Stability and boundedness of solutions of the initial value problem for a class of time-fractional diffusion equations
- Yanhua Wen^{1}Email author,
- Xian-Feng Zhou^{1} and
- Jun Wang^{1}
https://doi.org/10.1186/s13662-017-1271-6
© The Author(s) 2017
Received: 9 January 2017
Accepted: 10 July 2017
Published: 9 August 2017
Abstract
The aim of this paper is to study the stability and boundedness of solutions of the initial value problem for a class of time-fractional diffusion equations. We first establish a fractional Duhamel principle for the nonhomogeneous time-fractional diffusion equation. Then based on it and the superposition principle, the solution of the above initial value problem is represented. Finally, we obtain the stability and boundedness of the solution and present an illustrative example.
Keywords
stability boundedness fractional Duhamel principle initial value problem time-fractional diffusion equation Caputo derivative1 Introduction
Fractional differential equations have received considerable attentions during the past few decades because they are useful for modeling many practical phenomena. And a large amount of results such as existence, uniqueness, stability, etc. of the solution have been obtained for the fractional differential equations (see [1–5] and the references therein).
In recent years, fractional partial differential equations have been applicated in the study of viscoelasticity, biology, anomalous diffusion, such as [6–10]. Based on the existing inequalities, Jleli [6] presented the Lyapunov inequalities for fractional partial differential equations. The authors in [8] obtained the approximate analytical solutions for two different types of nonlinear time-fractional systems of partial differential equations using the fractional natural decomposition method. And a maximum principle for the generalized time-fractional diffusion equation with the Caputo fractional derivative is established by Luchko [7].
Furthermore, initial-boundary value problems for both ordinary fractional differential equations and fractional partial differential equations are studied in the literatures (see [11–15] and the references therein). The authors in [14] established an existence result for a class of nonlinear fractional partial differential equations with the standard Caputo fractional derivative of order \(1<\alpha\leq2\). In [15], Wang discussed the nonlocal initial value problem for fractional differential equations with the Hilfer fractional derivative.
The rest of this article is organized as follows. Section 2 is devoted to some preliminaries. In Section 3, we present our main results of this paper. An illustrative example is provided in Section 4.
2 Preliminaries
In this section, we introduce some definitions and lemmas which will be used later.
Definition 2.1
[1]
Definition 2.2
[1]
Definition 2.3
[1]
Lemma 2.1
[18]
Lemma 2.2
[19]
Remark 2.1
Obviously, \(0< E_{\alpha,1}(-x)<1\), for any \(x>0\) by Lemma 2.2.
Lemma 2.3
[20]
Lemma 2.4
[21]
Lemma 2.5
[21]
Lemma 2.6
[21], Hausdorff-Young inequality
3 Main results
Lemma 3.1
Proof
Property 3.1
Proof
3.1 Fractional Duhamel principle
A fractional Duhamel principle is firstly given, which can reduce the nonhomogeneous the IVP (33)-(34) to the corresponding homogeneous IVP.
Theorem 3.1
Fractional Duhamel principle
Proof
Corollary 3.1
Combining Lemma 3.1 with Corollary 3.1, we can get the following theorem.
Theorem 3.2
Theorem 3.3
Proof
3.2 Stability of solution
This section presents the stability of the solution of the nonhomogeneous the IVP (5)-(6).
Definition 3.1
Suppose that H is a linear normed space with the norm \(\Vert \cdot \Vert _{H}\), \(u_{1}(x,t)\), \(u_{2}(x,t)\) are solutions of the IVP (5)-(6) corresponding to initial datum \(\varphi_{1}(x)\), \(\varphi_{2}(x)\), respectively. For any \(\varepsilon >0\), if there exists a constant \(\delta>0\) such that \(\Vert \varphi_{1}(x)-\varphi_{2}(x) \Vert <\delta\) implies \(\Vert u_{1}(x,t)-u_{2}(x,t) \Vert <\varepsilon\), then we say that the solution of the IVP (5)-(6) is stable.
Theorem 3.4
Stability
Assume \(\varphi(x)\in L^{p}(R)\), \(p\geq1\). Then the solution \(u(x,t)\) of the nonhomogeneous IVP (5)-(6) is stable.
Proof
4 Illustrative example
In this section, we provide an example to show the application of our stability result.
Example 4.1
Declarations
Acknowledgements
This paper is supported by National Natural Science Foundation of China (11371027, 11471015 and 11601003), Natural Science Foundation of Anhui Province (1508085MA01, 1608085MA12 and 1708085MA15) and Program of Natural Science Research for Universities of Anhui Province (KJ2016A023).
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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