离散分数阶扩散方程

is the Caputo derivative [14]. It is a macroscopic model of continuous-time random walk. Various analytical and numerical methods have been suggested to investigate the model, such as the predictor–corrector algorithm [15], the finite difference method [16], and the integral method [17] On the other hand, the porous media is of discrete structure and the theories in continuum mechanics cannot be applied to analysis the behaviors directly. In this aspect, some real applications have been suggested in discrete systems. Machado used the Grünwald– Letnikov (G–L) difference to digital control systems in [18] and designed the discrete time fractional controller in [19]. The performance was investigated. Ortigueira et al. [20] investigated the scale conversion of discretetime signals and discussed applications to linear prediction. They also proposed the Laplace and Fourier transforms [21] among which the generalized G-L difference is analyzed and the Laplace transform of the forward difference of a sinusoid was presented. Edelman and Tarasov [22,23] suggested the application of the fractional derivative to differential equations and proposed fractional maps from numerical discretization’s view. The obtained maps hold discrete memory effects which can model the long-rang interaction of discrete systems. Pu et al. [24] applied the G-L difference to texture segmentation and reported that the ability for preserving high-frequency edge and complex texture information of the proposed fractional denoising model is obviously superior to traditional algorithms. Recently, it was considered the application of the discrete fractional calculus (DFC) [25–28] to the discretization of the chaotic systems and presented the fractional logistic map [29], the fractional standard maps [30] as well as the discrete fractional synchronization [31]. Since the memory effect and the fractional order are introduced, these fractional maps show more complicated dynamical behaviors compared with the classical maps. More recently, from the theories of timescales [3], Bastos, Ferreira, and Torres [32,33] defined the fractional h -sum and presented properties for fractional h -
123
282
G.-C. Wu et al.
diffusion [6–11] and the time-fractional diffusion equation was proposed [12,13]
C ν a Dt u ( x , t )
= K u x x (x , t ), u (x , a ) = f (x ), u (0, t ) = φ(t ), u ( L , t ) = ψ(t ), a < t Biblioteka Baidu 0≤x≤L (1.2)
Received: 26 July 2014 / Accepted: 15 December 2014 / Published online: 13 January 2015 © Springer Science+Business Media Dordrecht 2015
Abstract The tool of the discrete fractional calculus is introduced to discrete modeling of diffusion problem. A fractional time discretization diffusion model is presented in the Caputo-like delta’s sense. The numerical formula is given in form of the equivalent summation. Then, the diffusion concentration is discussed for various fractional difference orders. The discrete fractional model is a fractionization of the classical difference equation and can be more suitable to depict the random or discrete phenomena compared with fractional partial differential equations.
Keywords Discrete fractional calculus · Discrete anomalous diffusion · Discrete fractional partial difference equations
1 Introduction Many nonlinear phenomena in nature possess the discrete characteristics, such as population model, neural network, and gene information. In view of this point, the discrete models can be used for parameter identification directly from experimental data. The difference equations have been suggested, and several excellent contributions have been made to the theories [1–3]. The fractional calculus has been extensively applied in both theories and real world applications [4,5]. Particularly, it provides an efficient tool in modeling the anomalous diffusion arising in flow through the porous medium. The classical diffusion equation with the initial boundary conditions reads ∂ u (x , t ) = K u x x (x , t ), u (x , a ) = f (x ), u (0, t ) ∂t = φ(t ), u ( L , t ) = ψ(t ), a < t , 0 < L , 0≤x≤L (1.1) where a is the initial point, and K is the diffusion coefficient. Considering the flow’s complexity through porous medium as well as the speed or the concentrations, memory effect (or long history dependence), the fractional calculus has been suggested to depict anomalous
Nonlinear Dyn (2015) 80:281–286 DOI 10.1007/s11071-014-1867-2
ORIGINAL PAPER
Discrete fractional diffusion equation
Guo-Cheng Wu · Dumitru Baleanu · Sheng-Da Zeng · Zhen-Guo Deng
G.-C. Wu · S.-D. Zeng Data Recovery Key Laboratory of Sichuan Province, Neijiang Normal University, Neijiang 641100, Sichuan, China G.-C. Wu Institute of Applied Nonlinear Science, College of Mathematics and Information Science, Neijiang Normal University, Neijiang 641100, Sichuan, China e-mail: wuguocheng@gmail.com D. Baleanu (B) Department of Mathematics and Computer Sciences, Cankaya University, 06530 Balgat, Ankara, Turkey e-mail: dumitru@cankaya.edu.tr D. Baleanu Institute of Space Sciences, Magurele-Bucharest, Romania Z.-G. Deng School of Mathematics and Information Science, Guangxi University, Nanning 530004, China