the mismatch of space–time symmetry can induce fractional-order operators.

Owing to the inherent connection between fractional operators and nonlocal problems, this research offers a novel and comprehensive perspective for a deeper understanding of nonlocal phenomena. Not only that, the mismatch of time and space differential orders is one of the forms of symmetry breakage in physics and mechanics. Therefore, in-depth study of the mismatch of time and space orders could open up new avenues for us to understand the effect of symmetry breakage.

The differences in the differential orders, as well as the dimensions, of time and space can all be regarded as space–time symmetry breakages. This paper primarily focuses on the differences in the orders of differentiation.

2. Operator Differential Equations Induced by the Heat Conduction Equation

To reveal the fractional-order effect induced by the order mismatch, this paper introduces the operator differential equation. When investigating the telegraph equation, Heaviside formulated operational calculus [12]. Notably, within this framework, a time fractional-order operator, specifically, the 1/2-order derivative with respect to time, emerged. Historically, not only did Heaviside invent the operational calculus, but he also uncovered the profound intrinsic relationship between fractional-order operators and physical phenomena.

Courant and Hilbert [13] commended Heaviside’s operational calculus method in high regard. While systematizing the boundary value problems associated with the heat conduction differential equation, they demonstrated the power of the operational calculus method and extracted the operator differential equation pertinent to the boundary value problem of heat conduction. In the treatise “Operational Calculus” [14], Mikusiński analyzed both the wave equation and the heat conduction equation in detail. Mikusiński transformed the functions into the operator domain for solving the equations, and also formalized the operator differential equations.

Based on Mikusiński’s operator theory, the structure and operation of fractional-order operators have been further discussed [15,16,17]. Yu et al. [18] discussed the method for obtaining the kernel functions, and, then, Jiang et al. [19] introduced abstract algebra and demonstrated the rationality of obtaining the kernel function by inverse Laplace transform, from the perspective of homomorphism mapping. All of this progress constitutes the logical basis of our recent work, including this paper.

We have observed that the operator differential equation is a concept of universal significance, applicable not only to boundary value problems but also to initial value problems. Through the use of operator differential equations, we can rigorously derive the conditions for the existence of fractional operators. For the purpose of simplifying the problem, this paper focuses on the homogeneous heat conduction differential equation and its corresponding operator differential equation.

3. Operator Formulation of the Homogeneous Heat Conduction Equation

The homogeneous heat conduction equation on a one-dimensional semi-infinite interval is

$${\displaystyle \frac{\partial u}{\partial t}}-a^2{\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}=0\hspace{1em}0\le x<+\infty .$$

(1)

For the convenience of analysis, imitating Heaviside, we define the time differential operator p and the space differential operator D_x as follows:

$$p={\displaystyle \frac{\partial }{\partial t}}, D_x^2={\displaystyle \frac{{\partial }^2}{\partial x^2}}.$$

(2)

Then, the homogeneous differential Equation (1) is transformed into

$$\left(p-a^2{D}_x^2\right)u\left(x,t\right)=0.$$

(3)

The following sections focus on the boundary value problem and initial value problem corresponding to Equation (3), and induce the boundary value problem and initial value problem of operator differential equations from the aforementioned problems.

4. The Boundary Value Problem of the Operator Differential Equation

In their seminal work “Methods of Mathematical Physics” [13], Courant and Hilbert explored the boundary value problem associated with the homogeneous heat conduction equation, thereby elucidating the foundational concepts of operator differential equations. Their approach was rooted in Heaviside’s operational calculus, and they emphasized its substantial applicability to transient problems.

On the one-dimensional semi-infinite domain, Equation (3) satisfies the upper boundary condition:

$$\begin{gathered} \left(p-a^2{D}_x^2\right)u\left(x,t\right)=0\hspace{1em}0\le x<+\infty \\ u\left(0,t\right)=f\left(t\right), u\left(\infty ,t\right)=0 \end{gathered}$$

(4)

Unlike Heaviside’s direct solution, Courant and Hilbert [13] posited the existence of an undetermined operator T depending on the coordinate x, i.e., $T=T\left(x,p\right)$. Consequently, the solution to Equation (4) can be represented in the following form:

$$u\left(x,t\right)=T\left(x,p\right)u\left(0,t\right)=T\left(x,p\right)f\left(t\right).$$

(5)

We can interpret Courant and Hilbert’s foundational idea as follows: Equation (5) represents an extension of the classical method of separation of variables, which may be referred to as the generalized method of separation of variables. The classical method of separation of variables decomposes the bivariate function $u\left(x,t\right)$ into the product of two univariate functions. In contrast, the generalized method of separation of variables (as shown in Equation (5)) decomposes the bivariate function $u\left(x,t\right)$ into the product of the univariate operator $T\left(x,p\right)$ and the univariate function $f\left(t\right)$.

