Fourier Methods - Core Definitions

Fourier methods transform PDEs into algebraic equations in frequency space, providing powerful tools for both analysis and computation. These techniques exploit the diagonalizing property of Fourier bases for differential operators.

DefinitionFourier Series

For a $2\pi$ -periodic function $f(x)$ , the Fourier series is: $f(x) = \frac{a_0}{2} + \sum_{n=1}^\infty [a_n\cos(nx) + b_n\sin(nx)]$ where the Fourier coefficients are: $a_n = \frac{1}{\pi}\int_{-\pi}^\pi f(x)\cos(nx)\,dx, \quad b_n = \frac{1}{\pi}\int_{-\pi}^\pi f(x)\sin(nx)\,dx$

In complex form: $f(x) = \sum_{n=-\infty}^\infty c_n e^{inx}$ where $c_n = \frac{1}{2\pi}\int_{-\pi}^\pi f(x)e^{-inx}\,dx$ .

DefinitionFourier Transform

For functions on $\mathbb{R}^n$ , the Fourier transform is: $\hat{f}(\xi) = \mathcal{F}[f](\xi) = \int_{\mathbb{R}^n} f(x)e^{-2\pi i x \cdot \xi}\,dx$

The inverse Fourier transform is: $f(x) = \mathcal{F}^{-1}[\hat{f}](x) = \int_{\mathbb{R}^n} \hat{f}(\xi)e^{2\pi i x \cdot \xi}\,d\xi$

Key Property: Fourier transform converts differentiation to multiplication: $\mathcal{F}\left[\frac{\partial f}{\partial x_j}\right] = 2\pi i\xi_j\hat{f}(\xi)$

This transforms the PDE $\nabla^2 u = f$ to the algebraic equation $-4\pi^2|\xi|^2\hat{u} = \hat{f}$ .

ExampleSolving the Heat Equation via Fourier Transform

For $u_t = k\nabla^2 u$ with initial data $u(x,0) = f(x)$ , taking the Fourier transform: $\frac{\partial\hat{u}}{\partial t} = -4\pi^2k|\xi|^2\hat{u}$

This ODE has solution $\hat{u}(\xi,t) = \hat{f}(\xi)e^{-4\pi^2k|\xi|^2t}$ , giving: $u(x,t) = \mathcal{F}^{-1}[\hat{f}(\xi)e^{-4\pi^2k|\xi|^2t}] = f * G_t$ where $G_t$ is the heat kernel.

Remark

The exponential decay $e^{-4\pi^2k|\xi|^2t}$ in frequency space corresponds to smoothing in physical space: high frequencies (rough features) are damped more rapidly. This explains the smoothing property of the heat equation from a frequency perspective.

Fourier methods are particularly powerful for PDEs with constant coefficients on simple domains (whole space, periodic domains, or domains where separation of variables applies).