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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.

In the last few decades, anomalous diffusion has been extensively studied in a variety of physical applications, such as turbulent diffusion [

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 [

∂ ρ ∂ t = K 0 1 r d − 1 ∂ ∂ r ( r d − 1 − θ ∂ ∂ r ρ ν ) , (1)

with the initial and boundary conditions

ρ ( r , t 0 ) = ρ 0 ( r ) , (2)

ρ ( ∞ , t ) = 0, (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 [

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 Φ n transform) is a new family of integral transforms developed by Willams et al. [

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.

The generalized Fourier transform Φ n is defined as

Φ n f ( k ) = ∫ φ n ( k x ) f ( x ) d x , (4)

where the integral kernel φ n ( ω x ) = c n ( k x ) + i s n ( k x ) is,

c n ( η ) = 1 2 | η | n − 1 / 2 J − 1 + 1 2 n ( | η | n n ) , (5)

and

s n ( η ) = 1 2 sgn ( η ) | η | n − 1 / 2 J 1 − 1 2 n ( | η | n n ) , (6)

where J ν ( η ) is the cylindrical Bessel function, and n is the transform order, i.e., n = 1 , 2 , ⋯ .

The Fourier transform F is the special case with n = 1 . The Φ n 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 [ g 1 ] = g 1 if g 1 ( x ) = exp ( − x 2 / 2 ) . For the generalized Fourier transform, we have Φ n g n = g n , if g n ( x ) = e − x 2 n 2 n .

In addition, the generalized Fourier transform also has the following derivative property:

Φ n [ ∂ ∂ x x 2 − 2 n ∂ ∂ x f ] = k − 2 n Φ n f . (7)

In [

It is well known that the Fourier transform can be used to find the solution for the standard diffusion equation [

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. ν = 1 ) [

∂ ρ ( r , t ) ∂ t = K 0 r d − 1 ( ∂ ∂ r r d − 1 − θ ∂ ∂ r ρ ( r , t ) ) . (8)

In order to perform the Φ n transform, we apply the following scaling relationship

∂ ∂ r ( ⋅ ) = d r d − 1 ∂ ∂ r d ( ⋅ ) , (9)

to Equation (8); and with some simplification, we obtain

∂ ρ ( r ˜ , t ) ∂ t = K ˜ 0 ∂ ∂ r ˜ r ˜ 2 − λ / d ∂ ∂ r ˜ ρ ( r ˜ , t ) , (10)

where K ˜ 0 = K 0 d 2 , r ˜ = r d , and λ = 2 + θ .

By applying the Φ n transform to both sides and employing the derivative identity (Equation (7)), we obtain the diffusion equation in the wavenumber domain

∂ ρ ˜ ∂ t = − K ˜ 0 k λ / d ρ ˜ , (11)

with ρ ˜ = Φ n ρ .

Equation (11) can be exactly solved as

ρ ˜ ( k , t ) = e − K ˜ 0 k λ / d t ρ ˜ 0 . (12)

The solution to Equation (8) is then obtained by applying the inverse Φ n transform to ρ ˜ ( k , t ) .

We validate the Φ n transform method by comparing the numerical results with the analytical solution or a point source at the origin (i.e. ρ ( r , t 0 ) = δ ( r ) ), which is given as [

ρ a ( r , t ) = λ d Γ ( d / λ ) ( 1 K 0 λ 2 t ) d / λ exp ( − r λ K 0 λ 2 t ) . (13)

Now we consider the generalized nonlinear diffusion equation with ν ≠ 1 . For the point source (or Dirac delta initial condition), Equation (1) was analytically solved using a generalized stretched Gaussian function approach in [

ρ ( r , t ) = ( [ 1 − ( 1 − q ) β ( t ) r λ ] 1 / ( 1 − q ) / Z ( t ) , if 1 − ( 1 − q ) β ( t ) r λ ≥ 0 ; 0, otherwise .

Here q = 2 − ν , and β ( t ) and Z ( t ) are functions given in Equation (12) in [

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 Φ n transform. Instead of Equation (11), the wavenumber domain diffusion equation becomes

∂ ρ ˜ ∂ t = − K ˜ 0 k λ / d ρ ν ˜ , (14)

with ρ ν ˜ = Φ n ( ρ ν ) .

Due to the presence of nonlinearity term in the right hand side of Equation (13) (i.e. ρ ν ), 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 Δ t = 0.01 s . An equally spaced mesh with N r = 1001 is used over the domain r ˜ = [ 0,30 ] . The comparisons between the exact [

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

ρ 0 ( r , t 0 ) = 1 4 π t 0 exp ( − r 2 4 K 0 t 0 ) , (14)

where t 0 = 0.1 .

As we can see, the diffusion process finally approaches the same generalized

Gaussian shape as in the point source case (

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 Φ n 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.

The authors declare no conflicts of interest regarding the publication of this paper.

Yao, J., Williams, C.L., Hussain, F. and Kouri, D.J. (2019) Generalized Fourier Transform Method for Solving Nonlinear Anomalous Diffusion Equations. Applied Mathematics, 10, 1039-1047. https://doi.org/10.4236/am.2019.1012072