In Equation (5), $T\left(x,p\right)$ represents an undetermined operator on a spatial field. $f\left(t\right)$ denotes a known boundary condition, which can be considered as a predetermined input signal. Upon transformation of the input signal $f\left(t\right)$ by the undetermined operator $T\left(x,p\right)$, the resultant output signal $u\left(x,t\right)$ is produced.

The output $u\left(x,t\right)$ signal satisfies the partial differential Equation (3). By substituting Equation (5) into Equation (3), we derive the differential equation that the operator $T\left(x,p\right)$ satisfies:

$$\left(p-a^2{D}_x^2\right)T\left(x,p\right)=0\hspace{1em}0\le x<+\infty .$$

(6)

Equation (6) is the operator differential equation, where $p=\partial /\partial t$ may be considered as a parameter. Therefore, Equation (6) is a second-order ordinary differential equation of the undetermined operator $T\left(x,p\right)$.

It is noted that the differential Equation (6) of operator $T\left(x,p\right)$ exhibits an identical apparent form to the homogeneous differential Equation (3) of function $u\left(x,t\right)$. We hypothesize that this phenomenon should be ubiquitous, suggesting that within boundary value problems, there invariably exists an operator differential equation, which is formally identical to the homogeneous differential Equation (3), and the undetermined operator of the operator differential equation is $T\left(x,p\right)$.

Despite their similar outward appearance, Equations (3) and (6) possess fundamental differences. Specifically, while Equation (3) is a partial differential equation, Equation (6) is an ordinary differential equation. That is to say, following the separation of variables in Equation (5), the partial differential Equation (3) has been transformed into the ordinary differential Equation (6). Not only this, but the 0 on the right side of Equation (3) is the numerical value 0; while the 0 on the right side of Equation (6) is the 0 operator.

The undetermined operator $T\left(x,p\right)$ satisfies the boundary value condition (Equation (4)):

$$T\left(0,p\right)=1, T\left(\infty ,p\right)=0.$$

(7)

The general solution to the second-order ordinary differential Equation (6) is

$$T\left(x,p\right)=c_1{e}^{{\displaystyle \frac{x}{a}}\sqrt{p}}+c_2{e}^{-{\displaystyle \frac{x}{a}}\sqrt{p}}.$$

(8)

Under the constraint of boundary condition (7), the solution given by Equation (8) is

$$T\left(x,p\right)=e^{-{\displaystyle \frac{x}{a}}\sqrt{p}}.$$

(9)

It is evident that operator $T\left(x,p\right)$ varies with the coordinate. It is an exponentially decaying operator in space that incorporates a 1/2-order derivative $\sqrt{p}$.

Equation (9) raises a fundamental question: Is operator $T\left(x,p\right)$ itself a fractional-order operator? The answer is yes. Equation (9) is expanded into an operator series:

$$\begin{array}{ll}T\left(x,p\right)& =e^{-{\displaystyle \frac{x}{a}}\sqrt{p}}={{\displaystyle \sum _{n=0}^{\infty }\frac{1}{n!}\left(-\frac{x}{a}\sqrt{p}\right)}}^n\\ & =1-\left({\displaystyle \frac{x}{a}}\sqrt{p}\right)+{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{x}{a}}\sqrt{p}\right)}^2-\dots \dots \end{array}.$$

(10)

Equation (10) is an alternating series. The first term of the alternating series is the identity operator 1. $-\left({\displaystyle \frac{x}{a}}\sqrt{p}\right)$, the second term of this series, is a fractional operator of 1/2 order. Consequently, $T\left(x,p\right)$ is confirmed to be a fractional operator as well. However, $T\left(x,p\right)$ is not merely a simple 1/2-order fractional operator but rather a power series operator involving a 1/2-order derivative.

