Partial Differential Equations: Separation of Variables

This lecture covers the method of separation of variables to solve partial differential equations with boundary conditions, focusing on finding conditions for the convergence of the solution. The instructor explains the process step by step, from defining the problem to deriving the conditions for the convergence of the series. The lecture also delves into the theory of measures, discussing convergence properties of functions in different spaces and demonstrating the density of functions in Lp spaces. The instructor showcases the application of the separation of variables method through detailed mathematical derivations and proofs, emphasizing the importance of regularity conditions for ensuring convergence. The lecture concludes with a discussion on the convergence uniformity of solutions and the necessity of continuity and boundedness for achieving convergence.