Solving PDEs with the Laplace Transform - The Heat Equation

Learn how to solve Partial Differential Equations (PDEs) using Laplace Transforms, focusing on the heat equation for a semi-infinite domain. Explore the relationship between classic methods and modern problems, apply the Laplace Transform with respect to time, and solve ODEs with forcing through homogeneous and particular solutions. Discover how to handle initial and boundary conditions, and understand the solution in both frequency and time domains. This 40-minute video tutorial by Steve Brunton provides a comprehensive breakdown of the problem-solving process, offering valuable insights for students and professionals in applied mathematics and engineering.

Overview and Problem Setup
How Classic Methods e.g., Laplace Relate to Modern Problems
Laplace Transform with respect to Time
Solving ODE with Forcing: Homogeneous and Particular Solution
The Particular Solution and Initial Conditions
The Homogeneous Solution and Boundary Conditions
The Solution in Frequency and Time Domains