This result provides a significant insight: even the most fundamental heat conduction and diffusion equations exhibit fractional effects in the time domain.

Yu et al. [18] demonstrated that operator $T\left(x,p\right)$ presented in Equation (9), possesses a kernel function $K\left(x,t\right)$, which means

$$T\left(x,p\right)f\left(t\right)={\displaystyle \underset{0}{\overset{t}{\int }}K\left(x,t-\tau \right)}f\left(\tau \right)\mathrm{d}\tau ,$$

(11)

where

$$\begin{array}{ll}K\left(x,t\right)& =T\left(x,p\right)\delta \left(t\right)={\mathcal{L}}^{-1}\left[T\left(x,s\right)\right]={\mathcal{L}}^{-1}\left[e^{-{\displaystyle \frac{x}{a}}\sqrt{s}}\right]\\ & ={\displaystyle \frac{x}{2a\sqrt{\pi t^3}}}{e}^{-{\displaystyle \frac{x^2}{4a^{2}t}}}\end{array}.$$

(12)

In Equation (12), $\delta \left(t\right)$ denotes the Dirac pulse function, and ${\mathcal{L}}^{-1}\left(\cdot \right)$ signifies the inverse Laplace transform. Equation (12) demonstrates that the fractional effect of the operator $T\left(x,p\right)$ is ultimately manifested as a 3/2-order singularity within the kernel function $K\left(x,t\right)$.

From Equations (5) and (12), we can obtain the solution to the differential Equation (4):

$$\begin{array}{ll}u\left(x,t\right)& =T\left(x,p\right)f\left(t\right)={\displaystyle \underset{0}{\overset{t}{\int }}K\left(x,t-\tau \right)}f\left(\tau \right)\mathrm{d}\tau \\ & ={\displaystyle \underset{0}{\overset{t}{\int }}\frac{x}{2a\sqrt{\pi {\left(t-\tau \right)}^3}}{e}^{-\frac{x^2}{4a^2\left(t-\tau \right)}}}f\left(\tau \right)\mathrm{d}\tau \end{array}.$$

(13)

Equation (13) is completely consistent with the classical solution. It is the singularity of the kernel function $K\left(t\right)$ that leads to the singular integral in Equation (11).

At the end of this section, we will examine the 1/2 order of the derivative $\sqrt{p}$ in Equation (9). It is important to note that in the differential Equation (5) of operator $T\left(x,p\right)$, the order of the operator p is m = 1, while the order of the operator $D_x^2$ is n = 2. We hypothesize that the fractional order of 1/2 for $\sqrt{p}$ may originate from the ratio m/n = 1/2. In other words, the mismatch in the orders of space and time derivatives could be the underlying cause of the fractional order. This hypothesis will be further investigated and demonstrated in subsequent sections.

5. The Initial Value Problem of the Operator Differential Equation

To validate the aforementioned hypothesis, we further investigate the initial value problem associated with the heat conduction equation.

We continue to consider the homogeneous heat conduction equation on a one-dimensional infinite interval, as given by Equation (3). We then associate the initial conditions with Equation (3):

$$\begin{gathered} \left(p-a^2{D}_x^2\right)u\left(x,t\right)=0\hspace{1em}-\infty \le x<+\infty , \\ u\left(x,0\right)=g\left(x\right) \end{gathered}$$

(14)

For the initial value problem of the homogeneous heat conduction equation, Chinese scholar Ke Honglu [20] proposed a solution method utilizing operator series. In this paper, Ke’s operator series method is regarded as a reference. For the convenience of the comparison, we still apply the operator differential equation method discussed in the preceding section.

We observe that the operator differential equation method introduced by Courant and Hilbert [13] possesses a broad applicability, extending its utility from boundary value problems to initial value problems. Consequently, paralleling the thought of Courant and Hilbert, we assume the existence of an undetermined operator T depending on time t, i.e., $T=T\left(t,D_x\right)$, which allows the solution to the initial value problem (Equation (14)) to be formulated as follows:

$$u\left(x,t\right)=T\left(t,D_x\right)u\left(x,0\right)=T\left(t,D_x\right)g\left(x\right)$$

(15)

Substituting Equation (15) into Equation (14), we can obtain the differential equation of the undetermined operator $T\left(t,D_x\right)$:

$$\left(p-a^2{D}_x^2\right)T\left(t,D_x\right)=0$$

(16)

In Equation (16), $D_x^2={\partial }^2/\partial x^2$ can be treated as a parameter. Equation (16) is a first-order ordinary differential equation for the undetermined operator $T\left(t,D_x\right)$. That is, through variable separation (Equation (15)), the second-order partial differential Equation (3) is transformed into a first-order ordinary differential Equation (16) for the operator $T\left(t,D_x\right)$.

