Initial boundary value problems for space-time fractional conformable differential equation

Abstract: In this paper, the authors study a initial boundary value problems (IBVP) for spacetime fractional conformable partial differential equation (PDE). Several inequalities of fractional conformable derivatives at extremum points are presented and proved. Based on these inequalities at extremum points, a new maximum principle for the space-time fractional conformable PDE is demonstrated. Moreover, the maximum principle is employed to prove a new comparison principle and estimation of solutions. Beside that, the uniqueness and continuous dependence of the solution of the space-time fractional conformable PDE are demonstrated.

Jarad et al. [40] introduced the fractional conformable derivatives in the sense of Caputo and Riemann-Liouville and stated their properties. To the best of our knowledge, the mathematical literature on the maximum principles and their applications for Caputo fractional conformable derivatives is rarely mentioned. Inspired by the above works, in this paper we investigate a IBVP for space-time Caputo fractional conformable PDE. First, we present several inequalities of Caputo fractional conformable derivatives at extremum points and give detailed proof of two inequalities. After that, by using these inequalities, a new maximum principle is established. The maximum principle is employed to show that estimation of solutions, comparison principle and the uniqueness and continuous dependence of solutions on the initial and boundary conditions.
The rest of this article is organized as follows: In Section 2, we introduce some definitions about Caputo fractional conformable derivatives and establish several extremum principles. In Section 3, these extremum principles are employed to derive maximum principle. Finally, the maximum principle is applied to obtain estimation of solutions, comparison principle and the properties of the solution of the space-time fractional conformable differential equations in Section 4.

Problem formulation and extreme principles
In this paper, we shall investigate the following space-time Caputo fractional conformable PDE Here x and t are the space and time variables, a(x, t) ∈ C 1,1 ([a, b]×[T, T 1 ]), and 0 < , β < 1, 1 < γ < 2.
Cβ T D t is the left Caputo fractional conformable derivative of order β. Cγ a D x and Cγ D bx are the left and right Caputo fractional conformable derivatives of order γ. For f ∈ C n ,T ([T, T 1 ]), the left Caputo fractional conformable derivative of order β is defined by , the left (right) Caputo fractional conformable derivatives of order γ can be written, respectively, as and and I[a, b] are defined in Definition 3.1 in [41]). The detailed information of Caputo fractional conformable derivative, see [40]. Denote For our maximum principle, we make use of the following three Caputo fractional conformable extremum principles.
,a ([a, b]) reaches its maximum at a point x 0 ∈ (a, b). Then the inequality Therefore, the formula (2.8) becomes Therefore, the formula (2.10) becomes We can obtain Cγ a D x 0 f (x 0 ) ≤ 0. The lemma is proved. (2.14) . By calculation, we notice that Therefore, the formula (2.15) becomes Therefore, the formula (2.17) becomes We can obtain Cγ D bx 0 f (x 0 ) ≤ 0. The lemma is proved.
Using the same method, it is easy to obtain the following lemmas. holds. holds. holds. holds.

Maximum principle
In this section, we shall consider the linear space-time Caputo fractional conformable PDE (2.1) on the initial-boundary conditions: holds.
Proof. Arguing by contradiction, assume that there exists a point (x 0 , t 0 ) ∈ U satisfies According to the definition of w, we have The latter property implies that the maximum of w cannot be attained on S . Let w( By Lemma 2.1, 2.2 and 2.3, we know (3.5) By calculation, we can show . (3.8) which is in contradiction with (2.1).
This completes the proof of the theorem.
Similarly, the following minimum principle can be obtained by substituting −z for z in the Theorem 3.1.

If z(x, t) is a solution of the Eqs (2.1), (3.1) and (3.2), then w(x, t) is a solution of the problem (2.1) with
g * (x, t), µ * 1 (t) and µ * 2 (t) replace g(x, t), µ 1 (t) and µ 2 (t), respectively. Due to g * (x, t) ≤ 0, by using Theorem 3.1 (Maximum principle), we have Therefore, In a similar fashion, we get  . z(x, t) continuously depends on the data given in the problem in the sense that if then, the estimate for the corresponding classical solution z(x, t) and z * (x, t) true.
The demonstrate process is similar to Theorem 4.1.
Using the same way, the following Theorem holds.

Conclusion
In this paper, we have proved two extreme principles for the Caputo fractional conformable derivatives. Based on these extreme principles, a maximum principle for the space-time fractional conformable diffusion equation is established. Furthermore, the maximum principle is applied to show a new comparison principle, estimation of solutions and the uniqueness and continuous dependence of the solution for the IBVP to the space-time Caputo fractional conformable equations. Our results are new and contribute significantly to the literature on the topic.