The Maximum Principle for Variable-Order Fractional Diffusion Equations and the Estimates of Higher Variable-Order Fractional Derivatives

In this paper, the maximum principle of variable-order fractional diffusion equations and the estimates of fractional derivatives with higher variable order are investigated. Firstly, we deduce the fractional derivative of a function of higher variable order at an arbitrary point. We also give an estimate of the error. Some important inequalities for fractional derivatives of variable order at arbitrary points and extreme points are presented. Then, the maximum principles of Riesz-Caputo fractional differential equations in terms of the multi-term space-time variable order are proved. Finally, under the initial-boundary value conditions, it is verified via the proposed principle that the solutions are unique, and their continuous dependance holds.


INTRODUCTION
Fractional calculus Podlubny [1]; as a natural extension of traditional integer calculus, has become a classical and essential branch of mathematics through a long historical development. Recently Al-Refai and Baleanu [2], obtained the estimates of fractional derivatives with higher order for extreme points, providing an approach to the establishment of the maximum principles, as well as the results of the existence and uniqueness of solutions for the fractional differential equations (FDEs). As a kind of well-known technique for handling FDEs, the maximum principle may facilitate to acquire the key access to the solutions in the absence of any prior detailed knowledge about the solutions Protter and Weinberger [3]. Liu et al. [4] derived a maximum principle for fractional differential equations (VOFDEs, for short) with multi-term time variable order 0 < α(ζ, τ) ≤ 1 and space variable orders 0 < c(ζ, τ) ≤ 1 and 1 < β(ζ, τ) ≤ 2 in the sense of Riesz-Caputo, and showed the uniqueness of solutions as well as continuous of VOFDEs via the dependance. Ye et al. [5] investigated the solutions maximum principle. More researches in this area can be consulted in Luchko [6][7][8]; Li et al. [9]; Al-Refai and Luchko [10]; Yang et al. [11]; Coronelescamilla et al. [12]; Hajipour et al. [13].
However, the restriction for most of the aforesaid fractional diffusion equations is that their orders are constant. Such a restriction was relaxed by Samko and Ross [14] via a proposed variable-order (VO) operator to describe the diffusion process. In fact, VOFDEs are widerly used as powerful tools in many research topics, such as visco-elasticity Coimbra [15]; oscillation Ingman and Suzdalnitsky [16]; anomalous diffusion Sun et al. [17]; etc. For more applications of fractional differential equations, please refer to Cooper and Cowan [18]; Liu [19]; Sun et al. [20]; Liu and Li [21]; Yang [22], etc.
The contributions of this paper can be summarized as follows: (1) The higher derivative of fractional function with variable order is given. On the basis of it, three useful theorems are given, which provide theoretical guarantee for the applications. (2) The maximum principle for one-dimensional multi-term space-time higher VOFDEs is given. (3) Based on the proposed method, a concrete example is given for the practical applications.
The paper is structured as the following. In Section 2, we recall some fundamental definitions that will be used in this paper. In Section 3, we derive some equalities and inequalities of the higher VOFDEs at arbitrary points and extreme points. We also give an estimate of the error. In Section 4, by virtue of these important inequalities, we establish the maximum principle for Riesz-Caputo FDEs with multi-term time variable order and space variable orders. In Section 5, based on the given principle, the uniqueness of solutions with their continuous dependance in the present of initial-boundary value conditions are strictly proved.
Definition 2. Let f ∈ C n [0, T] and α : [0, L] × [0, T] → R + be a VO function. The Caputo fractional derivatives of left-side VO and right-side VO are defined respectively as Definition 3. The VO Riesz-Caputo fractional operator In this paper, we are interested in the following VOFDEs: where P α,α1 ,...,αn ( C 0 D τ ) denotes the multi-term time VO Caputo fractional derivative operator, i.e.,
By the induction hypothesis, one obtains Obviously, we have: Hence, Thus, This complete the proof.

Theorem 4. Given a VO function
It can be easily verified that By Theorem 1, we obtain Frontiers in Physics | www.frontiersin.org November 2020 | Volume 8 | Article 580554 Consequently,

THE MAXIMUM PRINCIPLE
In this section, we will display and show the maximum principle for one-dimensional multi-term space-time higher VOFDEs.
For convenience, the symbol Q β,c is used to denote the operator given by It is easy to see that Q β,c is a space VO operator on ζ.
Frontiers in Physics | www.frontiersin.org November 2020 | Volume 8 | Article 580554 5 If we substitute −w for w in Theorem 5, the minimum principle is obtained as follows.

APPLICATIONS
In this section, we discuss multi-term space-time higer VOFDEs in the one-dimensional case: with the initial conditions The boundary conditions are taken into consideration as below: By Theorems 5 and 6, we can get the following theorems.
Remark 2. If f (ζ, τ) 0, then, according to Theorem 7 and 8, we know that the diffusion problem Eqs 7-9 with zero initial and boundary conditions permits only zero solution in C 2,2 (Ω T ).

CONCLUSIONS
This paper serves as a survey on the maximum principle and the estimates of time higher VOFDEs. The proposed maximum principle contributes to verify some important properties of solutions, including the uniqueness and the continuous dependance with initial-boundary value conditions being taken account. In the future, we will put attention to the solutions for problem Eq. 1 in more general forms, and investigate the numerical solutions with their applications.

DATA AVAILABILITY STATEMENT
All datasets presented in this study are included in the article.

AUTHOR CONTRIBUTIONS
GX, FL, and GS contributed conception and layout of the research; GX organized the literature; FL completed the initial draft of the paper; GS carried out the proof; The main idea of this paper was proposed by GX; All authors approved the submitted paper.

FUNDING
The authors would like to express their thanks to the reviewers and the editors for their insightful recommendations. This work is supported by the Young and Middle-aged Researchers' Basic Ability Promotion Project of Guangxi Colleges and Universities (Grant No. 2019KY0669).