Solving PDEs with the Laplace Transform - The Wave Equation

Learn how to solve Partial Differential Equations (PDEs) using Laplace Transforms, focusing on the wave equation in a semi-infinite domain. Explore the problem setup, including initial and boundary conditions, before applying the Laplace Transform in time to convert the PDE to an ODE. Solve the resulting ODE in space and derive the general solution of the wave equation. Understand the Heaviside function and its application in this context. Conclude with an illustration of the solution and an introduction to the Method of Characteristics. This comprehensive video tutorial, presented by Steve Brunton, provides a step-by-step approach to mastering PDE solutions with Laplace Transforms.

Overview and Problem Setup Initial Conditions and Boundary Conditions
Laplace Transform in Time: PDE to ODE
Solving the ODE in Space
General Solution of the Wave Equation
The Heaviside Function
Illustration and Method of Characteristics