It is observed that the differential Equation (16) of the operator $T\left(t,D_x\right)$ exhibits complete formal consistency with the homogeneous differential Equation (3). We hypothesize that this phenomenon is universal, meaning that in initial value problems, there always exists an operator differential equation formally identical to the homogeneous differential Equation (3), where the undetermined operator is $T\left(t,D_x\right)$. It should be noted that a similar phenomenon has been observed in boundary value problems; see Equation (6) for details.

From Equations (14) and (15), it can be seen that the undetermined operator $T\left(t,D_x\right)$ satisfies the following initial value condition:

$$T\left(0,D_x\right)=1$$

(17)

The general solution to Equation (16) is

$$T\left(t,D_x\right)=c{e}^{a^{2}t{D}_x^2}$$

(18)

The solution satisfying the initial value condition (17) is

$$T\left(t,D_x\right)=e^{a^{2}t{D}_x^2}$$

(19)

This result is the same as that presented in Ke’s operator series method [20], and demonstrates that the operator $T\left(t,D_x\right)$ undergoes continuous evolution over time. It is an exponential operator that incorporates integer-order spatial derivative $D_x^2$.

Interestingly, operator $T\left(t,D_x\right)$ does not exhibit fractional-order effects in the spatial domain. This observation highlights a significant distinction between operators $T\left(t,D_x\right)$ and $T\left(x,p\right)$: despite their differential equations (Equations (6) and (16)) having identical forms and both displaying a space–time order mismatch, operator $T\left(x,p\right)$ (Equation (9)) manifests fractional-order effects, whereas operator $T\left(t,D_x\right)$ (Equation (19)) does not. How can this discrepancy be interpreted?

We can interpret this in the following way. In the differential Equation (16) of operator $T\left(x,p\right)$, the order of operator p is m = 1, while that of operator $D_x^2$ is n = 2. We hypothesize that the order 2 of $D_x^2$ in $T\left(x,p\right)$ arises from the ratio n/m = 2/1 = 2. Consequently, despite the mismatch in the differential orders of space and time in the initial value problem, the fractional-order effect is expected to diminish. This hypothesis will be rigorously examined in subsequent sections.

Operator $T\left(t,D_x\right)$ possesses a kernel function $K\left(t,x\right)$, which means

$$T\left(t,D_x\right)g\left(x\right)={\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}K\left(t,x-\xi \right)}g\left(\xi \right)\mathrm{d}\xi$$

(20)

where

$$\begin{array}{ll}K\left(t,x\right)& =T\left(t,D_x\right)\delta \left(x\right)={\mathcal{F}}^{-1}\left[T\left(t,is\right)\right]={\mathcal{F}}^{-1}\left(e^{-a^{2}t{s}^2}\right)\\ & ={\displaystyle \frac{1}{2a\sqrt{\pi t}}}{e}^{-{\displaystyle \frac{x^2}{4a^{2}t}}}\end{array}$$

(21)

In Equation (21), ${\mathcal{F}}^{-1}\left(\cdot \right)$ denotes the inverse Fourier transform.

From Equations (15) and (21) we can obtain the solution to the initial value problem of the following differential equation:

$$\begin{array}{ll}u\left(x,t\right)& =T\left(t,D_x\right)g\left(x\right)={\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}K\left(t,x-\xi \right)}g\left(\xi \right)\mathrm{d}\xi \\ & ={\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}\frac{1}{2a\sqrt{\pi t}}{e}^{-\frac{{\left(x-\xi \right)}^2}{4a^{2}t}}}g\left(\xi \right)\mathrm{d}\xi \end{array}$$

(22)

Equation (22) is precisely the solution to the classical initial value problem. It is noted that although the operator $T\left(t,D_x\right)$ is an integer-order operator, singular integrals still appear in Equation (22).

