Abstract
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.
1. 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 and space variable orders and 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 [6–8]; 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:
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.
The maximum principle for one-dimensional multi-term space-time higher VOFDEs is given.
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.
Notations: Throughout this paper, ζ denotes the space variable and τ denotes the time variable. and are the closure and the boundary of , respectively. , and represent binary VO functions. It is supposed that the VO functions , β and γ satisfy thatwhere , and . Also, the functions , , and are supposed to be all continuous on with , and .
2. Preliminaries
Throughout this paper, denotes the set of all positive real numbers. Let be a Banach space with the norm . For more details about the relevant concepts and results, please see Podlubny [1]; Liu et al. [4]; Kilbas et al. [23].
Definition 1. Let and be a VO function. The Riemann-Liouville fractional integrals of left-side VO and right-side VO are defined asrespectively, where and is the smallest integer not less than .
Definition 2. Let and 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 of VO with and is defined aswhere , is the coefficient with andMoreover, if , .
In this paper, we are interested in the following VOFDEs:where denotes the multi-term time VO Caputo fractional derivative operator, i.e.,
3. The Varable-Order Fractional Derivtives at Arbitrary Points and Extreme Points
In this section, we are in position to give some basic results.
Theorem 1. Let . and be a VO function. If satisfiesthen for any arbitrary point , the following equation holdswhere .
Proof. We shall prove this by induction argument. If , the result has been obtained in Liu et al. [4]. Assume that this is true for . Now we check that it still holds whenever .
Let , where . Then Define . Then.
By the induction hypothesis, one obtainsSubstituting for in the preceding equation, one haswhere .
Obviously, we have:
,
.
Hence,where and .
Integrating by parts, we haveSoandThus,Hence . This complete the proof.
Remark 1. If in () and is an extreme point, then Theorem 1 coincides with Al-Refai and Baleanu [2]’s result. Thus, our result generalizes AL-Refai and Baleanu’s original idea.
Theorem 2.
Let
. Suppose that the VO function
satisfies
For any arbitrary point
, one gets
(1) For any nonnegative with , then
(2) For any non-positive with , then
where
.
Proof. Employing the Taylor series expansion, we know that there is some with such thatSo, we haveNote that , andTherefore, we get , and
Theorem 3. Let , and , for all . If the VO function satisfiesthen for any arbitrary point , the following equation holds:where , andProof. According to Eq. 3, one hasAs a result,
Theorem 4. Given a VO function with for all . If attains its maximum at , then it holds thatMoreover, if , then .
P
roof.
Let
Obviously, we have
;
and ;
where and , .
It can be easily verified that
By Theorem 1, we obtainSince for all and , it follows that .
Hence,ThereforeConsequently, for all whenever ,
4. 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 is used to denote the operator given byIt is easy to see that is a space VO operator on ζ.
Theorem 5. Suppose andIf but whenever , thenProof. We prove this by contradiction. Assume that there exits such thatLet for all , where .
Precisely, we haveand
This implies that
Thus,This means fails to reach the maximum value on the boundary . Assume that obtains the maximum value at . It follows that
Trivially, one hasandNote that and , which follow by applying Theorem four in this paper along with Theorems 3.2 and 3.3 in Liu et al. [4]. By virtue of Eqs 4 and 5, we have
This is a contradiction to our assumption that
This completes the proof.
If we substitute for w in Theorem 5, the minimum principle is obtained as follows.
Theorem 6. Suppose , andIf and , for all , thenwhere is the boundary of .
5. 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.
Theorem 7. Suppose . If is a solution of the problem Eqs 7–9 with and for all , then .
Theorem 8. Suppose . If is a solution of the problem Eqs 7–9 with and for all , then .
Remark 2. If , 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 .
Consider the next nonlinear diffusion equation
Theorem 9. Assume that the partial derivative exists and satisfies for all . If and for all , then the problem Eqs 8–10 has at most one solution in .
Proof. Suppose that are two solutions of the problem Eqs 8–10. Let . Then
Since the homogeneous initial and boundary conditions are fulfilled by w, one hasOwing to the existence of , it holds thatfor all , where for some .
Consequently,where for all .
By Theorem 7, holds for all . Conversely, is also true by using Theorem 8. So, , i.e.,
This completes the proof.
6. 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.
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).
Statements
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.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
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Summary
Keywords
maximum principle, fractional diffusion equation, fractional derivative with variable order, extreme point, boundary value problem
Citation
Xue G, Lin F and Su G (2020) The Maximum Principle for Variable-Order Fractional Diffusion Equations and the Estimates of Higher Variable-Order Fractional Derivatives. Front. Phys. 8:580554. doi: 10.3389/fphy.2020.580554
Received
06 July 2020
Accepted
12 August 2020
Published
24 November 2020
Volume
8 - 2020
Edited by
Jia-Bao Liu, Anhui Jianzhu University, China
Reviewed by
Dongyan Li, Xi'an Polytechnic University, China
Lin Wang, Anhui University of Science and Technology, China
Updates
Copyright
© Xue, Lin and Su.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Guangwang Su, 617326891@qq.com
This article was submitted to Mathematical and Statistical Physics, a section of the journal Frontiers in Physics
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