In this paper we explain how to use the Fast Fourier Transform (FFT) to solve partial differential equations (PDEs). We start by defining appropriate discrete domains in coordinate and frequency domains. Then describe the main limitation of the method arising from the Sampling Theorem, which defines the critical Nyquist frequency and the aliasing effect. We then define the Fourier Transform (FT) and the FFT in a way that can be implemented in one and more dimensions. Finally, we show how to apply the FFT in the solution of PDEs related to problems involving two spatial dimensions, specifically the Poisson equation, the diffusion equation and the wave equation for elliptic, parabolic and hyperbolic cases respectively.

翻译：本文阐述了如何利用快速傅里叶变换（FFT）求解偏微分方程（PDEs）。我们首先在坐标域和频率域中定义合适的离散域，随后阐述该方法受采样定理制约的主要局限性——该定理定义了临界奈奎斯特频率与混叠效应。接着，我们以可在一维及多维中实现的方式定义了傅里叶变换（FT）与FFT。最后，我们展示了如何将FFT应用于涉及二维空间问题的PDE求解，具体包括分别对应椭圆型、抛物型和双曲型问题的泊松方程、扩散方程和波动方程。