In the last few decades, anomalous diffusion has been extensively studied in a variety of physical applications, such as turbulent diffusion , surface growth , transport of fluid in porous media , hydraulics problems , etc. The 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, and nonlinearly with time for the anomalous diffusion. 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 . 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  . It can also be achieved by incorporating nonlinear terms in the diffusion term, or external forces    . In some approaches, fractional equations have been employed to analyze anomalous diffusion and related phenomena     .
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   
with the initial and boundary conditions
where r is the radial coordinate, and and are real parameters. When the diffusion coefficient is a function of r only, it is a generalization of the diffusion equation for fractal geometry . It is the traditional nonlinear diffusion equation when the diffusion coefficient depends on only  . Analytical solutions of Equation (1) with a point source have been reported in  , where an ansatz for is proposed as a general stretched Gaussian function. In , 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 Equation (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.  . These transforms share all the properties of the Fourier transform; hence can be employed to perform more general frequency and time-frequency analysis  .
In Section 2, 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 3.1 and 3.2. The method is validated by comparison between analytical and numerical results. Then, some numerical results for a non-Delta function initial condition are given in 3.3. Conclusions are drawn in 4.
2. Generalized Fourier Transform
The generalized Fourier transform is defined as
where the integral kernel is,
where is the cylindrical Bessel function, and n 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 , the transform is developed for integer order case. However, it can be easily extended to the non-integer case; see  for additional discussion of the properties of the transform.
3. 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 . Motivated by this idea, here we explore employing the generalized Fourier transform for solving the generalized nonlinear diffusion equation.
3.1. 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 r only (i.e. ) . Equation (1) can be reduced to
In order to perform the transform, we apply the following scaling relationship
to Equation (8); and with some simplification, we obtain
where , , and .
By applying the transform to both sides and employing the derivative identity (Equation (7)), we obtain the diffusion equation in the wavenumber domain
Equation (11) can be exactly solved as
The solution to Equation (8) is then obtained by applying the inverse transform to .
We validate the transform method by comparing the numerical results with the analytical solution or a point source at the origin (i.e. ), which is given as 
Figure 1 shows the analytical and the numerical solution for , , and at different times. According to the classification discussed in , this example is a subdiffusion case with . From Figure 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 .
Figure 1. Comparison between exact (line) and numerical (symbols) solutions with , and .
3.2. Generalized Nonlinear Equation
Now we consider the generalized nonlinear diffusion equation with . For the point source (or Dirac delta initial condition), Equation (1) was analytically solved using a generalized stretched Gaussian function approach in  :
Here , and and are functions given in Equation (12) in . The same solution is derived in  using Lie group symmetry method.
In order to solve the generalized nonlinear diffusion equation numerically, we follow the procedure in 3.1, transforming the spatial domain equation to the wavenumber domain using the transform. Instead of Equation (11), the wavenumber domain diffusion equation becomes
Due to the presence of nonlinearity term in the right hand side of Equation (13) (i.e. ), an analytical solution in the form of Equation (12) is difficult to be obtained. However, Equation (13) can be numerically solved by employing certain types of time-stepping discretization methods for the time derivative. Here, the simple forward Euler finite difference scheme is employed for time discretization with . An equally spaced mesh with is used over the domain . The comparisons between the exact   and numerical solution for the point source (Dirac delta function ) initial condition in scaled ( ) and original (r) coordinates are shown in Figure 2(a) and Figure 2(b), respectively. The parameters used here are , , , and . Note that to avoid performing a transform for the fractional order of Dirac delta function (as the definition of is also an ongoing research topic  , we use as the initial condition for our numerical simulation. Again, good agreement between the numerical and analytical solutions can be observed.
3.3. 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 generalized diffusion equation for , and with the Gaussian initial condition
As we can see, the diffusion process finally approaches the same generalized
Figure 2. Comparison between exact (line) and numerical (symbols) solutions in (a) scaled coordinate and (b) original coordinate with , , and .
Figure 3. Numerical solution with , and for the initial condition Equation (14). The solution for normal diffusion with the same initial condition ( symbol) is also included for comparison.
Gaussian shape as in the point source case (Figure 1). In , 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  . A rigorous proof of the existence of the attractor of the generalized Gaussian diffusion has been done  ] for linear diffusion. This is a consequence of the negative semi-definite spectrum of the usual Laplacian. However, as mentioned in , 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.
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 transform. This issue will be investigated in future study.
Discussions with Bernhard G. Bodmann are appreciated. Partial support for this work was provided by resources of the uHPC cluster managed by the University of Houston under NSF Award Number 1531814.
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