It can be seen that the solution process of the initial value problem in this section is completely symmetrical to that of the boundary value problem in Section 4. The symmetry stems from the consistency of the apparent forms of the operator differential equations (Equations (6) and (16)). However, as shown in

Table 1. Comparison of the initial value problem and boundary value problem of the heat conduction equation.

	Boundary Value Problem (Section 4)	Initial Value Problem (Section 5)
Generalized Variable Separation	$u\left(x,t\right)=T\left(x,p\right)f\left(t\right)$	$u\left(x,t\right)=T\left(t,D_x\right)g\left(x\right)$
Operator Differential Equation	$\left(p-a^2{D}_x^2\right)T\left(x,p\right)=0$	$\left(p-a^2{D}_x^2\right)T\left(t,D_x\right)=0$
Operator Solution	$T\left(x,p\right)=e^{-{\displaystyle \frac{x}{a}}\sqrt{p}}$	$T\left(t,D_x\right)=e^{a^{2}t{D}_x^2}$
Exponent of the Fundamental Operator (p or Dx) in the Operator Solution	1/2	2
Fractional Order	Exist	Not exist

, there are differences between the boundary value problem and the initial value problem in terms of the fractional-order effect. This difference is also a phenomenon of symmetry breakage, which is inferred to result from the different orders of spatial and temporal derivatives: Equation (6) is a second-order ordinary differential equation in the space domain, while Equation (16) is a first-order ordinary differential equation in the time domain.

6. The General Proposition of the Fractional-Order Operator

To elucidate the aforementioned differences between operator $T\left(t,D_x\right)$ and operator $T\left(x,p\right)$, we attempt to examine the operator differential equation from a more general perspective. For this purpose, we fabricate a homogeneous differential equation on a one-dimensional infinite interval, as follows:

$${\displaystyle \frac{{\partial }^{m}u}{\partial t^m}}-a^n{\displaystyle \frac{{\partial }^{n}u}{\partial x^n}}=0,$$

(23)

where m and n are both positive integers. When m = 1 and n = 2, Equation (23) reduces to the heat conduction equation or the diffusion equation. It is quite obvious that when $m>2$ or $n>2$, such an equation is entirely a product of fiction or imagination and not a mathematical–physical reality.

However, when $m,n\le 2$, Equation (23) degenerates into several well-known basic differential equations. When m = 1 and n = 2, we can obtain the classic parabolic heat conduction equation or diffusion equation; when m = 2 and n = 2, it is the well-known hyperbolic wave equation. Thanks to the arbitrariness of the variables, we can even substitute the time parameter t with the spatial parameter y into the form. At this time, taking m = 2 and n = 2, a = ki, the obtained result is the elliptic harmonic equation.

According to the basic classification method of mathematical physics equations, arbitrary second-order linear homogeneous partial differential equation with constant coefficients can always be transformed into the above three forms through regular conformal transformation. Therefore, although Equation (23) is a fictitious or imagined product in a certain sense, it is also a result of generalization on mature mathematical physics equations, with clear physical significance in various degenerate cases. Objectively, it is the result of the reflection of physical laws. Physically, the conclusion of this section is applicable to all kinds of linear differential equations with the derivative orders of space and time less than or equal to 2.

In general, when $m>2$ or $n>2$, the aforementioned hypothetical equation may appear devoid of practical significance and unworthy of in-depth study. However, the objective of this section is to elucidate the distinctions between operator $T\left(t,D_x\right)$ and operator $T\left(x,p\right)$, and examine the influencing factors of fractional order. Therefore, if the equation can qualitatively illuminate the essence of the issue at hand, they acquire considerable value for illustrative purposes.

By introducing operators p and D_x, the hypothetical homogeneous differential Equation (23) is transformed into

$$\left(p^m-a^n{D}_x^n\right)u\left(x,t\right)=0.$$

(24)

The phenomenon of “apparent form consistency” observed in the aforementioned sections, given its universality, can be elevated to the status of a postulate. We refer to this as the “Postulate of Apparent Form Consistency”. The postulate asserts that for any linear homogeneous partial differential equation, there invariably exists an operator differential equation that exhibits complete formal congruence with it.

