Maximum Principle for the Space-Time Fractional Conformable Differential System Involving the Fractional Laplace Operator

In this paper, the authors consider a IBVP for the time-space fractional PDE with the fractional conformable derivative and the fractional Laplace operator. A fractional conformable extremum principle is presented and proved. Based on the extremum principle, a maximum principle for the fractional conformable Laplace system is established. Furthermore, the maximum principle is applied to the linear space-time fractional Laplace conformable differential system to obtain a new comparison theorem. Besides that, the uniqueness and continuous dependence of the solution of the above system are also proved.


Introduction
Many fractional partial differential equations were used for modeling complex dynamic systems of engineering, physics, biology, and many other fields [1][2][3][4]. As a significant tool, the maximum principle plays an important role in the study of the complex dynamic systems without certain knowledge of the solutions [5][6][7][8][9][10][11][12][13]. In 2016, by using the maximum principle, Luchko and Yamamoto [14] obtained the uniqueness of both the strong and the weak solutions of the IBVP for a general time-fractional distributed order diffusion equation. In 2016, Jia and Li [15] applied the maximum principle to the classical solution and weak solution of a time-space fractional diffusion equation. Furthermore, they also deduced the maximum principle for a full fractional diffusion equation other than time-fractional and spatialinteger order diffusion equations. In 2019, Wang et al. [16] investigated the IBVP for Hadamard fractional differential equations with fractional Laplace operator (− Δ) β by using the maximum principle.
ere are diverse fractional derivatives, such as the Riemann-Liouville derivative, the Caputo fractional derivative, the left and right conformable derivatives, and other fractional derivatives . In 2015, Abdeljawad [34] defined the left and right conformable derivatives.
Depending on [34], Jarad et al. [35] introduced the fractional conformable derivatives and presented the fractional conformable derivative in the sense of Caputo. e extremum principle of the Caputo fractional conformable derivative is seldom regarded in the existing literature. In addition, the papers which mentioned the fractional conformable derivative do not include the fractional Laplace operator.
Motivated by the above works, in this context, the authors investigate the IBVP for a space-time Caputo fractional conformable diffusion system with the fractional Laplace operator. First, we provide a detailed proof of the Caputo fractional conformable extremum principle. en, the new maximum principle is obtained by applying the extreme principle. As some applications of the maximum principle, a comparison principle for the space-time fractional Laplace conformable differential system is developed, and the properties of the solution of the system are given, such as the uniqueness and continuous dependence on the initial and boundary condition. e article is organized as follows: in Section 2, the extremum principle for the Caputo fractional conformable derivative is established. In Section 3, the maximum principle of the space-time fractional Laplace conformable differential system is derived, which is used to obtain the comparison principle for the space-time fractional Laplace conformable differential system, and the properties of the solution of the above system are given in Section 4.

Problem Formulation and Extremum Principles
In this paper, we focus on a space-time Caputo fractional conformable system with the fractional Laplace operator: where Ω represents an open and bounded domain in  [34]). For detailed information of the Caputo fractional conformable derivative, see [35].
Proof. First, we introduce an auxiliary function Concurrently, is is because erefore, formula (9) becomes 2

Journal of Mathematics
We can obtain Cβ a D α t 0 f(t 0 ) ≤ 0. e lemma is proved. Using the same method, it is easy to obtain the following lemma. holds.
Proof. We first suppose that inequality (14) is false; then, there exists a point ( Besides, w implies e latter property implies that the maximum of w cannot be attained on By Lemma 1, we know By calculation, we can show Assuming u � (τ − a/t − a) α and substituting into formula (21), we get Applying (19)- (22), it holds that

Theorem 3. Let u(x, t) ∈ H(Ω) be a solution of system (1) with initial-boundary values
where Proof. We first present a function If u(x, t) is a solution of system (1), (12), and (13), then w(x, t) is a solution of problem (1) with Substitute g 1 (x, t) and μ 1 (x, t) for g(x, t) and μ(x, t), respectively. Owing to g 1 (x, t) ≤ 0, applying eorem 1 (maximum principle), we have In a similar manner, we can get Combining (30) and (31), the theorem is proved.
Similarly, the following theorem holds.
Journal of Mathematics satisfies the following linear space-time fractional Laplace conformable differential system: Remark 2. Let (u, v) ∈ H(Ω) × H(Ω) satisfy the following linear space-time fractional Laplace conformable differential system: Next, we focus on the following linear space-time fractional Laplace conformable differential system:

Data Availability
No data were used to support this study.