Generalized Fourier transform method for nonlinear anomalous diffusion equation Jie Yao,1, ∗ Cameron L. Williams,2 Fazle Hussain,1 and Donald J. Kouri2, 3, † 1
arXiv:1701.03488v1 [physics.comp-ph] 12 Jan 2017
Texas Tech University, Department of Mechanical Engineering, Lubbock, Texas, USA, 79409 2 University of Houston, Department of Mathematics, Houston, Texas, USA,77204 3 University of Houston, Departments of Mechanical Engineering and Physics, Houston, Texas, USA,77204 (Dated: January 16, 2017) The solution of a nonlinear diffusion equation is numerically investigated using the generalized Fourier transform method. This equation includes fractal dimensions and power-law dependence on the radial variable and on the diffusion function. The generalized Fourier transform approach is the extension of the Fourier transform method used for the normal diffusion equation. The feasibility of the approach is validated by comparing the numerical result with the exact solution for a pointsource. The merit of the numerical method is that it provides a way to calculate anomalous diffusion with an arbitrary initial condition.
I.
INTRODUCTION
In the last few decades, anomalous diffusion has been extensively studied in a variety of physical applications, such as turbulent diffusion [1], surface growth [2], transport of fluid in porous media [2], hydraulics problems [3], etc. The anomalous diffusion is usually characterized by the time dependence of mean-square displacement (MSD) viz., hr2 i ∝ tσ . The MSD grows linearly with time (σ = 1) for the normal diffusion case. The process is called sub-diffusion for 0 < σ < 1 and super-diffusion for σ > 1. The standard normal diffusion described by the Gaussian distribution can be obtained from the usual Fokker-Planck equation with a constant diffusion coefficient and zero drift [4]. Extensions of the conventional Fokker-Planck equation have been used to study anomalous diffusion. For example, anomalous diffusion can be obtained by the usual Fokker-Planck equation, but with a variable diffusion coefficient [5, 6]. It can also be achieved by incorporating nonlinear terms in the diffusion term, or external forces [7–10]. In some approaches, fractional equations have been employed to analyze anomalous diffusion and related phenomena [11–14]. In this paper, we study the generalized nonlinear diffusion equation including a fractal dimension d and a diffusion coefficient which depends on the radial variable and the diffusion function ρ [15–17] ∂ρ 1 ∂ d−1−θ ∂ ν r (1) = K0 d−1 ρ , ∂t r ∂r ∂r with the initial and boundary conditions ρ(r, t0 ) = ρ0 (r), ρ(∞, t) = 0,
(2) (3)
where r is the radial coordinate, and θ and ν are real parameters. When the diffusion coefficient is a function
of r only, it is a generalization of the diffusion equation for fractal geometry [18]. It is the traditional nonlinear diffusion equation when the diffusion coefficient depends on ρ only [19, 20]. Analytical solutions of eq. (1) with a point source have been reported in [15, 16], where an ansatz for ρ is proposed as a general stretched Gaussian function. In [17], the same analytic solutions were also obtained by using Lie group symmetry analysis. Motivated by the research on generalized nonlinear diffusion, we propose here a numerical method for solving eq. (1) using a generalized Fourier transform. The generalized Fourier transform (also called the Φn transform), is a new family of integral transforms developed by Willams et. al. [21, 22]. These transforms share many properties of the Fourier transform. It can be employed to perform a more general frequency and time-frequency analysis. In section II, a brief introduction of the generalized Fourier transform is provided. The procedure of using the generalized Fourier transform for solving the generalized nonlinear diffusion equation is discussed in III A and III B. The method is validated by comparison between analytical and numerical results. Finally, some numerical results for a non-Delta function initial condition are given in III C.
II.
GENERALIZED FOURIER TRANSFORM
The generalized Fourier transform Φn is defined as Z Φn f (k) = ϕn (kx)f (x)dx, (4) where the integral kernel ϕn (ωx) = cn (kx) + isn (kx) is, |η|n 1 1 , (5) cn (η) = |η|n−1/2 J−1+ 2n 2 n and sn (η) =
∗ †
Previously at University of Houston, Department of Mechanical Engineering; Email:
[email protected],
[email protected] Email:
[email protected]
|η|n 1 1 , sgn(η)|η|n−1/2 J1− 2n 2 n
(6)
where Jν (η) is the cylindrical Bessel function, and n is the transform order, i.e., n = 1, 2, · · · .
2 The Fourier transform F is the special case with n = 1. The Φn transform shares many properties of the Fourier transform. Here we focus on two properties which will be used later. It is well-known that the Fourier transform preserves the functional form of a Gaussian; particularly, F [g1 ] = g1 if g1 (x) = exp(−x2 /2). For the generalized
t=0.2 Ex. Sol. t=0.2 Num. Sol. t=0.4 Ex. Sol. t=0.4 Num. Sol. t=0.6 Ex. Sol. t=0.6 Num. Sol.