Based on the aforementioned postulate, it can be inferred that there invariably exists an operator differential equation that is formally identical to the homogeneous Equation (24). These operator differential equations can be categorized into two types. The first type is the operator differential equation of the boundary value problem, where the undetermined operator is $T\left(x,p\right)$:

$$\left(p^m-a^n{D}_x^n\right)T\left(x,p\right)=0.$$

(25)

We need a qualitative proof that the solution of the unknown operator in Equation (25) possesses a fractional order.

If n is an even number, then Equation (25) admits at least hyperbolic sine and cosine solutions:

$$T\left(x,p\right)=\sinh \left({\displaystyle \frac{x}{a}}{p}^{{\displaystyle \frac{m}{n}}}\right), T\left(x,p\right)=\cosh \left({\displaystyle \frac{x}{a}}{p}^{{\displaystyle \frac{m}{n}}}\right).$$

(26)

If n is an odd number, then Equation (25) admits at least the exponential solution:

$$T\left(x,p\right)=e^{{\displaystyle \frac{x}{a}}{p}^{{\displaystyle \frac{m}{n}}}}.$$

(27)

The second type is the operator differential equation of the initial value problem, where the undetermined operator is $T=T\left(t,D_x\right)$:

$$\left(p^m-a^n{D}_x^n\right)T\left(t,D_x\right)=0.$$

(28)

If n is an even number, then Equation (28) admits at least hyperbolic sine and cosine solutions:

$$T\left(t,D_x\right)=\sinh \left(a^{{\displaystyle \frac{n}{m}}}t{D}_x^{{\displaystyle \frac{n}{m}}}\right), T\left(t,D_x\right)=\cosh \left(a^{{\displaystyle \frac{n}{m}}}t{D}_x^{{\displaystyle \frac{n}{m}}}\right).$$

(29)

If n is an odd number, then Equation (28) admits at least the exponential solution:

$$T\left(t,D_x\right)=e^{a^{{\displaystyle \frac{n}{m}}}t{D}_x^{{\displaystyle \frac{n}{m}}}}.$$

(30)

Obviously, the operator $T\left(x,p\right)$ consistently includes the term $p^{m/n}$, while the operator $T\left(t,D_x\right)$ consistently incorporates the term $D_x^{n/m}$. When $m\ne n$, and m and n are incommensurable, both $p^{m/n}$ and $D_x^{n/m}$ become fractional-order operators. That is to say, under general circumstances, $T\left(x,p\right)$ and $T\left(t,D_x\right)$ are both fractional-order operators. Of course, there are exceptional cases, as discussed in Section 5. Specifically, in the heat conduction equation, when m = 1 and n = 2, the operator $T\left(t,D_x\right)$ in the initial value problem no longer exhibits fractional-order behavior.

We can formulate a general proposition as follows: in partial differential equations, the discrepancy between the order m of the time derivative and the order n of the spatial derivative leads to the emergence of fractional-order operators. The order of these fractional-order operators is determined by the ratio m/n or n/m. Suppose that the order of time derivative m is equal to that of space derivative n, that is, the time and space are completely symmetrical and the differential orders are completely matched; then, the fractional-order effects in both the boundary value problem and the initial value problem will completely disappear.

It is both unexpected and profound that fractional operators are hidden within this classical integer-order partial differential equation.

We can consider the homogeneous wave Equation (31), which corresponds to the degenerate case of Equation (23) when m = n = 2:

$${\displaystyle \frac{{\partial }^{2}u}{\partial t^2}}-a^2{\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}=0.$$

(31)

Consider the following initial value condition:

$$u\left(x,0\right)=\phi \left(x\right).$$

(32)

Introducing the operators p and $D_x$, separating variables according to (15), and following the solution process in Section 5, we can obtain the operator solutions

$$T\left(t,D_x\right)=c_1\cosh \left(at{D}_x\right)+c_2{\displaystyle \frac{1}{a{D}_x}}\sinh \left(at{D}_x\right).$$

(33)

By following and performing a series expansion similar to Equation (10) on the two terms in Equation (33), it can be concluded that the operator in Equation (33) is indeed not of fractional orders.

The wave equation and the heat conduction equation form a pair of contrasts, providing evidence for the conclusion that the time–space mismatch can induce fractional operators.

