PDE as a system of ODEs

Question

It has been said that a PDE is like an infinite dimensional system of ODEs. How can one see this by a clear example? Can any PDE be transformed into an infinite dimensional system of ODEs?

Answer 1

Consider a PDE with two independent variables $x$ and $y$ and dependent variable $u$. You can approximate derivatives with respect to $x$ by difference quotients, e.g. $$ \dfrac{\partial u}{\partial x} \approx \frac{u(x+h,y)-u(x,y)}{h}, \ \dfrac{\partial^2 u}{\partial x^2} \approx \dfrac{u(x+h,y) - 2 u(x,y) + u(x-h,y)}{h^2}$$ Then with $x_k = h k$ for integers $k$, the PDE becomes a system for the infinitely many dependent variables $u_k(y) = u(x_k, y)$.