What is PDE?

A relation between the values taken at each $x \in Ω \subset R^{n}$ by a function $u : Ω \to R$ & its derivatives $u, D u, D^{2} u, \dots, D^{k} u$ : $F (x, u (x), D u (x), \dots, D^{k} u (x)) = 0$ OR $F (\cdot, u, D u, \dots, D^{k} u) = 0$ OR $F (J^{k} u) = 0.$ The order of (PDE) is $k$ .

For us, $k = 2$ .

Examples: Newtonian gravity Diffusion Vibration in a String Schrodinger Equation