PDE 101 - Separation of Variables or How I Learned to Stop Worrying and Solve Laplace's Equation

Explore the powerful technique of Separation of Variables for solving Partial Differential Equations (PDEs) in this 50-minute video lecture. Learn how to apply this method to solve Laplace's equation in two dimensions for steady-state heat distribution on a rectangle. Discover the wide applicability of this technique in various physics and engineering problems. Begin with an overview of Laplace's Equation in 2D, then delve into linear superposition and the core concept of separation of variables. Follow along as the PDE is reduced to a system of ODEs, leading to the solution of the PDE. Benefit from a comprehensive recap of the separation of variables method, and conclude with insights on the last boundary condition and the Fourier Transform. Gain valuable problem-solving skills applicable to numerous scientific and engineering challenges.

Overview and Problem Setup: Laplace's Equation in 2D.
Linear Superposition: Solving a Simpler Problem.
Separation of Variables.
Reducing the PDE to a system of ODEs.
The Solution of the PDE.
Recap/Summary of Separation of Variables.
Last Boundary Condition & The Fourier Transform.