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
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.
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:
[email protected] D. Baleanu (B) Department of Mathematics and Computer Sciences, Cankaya University, 06530 Balgat, Ankara, Turkey e-mail:
[email protected] D. Baleanu Institute of Space Sciences, Magurele-Bucharest, Romania Z.-G. Deng School of Mathematics and Information Science, Guangxi University, Nanning 530004, China
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
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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, 0≤x≤L
ν where C a Dt
(1.2)
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-
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difference equations and the discrete fractional variational problems. Mozyrska and Girejko et al. [34,35] gave the definition of the Caputo-like h-difference, and the initial problems are discussed. Compared with (1.2), we can conclude that there are some merits when applying the tool of the DFC: (a) the domain is defined as a time scale while the fractional calculus requires the defined function that should be continuous. (b) The DFC operators have memory effect and therefore it makes them attractive and more powerful than the classical discrete operators, mainly because they can describe better the dynamics of complex discrete systems with the long-range interaction traits. See the discrete fractional models equivalent form in the summuation [29] μ (i − j + ν) (ν) (i − j + 1) i
u(i) = u(0) +
j=1
×u( j − 1)(1 − u( j − 1)).
(1.3)
(c) There is the fractional difference order, which is a crucial parameter of the dynamical models. We can see that the fractional difference equation [2] are the fractional generalization of the classical difference models and they hold the merits (b) and (c). These are also the main reasons why we adopt the discrete fractional partial difference method for the diffusion problem in this paper. We consider the numerical simulation of the fractional diffusion model of time discretization C ν a h u(x, t)
= K u x x (x, t + (ν − 1)h), u(0, t + (ν − 1)h) = φ(t), u(L , t + (ν − 1)h) = ψ(t),
(1.4)
ν where u(x, a) = f (x), C a h is the Caputo-like difference, (hN)a = {a, a + h, . . .} and here t ∈ (hN)a+(1−ν)h = {a + (1 − ν)h, a + (1 − ν)h + h, . . .}, 0 < ν ≤ 1.
2 Preliminaries There are various versions of the fractional differences such as the G-L difference [4], the DFC [25– 28], the Caputo difference [36], and the piecewise constant argument [37]. In this paper, we use the fractional h-sum and h-difference developed by Torres and Mozyrska et al. [32–35]. The definitions are given as the following.
Discrete fractional diffusion equation
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Definition 2.1 (See [32]) Let u : (hN)a → R and 0 < ν be given. Then, the generalized sum of ν order is defined by −ν a h u (t)
t
h −ν h (ν−1) := u(sh), (t − σ (sh))h (ν) a
s= h
t ∈ (hN)a+νh
(2.1)
where a is the starting point, a ∈ R, σ (s) = s + h and th(ν) is called the h-factorial function (ν) th
=h
ν
( ht + 1) ( ht + 1 − ν)
.
C ν a h u
h (t) : = (1 − ν)
(t
− σ (sh))(−ν) h
s= ah
t ∈ (hN)a+(1−ν)h .
(2.3)
Property 2.3 (See [35]) For a ∈ R, h > 0 and α, β > 0, the following property holds −β −β −α −α a+βh h a h u (t) = a+αh h a h u (t) −(α+β) = a h u (t), t ∈ (hN)a+(α+β)h .
(2.4)
In, Mozyrska et al. [34] proved the following Leibniz sum law −ν C ν 0 h μ h u
t ∈ (hN)a+(1−ν)h , 0 < ν ≤ 1 (3.1) has an equivalent discrete form as t
h u(t) = u(a) + (ν)
h −ν
(ν−1) (t − σ (sh))h
s= ah +1−ν
F(u(sh + (ν − 1)h), sh + (ν − 1)h), t ∈ (hN)a+h .
t h −(1−ν)
×u(sh),
Generally, from the Property 2.4, the delta fractional difference equation C ν a h u (t) = F(u(t + (ν − 1)h), t + (ν − 1)h),
(2.2)
Definition 2.2 (See [34]) For 0 < ν ≤ 1, and u(t) defined on (hN)a , the Caputo-like delta difference is defined by
3 Numerical solutions of the discrete fractional diffusion equation
(t) = u(t) − u(μ), t ∈ (hN)νh (2.5)
where μ = (ν − 1)h. Similarly, we can have the following one. Property 2.4 For the initial point a ∈ R, the discrete Leibniz sum law holds −ν C ν a+(1−ν)h h a h u (t) = u(t) − u(a), t ∈ (hN)a+h .