0.9 0.8 0.7 0.6
2n
− x2n
Φn [
∂ 2−2n ∂ x f ] = k −2n Φn f. ∂x ∂x
It is well known that the Fourier transform can be used to find the solution for the standard diffusion equation [23]. Motivated by this idea, we explore employing the generalized Fourier transform to solve the generalized nonlinear diffusion equation. The O’Shaugnessy-Procaccia anomalous diffusion equation on fractals
Let us first consider the generalization of the diffusion equation for fractal geometry, where the diffusion coefficient is a function of r only ( or ν = 1)[18]. Eq. (1) can be reduced to ∂ρ(r, t) K0 ∂ d−1−θ ∂ = d−1 r ρ(r, t) . (8) ∂t r ∂r ∂r In order to satisfy the form of the Φn transform, we apply the following scaling relationship (9)
Then eq. (8) becomes ∂ρ(˜ r , t) ˜ 0 ∂ r˜2−λ/d ∂ ρ(˜ =K r , t), ∂t ∂˜ r ∂˜ r
(10)
˜ 0 = K0 d2 , r˜ = rd , and λ = 2 + θ. where K By applying the Φn transform to both sides and employing the derivative identity (eq. (7)), we obtain the diffusion equation in the wavenumber domain ∂ ρ˜ ˜ 0 k λ/d ρ˜. = −K ∂t
(11)
Eq. (11) can be exactly solved as ˜
ρ˜(k, t) = e−K0 k
λ/d
t
ρ˜0 .
0.3
0.1 0
III. SOLVING THE GENERALIZED DIFFUSION EQUATION WITH GENERALIZED FOURIER TRANSFORM
∂ ∂ (·) = drd−1 d (·). ∂r ∂r
0.4
0.2
(7)
In [21], the Φn transform is developed for integer order case. However, it can be easily extended to the noninteger case; see [21] for additional discussion of the properties of the Φn transform.
A.
0.5
ρ
. Fourier transform, we have Φn gn = gn , if gn (x) = e In addition, the generalized Fourier transform also has the following derivative property:
(12)
−0.1
0
1
2
3
4
5
r
FIG. 1: Comparison between exact (line) and numerical (symbols) solutions with K0 = 1, D = 1 and θ = 2.5.
The solution to eq. (8) is then obtained by applying the inverse Φn transform to ρ˜(k, t). The exact solution to eq. (8) for a point source at the origin (i.e., ρ(r, t0 ) = δ(r)) is given by [18] ρa (r, t) =
1 d/λ λ rλ .(13) exp − dΓ (d/λ)) K0 λ2 t K0 λ2 t
We validate the Φn transform method by comparing the numerical results with the analytical solution. Fig. (1) shows the analytical and the numerical solution for K0 = 1, D = 1, and θ = 2.5 at different times. According to the classification discussed in [15], this example is a subdiffusion case with θ > D(1 − ν). From fig. (1), it can be seen that the numerical solution is in good agreement with the analytical solution. In addition, we observe the short tail behaviours of the solution ρ(r, t).
B.
Generalized nonlinear equation
Now we consider the generalized nonlinear diffusion equation with ν 6= 1. In [15], eq. (1) is analytically solved using a generalized stretched Gaussian function approach: ρ(r, t) = [1 − (1 − q)β(t)rλ ]1/(1−q) /Z(t),
(14)
if 1−(1−q)β(t)rλ ≥ 0, and ρ(r, t) = 0 if 1−(1−q)β(t)rλ < 0. Here q = 2 − ν, and β(t) and Z(t) are functions given in eq. (12) in [15]. The same solution is derived in [16] using Lie group symmetry method. In order to solve the generalized diffusion equation numerically, we follow the procedure in III A, transforming the spatial domain equation to the wavenumber domain using the Φn transform. Instead of eq. (11), the wavenumber domain diffusion equation becomes ∂ ρe e 0 k λ/d f = −K ρν . ∂t
(15)
3 1
t0=0.1 Initial t=0.2 Ex. Sol. t=0.2 Num. Sol. t=0.4 Ex. Sol. t=0.4 Num. Sol. t=0.6 Ex. Sol. t=0.6 Num. Sol.
0.14 0.12 0.1
0.8 0.7 0.6
ρ
ρ
0.08
t=t0 Initial t=t0+0.05 t=t0+0.15 t=t0+0.25 Normal t=t0+0.25
0.9
0.06
0.5 0.4
0.04
0.3
0.02
0.2 0.1
0
0
−0.02 0
0
5
10
15
20
25
0.5
1
30
1.5
2
2.5
3
r
r
FIG. 3: Numerical solution with K0 = 1, D = 1 and θ = 2.5 for the initial condition eq. (16).
(a)
t0=0.1 Initial t=0.2 Ex. Sol. t=0.2 Num. Sol. t=0.4 Ex. Sol. t=0.4 Num. Sol. t=0.6 Ex. Sol. t=0.6 Num. Sol.
0.14 0.12 0.1
ρ
0.08
C.
