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An anomalous diffusion model based on a new general fractional operator with the Mittag-Leffler function of Wiman type
- Xin Liang†^{1, 2}Email author,
- Feng Gao†^{1, 2},
- Chun-Bo Zhou†^{1, 2},
- Zhen Wang†^{3} and
- Xiao-Jun Yang†^{1, 2, 3}
https://doi.org/10.1186/s13662-018-1478-1
© The Author(s) 2018
- Received: 27 September 2017
- Accepted: 7 January 2018
- Published: 18 January 2018
Abstract
In this article, an anomalous diffusion model via a new Liouville-Caputo general fractional operator with the Mittag-Leffler function of Wiman type is investigated for the first time. The convergence of the series solution for the problem is discussed with the aid of the Laplace transform. The anomalous diffusion processes are compared to the characteristics of the conventional diffusion graphically. The results show that the new Liouville-Caputo general fractional operator is effective in characterizing and solving the problems of the anomalous diffusion.
Keywords
- anomalous diffusion
- general fractional operator
- analytic solution
- Laplace transform
1 Introduction
In nature, the anomalous diffusion phenomena occur in multiple scientific fields including physical chemistry, bioscience and engineering technology [1–4]. When many researchers attempt to describe or characterize these phenomena by the conventional integer-order differential equations, the theoretical analyses are not in good agreement with the real observations [4]. With the development of fractional calculus theory (FCT), the scientists discovered that the fractional differential equations (FDEs) have great advantages in solving many anomalous physical phenomena [5–8]. Kumar et al. [9] proposed a time-fractional modified Kawahara equation based on the Caputo-Fabrizio operator and discussed a fractional model of convective radial fins with the aid of the Homotopy analysis transform method numerically [10]. Singh et al. [11] analyzed the El Nino-Southern Oscillation model via the iterative method and fixed point theorem, as well as a nonlinear fractional dynamical model of interpersonal and romantic relationships through the q-homotopy analysis Sumudu transform method [12]. Especially, the anomalous behaviors involving the diffusion problem, heat transfer, creep phenomena, fluid flow process, oscillating circuits and the convection dispersion have been the research hot spots (see [13–18]). For instance, Metzler and Klafter presented the kinetic diffusion equations based on the fractional derivative (FD) in [19]. Yang et al. proposed the rheological models with the aid of FD in [20]. Ren et al. solved the time-fractional convection dispersion equations in [21]. However, the methods for obtaining the analytic solutions of those FDEs are still lacking due to the complexity of FCT (see [22, 23]).
The Mittag-Leffler function (MLF) and its extended forms, which are closely related to the FCT, seem to provide us with new ideas in describing some anomalous phenomena and solving the FDEs [24, 25]. Based on the MLFs, many fractional differential operators (FDOs) have been suggested (see [18, 25–31]). For example, Caputo and Fabrizio [30] defined a new fractional derivative without singular kernel. Atangana et al. used the MLF to replace the exponent function in the integral kernel of the above definition and obtained its new form in [18]. Giusti et al. suggested the Prabhakar-like fractional derivative in [31]. A family of general FDOs based on the extensions of the classical MLFs (Gösta Mittag-Leffler, Wiman and Prabhakar functions) were proposed in [25, 28]. In fact, there are close relationships among these FDOs. As analyzed in [9] by Kumar, the Caputo-Fabrizio operator is much more efficient than the classical Caputo derivative. For Atangana’s suggestion to the new fractional derivative with one-parameter MLF, it is an extended version of the Caputo-Fabrizio operator [18]. Similarly, the Prabhakar-like fractional derivative can return to the Caputo-Fabrizio operator and the new FDOs defined by Yang when the parameters take the particular values (see [25, 28, 31]). These FDOs involving the MLFs in the integral kernel have been applied to model many physical phenomena, such as the anomalous relaxation, heat-transfer problems, viscoelastic problems, Euler-Lagrange equation and the boundary value problem, extensively (see [18, 28–36]). Especially, the new general FDOs with the aid of Laplace transform (LT) of the MLFs may be provided to describe different anomalous physical phenomena (see [28]). Inspired by this, this paper aims to model the anomalous diffusion problems by the new general FDOs and to obtain the analytic solution.
The remainder of this paper is structured as follows. In Section 2, the definitions of several MLFs, the new Liouville-Caputo general FDO with the extension of Wiman function, as well as the LT of the Wiman and Prabhakar MLFs with power-law functions, are reviewed. In Section 3, an anomalous diffusion model (ADM) based on the above general FDO is proposed, and its analytic solution is also given. In addition, the ADMs with different parameters are analyzed graphically. Finally, the conclusions are summarized in Section 4.
2 The MLF and a new general FDO
2.1 The MLF and LTs of its generations
In recent decades, owing to the successful applications of MLFs in multiple fields involving applied sciences and pure mathematics, they have received wide attention. Correspondingly, a variety of the extensions or generalizations of the MLF have been suggested. Here, we review them as follows.
Let \(\mathbb{C}\), \(\mathbb{ R}\) and \(\mathbb{N}\) be the sets of complex numbers, real numbers and positive integers, respectively.
Definition 1
Definition 2
Definition 3
Definition 4
Definition 5
The LT of the new extensions of Wiman and Prabhakar functions with the power-law functions [ 28 ]
The Wiman and Prabhakar functions with the power-law functions | LT |
---|---|
\(\chi ^{\varpi +\theta -1}E_{\vartheta ,\varpi +\theta } ( {\chi ^{\vartheta}} )\) | \(\xi ^{- ( {\varpi +\theta } )} ( {1-\xi ^{-\vartheta }} )^{-1}\) |
\(\chi ^{\varpi +\theta -1}E_{\vartheta ,\varpi +\theta }^{\tau}( {\chi ^{\vartheta}} )\) | \(\xi ^{- ( {\varpi +\theta } )} ( {1-\xi ^{-\vartheta }} )^{-\tau }\) |
2.2 The new Liouville-Caputo general FDO with the extension of Wiman function
Definition 6
Remark
3 Analyses of the ADM
3.1 Analytic solution of the ADM
3.2 Numerical analyses of the ADM
In this subsection, we illustrate the numerical analyses of the ADM from multiple angles. Firstly, we analyze the convergence of Eq. (22) by discussing the values of n. Secondly, the comparisons between the conventional diffusion and the anomalous diffusion are presented graphically. Next, the characteristics of the anomalous diffusion with the varied fractional orders are represented. Finally, the ADMs with several varied parameters are shown graphically.
4 Conclusions
In the current paper, we have solved a new ADM based on a new Liouville-Caputo general FDO with the extension of Wiman function. The analytic series solution was obtained and its convergence was discussed. The results show that the numerical solutions can satisfy the accuracy when \(n=4\), and the new FDO is effective in describing the anomalous diffusion phenomena. In addition, the anomalous diffusion processes exhibit different characteristics from the conventional diffusion, and they are greatly affected by the varied parameters. Specially, the smaller the orders are, the stronger the diffusion processes are.
Notes
Declarations
Acknowledgements
This research was financially supported by the Fundamental Research Funds for the Central Universities (No. 2017BSCXA21) and the Postgraduate Research & Practice Innovation Program of Jiangsu Province (No. KYCX17_1531).
Authors’ contributions
All authors drafted the manuscript, and they read and approved the final version of the 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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