I am wondering if there are any books providing intuitive explanations for PDEs. I have been using Evans' PDE, but it only focuses on how to solve PDEs and some other theoretical properties. I would like to find a book that discusses how to derive certain PDEs (such as the wave equation, Laplace's equation or some more advanced ones, etc.), as well as providing intuitive interpretations for each differential operator (For example: like what $\Delta$ is trying to do in the equation $u_t= \Delta u + u_x$). It doesn't need to be too mathematically rigorous.
My background is in mathematics, and I haven't studied much physics. So, books on physics are also great!