A variational formulation for PDEs

Definition 1. A classic solution (sometimes called strong solution) to the problem is a function u ∈ C2([a, b]) such that it vanishes at the endpoints and pointwise it satisfies the equation −u′′(x)+ u(x) = f(x). Let’s change approach. Suppose that we formally multiply the differential equation by an arbitrary function φ ∈ C1([a, b]) with φ(a) = φ(b) = 0 and we integrate over the interval [a, b]: − ∫ b