Heat equation Updated +Created
Besides being useful in engineering, it was very important historically from a "development of mathematics point of view", e.g. it was the initial motivation for the Fourier series.
Some interesting properties:
Wave equation solver Updated +Created
This section talks about solvers/simulators dedicated solving the wave equation. Of course, any serious solver will likely be able to solve a wider range of PDE, so this section contains mostly fun toys. For more serious stuff see: Section "PDE solver".
JavaScript toy solvers: