Fourier Transform PDE Examples

PDEs · Axiom Academy

EXAMPLE Fourier Transform PDE Examples Solving PDEs using Fourier transforms: heat equation, wave equation, and diffusion Example 1: Heat Equation on Infinite Line Excellent work! You've completed three Fourier transform PDE examples. Here's what we learned: Fourier Transform Method: Transform spatial derivatives into algebraic operations in frequency space, converting PDEs into ODEs that are easier to solve. Heat Equation: The Gaussian initial condition remains Gaussian but spreads over time. The solution width grows as √(1 + 4t), demonstrating diffusion. Wave Equation: Fourier methods reveal wave propagation in terms of superposition of harmonic oscillators with frequency ck. Step Function Diffusion: Discontinuous initial conditions smooth out exponentially fast through diffusion, with the solution approaching the error function profile. Convolution Theorem: Products in frequency space become convolutions in physical space, making inverse transforms tractable. The Fourier transform is a powerful tool for solving linear PDEs on infinite or periodic domains. Master these techniques to tackle a wide range of physics and engineering problems!

This is the written version of the interactive lesson above. See the full PDEs course.