Generalized Fourier transform method for nonlinear anomalous diffusion equation
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 point-source. The merit of the numerical method is that it provides a way to calculate anomalous diffusion with an arbitrary initial condition.
Previously at ]University of Houston, Department of Mechanical Engineering
In the last few decades, anomalous diffusion has been extensively studied in a variety of physical applications, such as turbulent diffusion gavrilov1995trial, surface growth spohn1993surface, transport of fluid in porous media spohn1993surface, hydraulics problems daly2004similarity, etc. The anomalous diffusion is usually characterized by the time dependence of mean-square displacement (MSD) viz., . The MSD grows linearly with time () for the normal diffusion case. The process is called sub-diffusion for and super-diffusion for . 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 risken1984fokker. 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 fa2005exact; fa2011solution. It can also be achieved by incorporating nonlinear terms in the diffusion term, or external forces bologna2000anomalous; assis2006nonlinear; lenzi2010some; zola2008exact. In some approaches, fractional equations have been employed to analyze anomalous diffusion and related phenomena metzler2000random; sokolov2002fractional; tsallis2002anomalous; liu2004numerical.
In this paper, we study the generalized nonlinear diffusion equation including a fractal dimension and a diffusion coefficient which depends on the radial variable and the diffusion function malacarne2001nonlinear; pedron2002nonlinear; abraham2005lie
with the initial and boundary conditions
where is the radial coordinate, and and are real parameters. When the diffusion coefficient is a function of only, it is a generalization of the diffusion equation for fractal geometry o1985analytical. It is the traditional nonlinear diffusion equation when the diffusion coefficient depends on only crank1979mathematics; bluman1980remarkable. Analytical solutions of eq. (1) with a point source have been reported in malacarne2001nonlinear; pedron2002nonlinear, where an ansatz for is proposed as a general stretched Gaussian function. In abraham2005lie, 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 transform), is a new family of integral transforms developed by Willams et. al. williams2014fourier; kouri2016canonical. 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.1 and III.2. 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.3.
Ii generalized Fourier transform
The generalized Fourier transform is defined as
where the integral kernel is,
where is the cylindrical Bessel function, and is the transform order, i.e., .
The Fourier transform is the special case with . The 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, if . For the generalized Fourier transform, we have , if .
In addition, the generalized Fourier transform also has the following derivative property:
In williams2014fourier, the transform is developed for integer order case. However, it can be easily extended to the non-integer case; see williams2014fourier for additional discussion of the properties of the transform.
Iii solving the generalized diffusion equation with generalized Fourier transform
It is well known that the Fourier transform can be used to find the solution for the standard diffusion equation haberman1983elementary. Motivated by this idea, we explore employing the generalized Fourier transform to solve the generalized nonlinear diffusion equation.
iii.1 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 only ( or )o1985analytical. Eq. (1) can be reduced to
In order to satisfy the form of the transform, we apply the following scaling relationship
Then eq. (8) becomes
where , , and .
By applying the transform to both sides and employing the derivative identity (eq. (7)), we obtain the diffusion equation in the wavenumber domain
Eq. (11) can be exactly solved as
The solution to eq. (8) is then obtained by applying the inverse transform to .
The exact solution to eq. (8) for a point source at the origin (i.e., ) is given by o1985analytical
We validate the transform method by comparing the numerical results with the analytical solution. Fig. (1) shows the analytical and the numerical solution for , , and at different times. According to the classification discussed in malacarne2001nonlinear, this example is a subdiffusion case with . 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 .
iii.2 Generalized nonlinear equation
Now we consider the generalized nonlinear diffusion equation with . In malacarne2001nonlinear, eq. (1) is analytically solved using a generalized stretched Gaussian function approach:
if , and if . Here , and and are functions given in eq. (12) in malacarne2001nonlinear. The same solution is derived in pedron2002nonlinear using Lie group symmetry method.
In order to solve the generalized diffusion equation numerically, we follow the procedure in III.1, transforming the spatial domain equation to the wavenumber domain using the transform. Instead of eq. (11), the wavenumber domain diffusion equation becomes
There is no exact solution for eq. (15) except for . However, it can be numerically solved by employing certain types of time-stepping discretization methods for the time derivative.
The comparisons between the exact malacarne2001nonlinear; pedron2002nonlinear and numerical solution for the Delta function initial condition in scaled and original coordinates are shown in figs. ((a)a) and((b)b) , respectively. To avoid performing a transform of the fractional order Delta function, we use as the initial condition for our numerical algorithm. The parameters used here are , , , and . The forward Euler finite difference scheme with s is employed for time discretization in the numerical simulation. Again, good agreement between the numerical and analytical solutions can been observed.
iii.3 Generalized diffusion for arbitrary initial condition
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 , and with the Gaussian initial condition
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 toscani2005central; schwammle2008q. By comparing with the solution for normal diffusion with the same initial condition, the subdiffusion process clearly exhibits the short tail behaviour.
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 transform. This issue will be investigated in a future study.