Basis Decomposition of PDEs: 1D and Multidimensional Problems

This lecture focuses on solving homogenous linear PDEs with constant coefficients using eigenmodes decomposition. It covers the extension to multi-dimensional problems, reconciliation with 'Separation of Variables', and treatment of initial profiles with Fourier series. The lecture also explores problems with finite-sized systems and the Fourier series representation of solutions. The basis decomposition approach is illustrated through the continuous wave equation and the solution's superposition of eigenmodes of vibrations.