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Analysis of the fractional diffusion equations with fractional derivative of non-singular kernel
- Mohammed Al-Refai^{1}Email author and
- Thabet Abdeljawad^{2}
https://doi.org/10.1186/s13662-017-1356-2
© The Author(s) 2017
Received: 10 May 2017
Accepted: 11 September 2017
Published: 10 October 2017
Abstract
In this paper we study linear and nonlinear fractional diffusion equations with the Caputo fractional derivative of non-singular kernel that has been launched recently (Caputo and Fabrizio in Prog. Fract. Differ. Appl. 1(2):73-85, 2015). We first derive simple and strong maximum principles for the linear fractional equation. We then implement these principles to establish uniqueness and stability results for the linear and nonlinear fractional diffusion problems and to obtain a norm estimate of the solution. In contrast with the previous results of the fractional diffusion equations, the obtained maximum principles are analogous to the ones with the Caputo fractional derivative; however, extra necessary conditions for the existence of a solution of the linear and nonlinear fractional diffusion models are imposed. These conditions affect the norm estimate of the solution as well.
Keywords
- fractional diffusion equations
- maximum principle
- fractional derivatives
1 Introduction
Fractional diffusion models (FDM) are generalization to the diffusion models with integer derivatives. In recent years there has been great interest in the study of FDM because of their appearance in modeling various applications in the physical sciences, medicine and biology; see, for instance, [2–10]. Therefore, analytical and numerical techniques have been implemented to study these models. The maximum principle is one of the common tools to study partial differential equations analytically, see [11, 12] for intensive survey and results. In recent years, maximum principles have been developed to study various types of fractional diffusion systems (see [13–22]), and we refer the reader to [23] for the recent development on the theory of fractional differential equations. In [13] and [14], two classes of eigenvalue problems of Caputo fractional order α, \(1< \alpha<2\), were considered. Maximum principles and the method of lower and upper solutions have been developed and used to establish certain existence and uniqueness results of the problems. In [19] and [21] Luchko has developed and implemented maximum principles to study the generalized fractional diffusion equation of Caputo fractional derivative. Existence and uniqueness results were established by the new maximum principles obtained by estimating the fractional derivative of a function at its extreme points. Analogous results were obtained in [20, 22] for the fractional diffusion systems of multi-term and distributed order fractional derivatives of Caputo type. Another maximum principle for the linear multi-term fractional differential equations with the modified Riesz fractional derivative of Caputo type was introduced and employed in [24]. The applicability of maximum principles for the linear and nonlinear fractional diffusion systems with the Riemann-Liouville fractional derivative was discussed and proved for the first time by Al-Refai and Luchko in [15], where existence, uniqueness and stability results were established. Analogous results for the fractional diffusion systems with the multi-term and distributed order fractional derivatives of Riemann-Liouville type were obtained in [16] and [18]. In [25] and [26], a maximum principle was used to analyze a type of fractional diffusion equation without an explicit formulation of this principle.
Recently [1] Caputo et al. have introduced a new type of fractional derivative with non-singular kernel. After then, the authors in [27–29] studied their discrete versions and analyzed the monotonicity properties for the fractional difference operator. Insisting on the importance of having fractional operators with non-singular kernels, later in [30–32] the authors introduced, explored and studied fractional operators of Mittag-Leffler kernels together with their discrete versions.
In this paper, we extend the results presented in [15] for the fractional diffusion equations with the Caputo fractional derivative of non-singular kernel. The rest of this paper is organized as follows. First, we give the basic definitions and results about fractional derivatives with exponential kernels. In Section 2, an estimate of the Caputo fractional derivative of non-singular kernel of a function at its extreme points is deduced in a form of certain inequality. This inequality is then employed to derive a weak and a strong maximum principles for the time-fractional diffusion equation with the Caputo fractional derivative of non-singular kernel. We apply the obtained maximum principles to analyze the solutions of linear and nonlinear time-fractional diffusion models in Sections 3 and 4, respectively. Uniqueness and stability results as well as norm estimates of solutions are obtained. Some illustrative examples are presented in Section 5. Finally, we close up with some concluding remarks in Section 6.
Definition 1.1
Lemma 1.1
[27]
2 Maximum principles
We start with estimating the fractional derivative of a function at its extreme points. These results are analogous to the ones obtained in [33] for the Caputo fractional derivative. We then use these results to establish new maximum principles for linear fractional equations with Caputo fractional derivative of non-singular kernel.
Lemma 2.1
Proof
Lemma 2.2
Proof
Theorem 2.1
Weak maximum principle
Proof
Theorem 2.2
Strong maximum principle
Suppose that \(u(x,t)\in C^{2} [0,\ell]\cap H^{1}(0,T]\) satisfies the inequality \(P_{\alpha}(u) \ge0\) in the rectangular region \(\Omega _{T}=(0,\ell)\times(0,T]\) and that u attains a maximum at \((x_{0},t_{0}) \in\Omega_{T}\). Then \(u(x_{0},t)=u(x_{0},t_{0})\) for all \(0 \le t \le t_{0}\).
Proof
3 Linear fractional diffusion problems
Theorem 3.1
The time-fractional initial-boundary value problem (3.1-3.3) has at most one solution \(u\in C^{2}[0,\ell]\cap H^{1}(0,T]\).
Proof
In the following we present essential results to guarantee the existence of a solution to the time-initial-boundary value problem (3.1)-(3.3) and to obtain analytical bound of the solution.
Lemma 3.1
Proof
We have the following necessary condition for the existence of solution to Eq. (3.1).
Lemma 3.2
Proof
Theorem 3.2
Proof
Corollary 3.1
Proof
We have the following stability result.
Theorem 3.3
Proof
4 Nonlinear fractional diffusion problems
Theorem 4.1
If \(F(x,t,u)\) is nonincreasing with respect to u, then the nonlinear time-fractional diffusion equation (4.1) subject to the initial and boundary conditions (3.2)-(3.3) possesses at most one solution \(u\in C^{2}[0,\ell]\cap H^{1}(0,T]\).
Proof
Theorem 4.2
Proof
By applying analogous steps in the proof of Lemma 3.2, we have the following necessary condition for the existence of solution of 4.1.
Lemma 4.1
5 Illustrated examples
Example 5.1
Example 5.2
6 Concluding remarks
We have considered linear and nonlinear fractional diffusion equations with Caputo fractional derivative of non-singular kernel. We have obtained an estimate of the Caputo fractional derivative of non-singular kernel of a function at its extreme points. We then have derived a weak and a strong maximum principles for the linear time-fractional diffusion equation. We have analyzed the solutions of the linear and nonlinear time-fractional diffusion models using the obtained maximum principles. Some examples are presented to illustrate the applicability of the obtained results. The obtained results will lead to better understanding of the time-fractional diffusion models with Caputo fractional derivative of non-singular kernel. In contrast to previous studies on fractional diffusion models with Caputo and Riemann-Liouville fractional derivatives, it is noticed that extra conditions have been imposed to guarantee the existence of solutions to the linear and nonlinear time-diffusion models.
Declarations
Acknowledgements
The first author acknowledges the support of the United Arab Emirates University under the Fund No. 31S239. The second author would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17.
Authors’ contributions
All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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