Fractional operators provide a new perspective for classifying second-order linear partial differential equations. Classical textbooks on mathematical physics analogize the standard form of partial differential equations with that of quadratic curves, and classify second-order partial differential equations into hyperbolic, elliptic, and parabolic types. Although this classification method is intuitive, it does not deepen our understanding of the physical essence. Now, from the perspective of fractional orders, we can re-examine the classification criteria. The following classification criteria should be more essential:

(a)

Are the differential orders of space and time mismatched?

(b)

If there is a mismatch, can the ratio of differential orders of space and time, as the exponent of the fundamental operator, be reduced to an integer?

If there is a mismatch and the exponent cannot be reduced to an integer, then the partial differential equation implies fractional orders; if there is a mismatch but the exponent can be reduced to an integer, then the fractional effect in the partial differential equation disappears. Specifically, in the classical heat conduction equation, for boundary value problems, the exponent of the fundamental operator in the operator solution is m/n = 1/2, where the spatial derivative order n = 2 cannot be divided by the time derivative order m = 1, thus inducing fractional orders. For initial value problems, the exponent of the basic operator in the operator solution is n/m = 2/1, where the time derivative order m = 1 can be divided by the spatial derivative order n = 2; thus, no fractional orders are induced.

Section 4 and Section 5 are symmetrical, and the boundary value and initial value problems in this section also exhibit symmetry, as shown in

Table 2. Comparison of the initial value problem and boundary value problem of the imaginary equation.

	Boundary Value Problem	Initial Value Problem
Generalized Variable Separation	$u\left(x,t\right)=T\left(x,p\right)f\left(t\right)$	$u\left(x,t\right)=T\left(t,D_x\right)g\left(x\right)$
Operator Differential Equation	$\left(p^m-a^n{D}_x^n\right)T\left(x,p\right)=0$	$\left(p^m-a^n{D}_x^n\right)T\left(t,D_x\right)=0$
Exponent of the Fundamental Operator (p or Dx) in the Operator Solution	${\displaystyle \frac{m}{n}}$	${\displaystyle \frac{n}{m}}$
Numerator of the Exponent	Time derivative order m	Space derivative order n
Denominator of the Exponent	Space derivative order n	Time derivative order m
Condition for Generating Fractional Orders	n is not divisible by m	m is not divisible by n

. The symmetry breakage caused by the mismatch of the differential orders in space and time leads to the difference between the presence and absence of the fractional effect.

Historically, physicists have attempted to unify general relativity and quantum mechanics, but have encountered significant difficulties. One of the root causes of these difficulties lies in the mismatch of the differential orders of space and time—the orders of space and time in general relativity and quantum mechanics are not unified. It is evident that the mismatch of space and time orders, as a manifestation of symmetry breakage, has far-reaching implications. Now, we can say that the symmetry breakage of space and time orders is one of the sources of fractional orders. And the degree of symmetry breakage (ratio m/n and n/m) determines the order of the fractional orders.

7. Generalization in the High-Dimensional Case

The conclusion in the previous section can also be naturally extended to the high-dimensional case. Taking the three-dimensional case as an example, Equation (23) is written as

$${\displaystyle \frac{{\partial }^{m}u}{\partial t^m}}-a^n\left({\displaystyle \frac{{\partial }^{n}u}{\partial x^n}}+{\displaystyle \frac{{\partial }^{n}u}{\partial y^n}}+{\displaystyle \frac{{\partial }^{n}u}{\partial z^n}}\right)=0.$$

(34)

Equation (34) can also be concisely expressed by using the Nabla operator as

$${\displaystyle \frac{{\partial }^{m}u}{\partial t^m}}-a^n{\nabla }^{n}u=0.$$

(35)

By introducing the operator p, Equation (35) is then changed into

$$\left(p^m-a^n{\nabla }^n\right)u\left(x,t\right)=0.$$

(36)

By comparing Equation (36) with Equation (24), apart from the substitution of the Nabla operator $\nabla$ for $D_x$, there is no difference between the two equations. Therefore, through the derivation process completely consistent with Equations (25)–(30), there is no difficulty in finding that the operator $T\left(x,p\right)$ always contains the term $p^{m/n}$, and the operator $T\left(t,\nabla \right)$ always contains the term ${\nabla }^{n/m}$. Generally, $T\left(x,p\right)$ and $T\left(t,\nabla \right)$ are both fractional-order operators, and the discussion in the one-dimensional case can be completely extended to the high-dimensional case.

