Generalized Fourier Transform Method for Solving Nonlinear Anomalous Diffusion Equations

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 exten-sion 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.


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 diffusion is usually characterized by the time dependence of mean-square displacement (MSD) viz., sion 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] [8] [9] [10]. In some approaches, fractional equations have been employed to analyze anomalous diffusion and related phenomena [11] [12] [13] [14] [15].
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 [16] [17] [18] with the initial and boundary conditions 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 [19]. It is the traditional nonlinear diffusion equa- where an ansatz for ρ is proposed as a general stretched Gaussian function. In [18], the same analytic solutions were also obtained by using Lie group symme- Conclusions are drawn in 4.

Generalized Fourier Transform
The generalized Fourier transform and ( ) ( ) where ( ) J ν η is the cylindrical Bessel function, and n is the transform order, i.e., 1, 2, n =  . The Fourier transform  is the special case with 1 n = . The n Φ 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, . For the generalized Fourier transform, we have n n In addition, the generalized Fourier transform also has the following derivative property: In [22], the n Φ transform is developed for integer order case. However, it can be easily extended to the non-integer case; see [22] for additional discussion of the properties of the n Φ transform.

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 [26]. Motivated by this idea, here we explore employing the generalized Fourier transform for solving 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 (i.e. 1 ν = ) [19].
In order to perform the n Φ transform, we apply the following scaling rela- to Equation (8); and with some simplification, we obtain , and 2 λ θ = + .
By applying the n Φ transform to both sides and employing the derivative identity (Equation (7)), we obtain the diffusion equation in the wavenumber Equation (11) can be exactly solved as The solution to Equation (8) is then obtained by applying the inverse We validate the which is given as [19] ( ) ( )

Generalized Nonlinear Equation
Now we consider the generalized nonlinear diffusion equation with 1 ν ≠ . For the point source (or Dirac delta initial condition), Equation (1) was analytically solved using a generalized stretched Gaussian function approach in [16]:  (12) in [16]. The same solution is derived in [17] using Lie group symmetry method.
In as the initial condition for our numerical simulation. Again, good agreement between the numerical and analytical solutions can be observed.

Generalized Diffusion for Arbitrary Initial Condition
The merit of the numerical approach using the generalized Fourier transform is that it provides a way for solving the generalized diffusion equation with arbitrary initial condition. In Figure 3, we present the numerical solution of the ge- where 0 0.1 t = .
As we can see, the diffusion process finally approaches the same generalized  Gaussian shape as in the point source case (Figure 1). In [29], it was analytically shown that the normal diffusion equation, when initialized with a generalized Gaussian distribution will asymptotically approach its final solution, i.e. a Gaussian distribution. Here, we present a numerical example of what amounts to the "generalized central limit'' behaviour in which the diffusion process will finally transform the arbitrary initial distribution to the corresponding generalized Gaussian distribution [30] [31]. A rigorous proof of the existence of the attractor of the generalized Gaussian diffusion has been done [24]] for linear diffusion.
This is a consequence of the negative semi-definite spectrum of the usual Laplacian. However, as mentioned in [31], the diffusion procedure, initialized with different distribution, may take very long time to reach its asymptotic behaviour.
In addition, by comparing with the solution for normal diffusion with the same initial condition, the sub-diffusion process clearly exhibits the short tail behaviour.

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 serve as a useful tool to study a variety of systems involving the anomalous diffusion. Currently, no fast transform algorithm has yet been developed for the n Φ transform. This issue will be investigated in future study.