Solving unbounded PDE's using Fourier Transforms and Convolution: Obtaining the d'Alembert solution of the wave equation

This lecture covers elementary properties of Fourier Transforms, including properties for 1D, 2D, and 3D transforms, the Fourier Transform of an integral, Fourier Convolution, Parseval's Theorem, self-convolution, and Energy Spectral Density (ESD). It then explores the ESD of a finite wave train and the application of Fourier Transforms and convolution to solve the one-dimensional wave equation in an infinite system, leading to the d'Alembert solution. The lecture emphasizes the importance of initial conditions in Fourier transform space and demonstrates the convenient deployment of convolution in solving PDEs, providing a general solution for a wave in a one-dimensional system.