In the three-dimensional flat space, the Nabla operator $\nabla$ holds a position equivalent to that of the operator $D_x$ of the one-dimensional case. According to the conclusion of tensor analysis, the Nabla operator $\nabla$ is an invariant entity and satisfies coordinate independence. Therefore, in flat space, the basic position of the Nabla operator cannot be replaced by others. However, in curved space, there exists the second type of invariant operators [21], whose operator algebraic properties demand further research.

For the wave Equation (31), when extended to higher dimensions, it can lead to some more profound and interesting conclusions, apart from the still valid conclusion in the last section. These will be demonstrated in our subsequent research.

For nonlinear equations, we can linearize them by regarding the general solution as the background solution plus small perturbations, and then apply the conclusion of the previous section. Therefore, it is reasonable to infer that the fractional-order effects induced by the mismatch of differential orders in space and time will also occur in nonlinear equations.

8. Operator Algebraic Equations and Operator Differential Equations in Mechanics

In classical mechanics, the unknowns are mostly unknown numbers and unknown functions, satisfying algebraic and differential equations, respectively. The conservation law equations in mechanics are mostly expressed as algebraic equations and differential equations. Therefore, it can be said that the objects to be solved in mechanics are mostly algebraic equations with unknown numbers or differential equations with unknown functions.

However, the research objects of a discipline are not immutable. The progress in our previous article [8] and this article has expanded the new boundaries of mechanics—the transformation of the objects to be solved. Our previous article [8] and this article show that the unknowns in mechanics can also be unknown operators. The description of the conservation law equations in mechanics with unknown operators can be expressed as the algebraic equations of operators, as in the previous article [8], or the differential equations of operators, as in this article. Therefore, it can be said that the objects to be solved in mechanics can also be algebraic equations or differential equations with unknown operators.

The impact of the transformation of the objects to be solved should not be underestimated. It is difficult for the algebraic equations and differential equations in classical mechanics to show fractional-order effects. However, once the unknowns in algebraic and differential equations are transformed into operators, the study of fractional-order effects can be carried out smoothly. More importantly, once the influencing factors of fractional order are clarified, this can conveniently guide the design and purposefully construct fractional-order systems, and achieve fractional-order control more accurately.

9. Conclusions

Even classical integer-order differential equations can lead to fractional-order operators. Given that fractional-order operators are potent tools for characterizing nonlocal phenomena, it is reasonable to infer that differential equations formulated based on localized mathematical and physical principles may also effectively describe nonlocal phenomena.

Based on the analytical structures of fractional operators, nonlocal phenomena can be categorized into two distinct types: spatial nonlocality and temporal nonlocality. The spatial nonlocality effect is characterized by the fractional operator $T\left(x,p\right)$, whereas temporal nonlocality is described by the fractional operator $T\left(t,D_x\right)$. It could be inferred that if fractional-order effects exist in both the time and the spatial domain, it is likely that they need to be characterized by the fractional-order operator $T\left(x,t,p,D_x\right)$.

The symmetry breakage caused by the mismatch of the order of time and space derivatives in differential equations leads to the emergence of fractional operators, giving rise to nonlocal phenomena. It is important to note that under appropriate constraints, the fractional effects brought about by this mismatch can be offset, which is the reason why no fractional effects occur in the initial value problem of the heat conduction equation. Thus, in this paper, symmetry breakage is a necessary condition for the generation of fractional operators, but not a sufficient condition. This conclusion will be further confirmed in our subsequent articles.

Equations of undetermined operators can be categorized into two types: algebraic operator equations and differential operator equations. Previous research [8] has established that algebraic operator equations can lead to fractional-order operators. This study further confirms that differential operator equations also have the potential to induce fractional-order operators. Consequently, it is reasonable to assert that a wide range of operator equations can lead to fractional-order operators. This finding underscores the objective existence and ubiquity of fractional-order operators in mechanics. The emergence of fractional-order operators is the result of symmetry breakage. It is anticipated that the employment of fractional-order operators in the study of mechanics and effects of symmetry breakage will constitute an effective approach.