The numerical method provides us a way to solve the generalized diffusion equation for the non-point-source initial condition. In fig. (3), we present the numerical solution of the generalized diffusion equation for K0 = 1, D = 1 and θ = 2.5 with the Gaussian initial condition
0.06 0.04 0.02 0 −0.02 0
0.5
1
1.5
2
2.5
Generalized diffusion for arbitrary initial condition
3
1 r2 ρ0 (r, t0 ) = √ ), exp(− 4K0 t0 4πt0
(16)
r
(b)
FIG. 2: Comparison between exact (line) and numerical (symbols) solutions in (a) scaled coordinate and (b) original coordinate withK0 = 1, D = 3, θ = 2.5 and ν = 0.8.
There is no exact solution for eq. (15) except for ν = 1. However, it can be numerically solved by employing certain types of time-stepping discretization methods for the time derivative. The comparisons between the exact [15, 16] and numerical solution for the Delta function initial condition in scaled and original coordinates are shown in figs. (2a) and(2b) , respectively. To avoid performing a Φn transform of the fractional order Delta function, we use ρa (r, t0 = 0.1) as the initial condition for our numerical algorithm. The parameters used here are K0 = 1, D = 3, θ = 2.5, and ν = 0.8. The forward Euler finite difference scheme with ∆t = 0.01s is employed for time discretization in the numerical simulation. Again, good agreement between the numerical and analytical solutions can been observed.
where t0 = 0.1. As we can see, the diffusion process finally approaches the same generalized gaussian shape as in the point source case (fig. (1)). This is a numerical example of what amounts to a “generalized central limit” behaviour in which the diffusion process will finally transform the arbitrary initial distribution to the corresponding generalized Gaussian distribution [24, 25]. By comparing with the solution for normal diffusion with the same initial condition, the subdiffusion process clearly exhibits the short tail behaviour.
IV.
CONCLUSION
In this paper, a numerical method for solving the generalized nonlinear diffusion equation has been presented and validated. The method is based on the generalized Fourier transform and has been validated by comparing the numerical solution with analytical solution for the point source. The presented method may be useful to study a variety of systems involving the anomalous diffusion. Currently, no fast transform algorithm has yet been developed for the Φn transform. This issue will be investigated in a future study.
4 ACKNOWLEDGMENTS
Discussion with Bernhard G. Bodmann are appreciated. Financial support of this research under R.A.
[1] V. Gavrilov, N. Klepikova, and H. Rodean, Atmospheric Environment 29, 2317 (1995). [2] H. Spohn, Journal de Physique I 3, 69 (1993). [3] E. Daly and A. Porporato, Physical Review E 70, 056303 (2004). [4] H. Risken, in The Fokker-Planck Equation (Springer, 1984) pp. 63–95. [5] K. S. Fa, Physical Review E 72, 020101 (2005). [6] K. S. Fa, Physical Review E 84, 012102 (2011). [7] M. Bologna, C. Tsallis, and P. Grigolini, Physical Review E 62, 2213 (2000). [8] P. Assis Jr, P. da Silva, L. da Silva, E. Lenzi, and M. Lenzi, Journal of mathematical physics 47, 3302 (2006). [9] E. Lenzi, M. Lenzi, T. Gimenez, and L. da Silva, Journal of Engineering Mathematics 67, 233 (2010). [10] R. Zola, M. Lenzi, L. Evangelista, E. Lenzi, L. Lucena, and L. da Silva, Physics Letters A 372, 2359 (2008). [11] R. Metzler and J. Klafter, Physics reports 339, 1 (2000). [12] I. M. Sokolov, J. Klafter, and A. Blumen, Physics Today 55, 48 (2002). [13] C. Tsallis and E. Lenzi, Chemical Physics 284, 341 (2002). [14] F. Liu, V. Anh, and I. Turner, Journal of Computational and Applied Mathematics 166, 209 (2004).
Welch Foundation Grant E-0608 is gratefully acknowledged. Partial support for this work was provided by resources of the uHPC cluster managed by the University of Houston under NFS Award Number 1531814.
[15] L. Malacarne, R. Mendes, I. Pedron, and E. Lenzi, Physical Review E 63, 030101 (2001). [16] I. T. Pedron, R. Mendes, L. C. Malacarne, and E. K. Lenzi, Physical Review E 65, 041108 (2002). [17] B. Abraham-Shrauner, Journal of Physics A: Mathematical and General 38, 2547 (2005). [18] B. O’Shaughnessy and I. Procaccia, Physical review letters 54, 455 (1985). [19] J. Crank, The mathematics of diffusion (Oxford university press, 1979). [20] G. Bluman and S. Kumei, Journal of Mathematical Physics 21, 1019 (1980). [21] C. L. Williams, B. G. Bodmann, and D. J. Kouri, arXiv preprint arXiv:1403.4168 (2014). [22] D. J. Kouri, C. L. Williams, N. Pandya, and H. Eisenberg, arXiv preprint arXiv:1610.04925 (2016). [23] R. Haberman, Elementary applied partial differential equations, Vol. 987 (Prentice Hall Englewood Cliffs, NJ, 1983). [24] G. Toscani, Journal of Evolution Equations 5, 185 (2005). [25] V. Schw¨ ammle, F. D. Nobre, and C. Tsallis, The European Physical Journal B 66, 537 (2008).
