I want to learn a very simple PDE to gain physical and mathematical intuition—a PDE with just one spatial dimension $x$ and a time dimension $t$. And, I want to write code for it in Matlab and use a standard PDE solver that Matlab might have, so that I can get a feel for the solution to the PDE and how the solution changes/behaves with a change in $x$ or $t$.
I have no experience with PDEs.
What could be a nice PDE to start with?
What's a nice Matlab solver for PDEs?
For ODEs, the Matlab solver ode45 is pretty nice to use.