(2.6)
(3.2)
which was given in [38] and existence results were discussed. The domain should be changed from (hN)a+(1−ν)h to (hN)a+h in order to satisfy the existence of solutions. One can see that this is totally different from the classical fractional differential equations. The fractional operators here are domain shifting ones, and various domains are summarized in [39]. Furthermore, we can rewrite the discrete fractional diffusion equation (1.4) in the h-sum as K hν u(x, t) = u(x, a) + (ν)
t
h −ν
s= ah +1−ν
∂ 2 u(x, sh + (ν − 1)h) , ∂x2 0 ≤ x ≤ L,
×
(ν−1)
(t − σ (sh))h
t ∈ (hN)a+h , (3.3)
where 0 < ν ≤ 1, u(x, a) = f (x), u(0, t +(ν−1)h) = φ(t), u(L , t + (ν − 1)h) = ψ(t). In this paper, we consider a = 0 and the following the boundary condition 10x, 0 ≤ x ≤ 1; φ(t) = ψ(t) = 0, f (x) = 10(x − 2), 1 ≤ x ≤ 2. Assume that
In fact, for 0 < ν ≤ 1, according to the Definition (2.2) and Property 2.3, we can have −ν C ν a+(1−ν)h h a h u (t) −(1−ν) = a+(1−ν)h −ν u (t) h a h
Then, using the central difference formula concerning 2 x, ∂ ∂u(x,t) can be approximately expanded as x2
= (−1 u)(t) = u(t) − u(a), t ∈ (hN)a+h . (2.7)
(3.4)
x0 = 0, xm = 2, xi+1 = xi + x, i = 0 . . . m − 1.
∂ 2 u(x, t) u i+1 (t) − 2u i (t) + u i−1 (t) ≈ . ∂x2 x 2
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Furthermore, along the discrete time direction, we can have a discrete system
×
ν=0.6 ν=0.7 ν=0.8 ν=0.9 ν=1
4
t
h −ν
3.5
(ν−1)
(t − σ (sh))h
s= ah +1−ν
u i+1 ((s +ν −1)h)−2u i ((s +ν −1)h)+u i−1 ((s +ν − 1)h) , x 2
(3.5) where t ∈ (hN)a+h , 0 ≤ x ≤ 2, 0 < ν ≤ 1. We can rewrite it in a iteration formula as
the concentration u
K hν u(x, t) = u(x, 0) + (ν)
t=0.5
4.5
3 2.5 2 1.5 1 0.5
K h ν (n − j + ν) u i (nh) = f i + (ν) (n − j + 1) n
0
0
0.5
1
u i+1 (( j −1)h)−2u i (( j −1)h)+u i−1 (( j −1)h) , x 2 (3.6)
where 1 ≤ i ≤ m − 1, f i = f (ix) and u i (nh) = u(ix, nh). In the numerical formula (3.6), since the initial condition u(x, 0) = f (x) is known, we can readily have u 0 (0) = f (0), u 1 (0) = f (x) = f 1 , . . . , u m (0) = f (mx) = f m . Also from the boundary condition, we get u 0 (h) = u m (h) = 0.
x=0.5
5
ν=0.6 ν=0.7 ν=0.8 ν=0.9 ν=1
4.5 4 3.5 3
(3.7)
2.5
As a result, we can have all the information of the u 0 (h), . . . , u m (h). Similarly, we obtain the numerical solution u i ( j h) successively. Now, with different fractional parameters, we can have the different behaviors of the discrete fractional diffusion equations. Set h = 0.01, x = 0.1, K = 0.2. We vary the fractional order μ and give the diffusion behaviors in Figs. 1 and 2. The results are carried out on the MATLAB. Here, we need to point out the differences between the above method and the classical fractional difference methods. For the fractional diffusion models [40,41], the G-L difference is often adopted as a discretization technique in the finite difference method. The obtained solutions are approximate ones defined on continuous time interval. Our diffusion equation (1.4) is given in the DFC’s sense and depicts the real evolution happens on the time scale. The solutions are explicit ones. This treatment is a fractionalization of the difference equation [2] and provides a tool to model random or discrete phenomenons holding long history or memory effects.
2
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2
Fig. 1 The diffusion concentration versus the continuous space x for t = 0.5
the concentration u
×
1.5
x
j=1
1.5
0
0.2
0.4
0.6
0.8
1
t
Fig. 2 The diffusion concentration versus the discrete time t for the x = 0.5
In view of this point, the discrete fractional diffusion equation (1.4) and the numerical discrete version of the fractional diffusion equation (1.2) are different models.
4 Conclusions This paper suggests the application of the discrete fractional calculus to the partial differential equations, and a time-discretization diffusion model is proposed. We then consider the numerical solution using its equivalent fractional sum. The space partial derivative term is discretized by the classical finite difference from which
Discrete fractional diffusion equation
the discrete model is reduced to discrete fractional difference equations with respect to t. Finally, the anomalous diffusion for various fractional difference orders is discussed and illustrated. We conclude that the proposed diffusion model has extensive applications when the media is of the discrete structure. It can better reflect the discrete natural phenomenons with memory effect compared with both classical difference equations and the fractional differential equations. Acknowledgments This work was financially supported by the National Natural Science Foundation of China (Grant No. 11301257), the Innovative Team Program of the Neijiang Normal University (Grant No. 13TD02), the Guangxi Natural Science Foundation (Grant No. 2013GXNSFBA019021), and the Scientific Research Foundation of GuangXi University (Grant No. XBZ120542).
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