Solving the heat equation, wave equation, Poisson equation using separation of variables and eigenfunctions 1 Review: Interval in one space dimension

Our domain G = (0,L) is an interval of length L. The boundary ∂G = {0,L} are the two endpoints. We consider here as an example the case (DD) of Dirichlet boundary conditions: Dirichlet conditions at x = 0 and x = L. For other boundary conditions (NN), (DN), (ND) one can proceed similarly. In one dimension the Laplace operator is just the second derivative with respect to x: ∆u(x, t) = uxx(x, t). We will consider three different problems: • heat equation ut −∆u = f with boundary conditions, initial condition for u • wave equation utt −∆u = f with boundary conditions, initial conditions for u, ut • Poisson equation −∆u = f with boundary conditions Here we use constants k = 1 and c = 1 in the wave equation and heat equation for simplicity. But the case with general constants k, c works in the same way.