Generalized Fourier transform method for nonlinear anomalous diffusion equation

TL;DR

This paper introduces a generalized Fourier transform method to numerically solve nonlinear anomalous diffusion equations involving fractal dimensions and power-law dependencies, extending traditional Fourier methods for normal diffusion.

Contribution

The paper develops and validates a generalized Fourier transform approach specifically designed for nonlinear anomalous diffusion equations with complex dependencies.

Findings

The method accurately reproduces known exact solutions for point-source cases.

It effectively handles arbitrary initial conditions in anomalous diffusion scenarios.

The approach extends Fourier transform techniques to more complex nonlinear diffusion equations.

Abstract

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 normal diffusion equation. The feasibility of the approach is validated by comparing the numerical result with the exact solution for point-source. The merit of numerical method is that it provide a way to calculate anomalous diffusion with an arbitrary initial condition.