Differential Equations With Fourier Ser... — Partial

To solve Partial Differential Equations (PDEs) like the Heat Equation or the Wave Equation , you use the method of separation of variables to turn a multivariable equation into several Ordinary Differential Equations (ODEs). Fourier Series are then used to combine these individual solutions to satisfy the initial and boundary conditions of the original problem. Assume the solution can be written as a product of two independent functions, . Substitute this into the PDE to isolate all terms on one side and all

). The spatial ODE is typically an eigenvalue problem (e.g., Partial Differential Equations with Fourier Ser...

), which you solve using the given boundary conditions (like ) to find specific values for and their corresponding eigenfunctions . To solve Partial Differential Equations (PDEs) like the