What does the solution of a PDE represent? So I took a course in PDE's this semester and now the semester is over and I'm still having issue with what exactly we solved for. I mean it in this sense, in your usual first or second calculus course you are taught the concept that your second derivative represents acceleration, your first derivative represents velocity and your fucntion is some of position function. So in the world of PDE's what does the function U(x,t) really capture? Your derivatives model rates of change of a phenomena in a broad sense, but what does the soultion model? So for example the solution of the Heat equation is your U(x,t) function modelling temperature? or or what is the solution to the wave equation modelling? Are these some sort of "position functions"? I ask because I'm going over Green's equations and the relationship with the Dirac function and I'm trying to wrap my head around what is being described. I understand the steps towards a solution, but it is the description of what is occurring which is faltering and essentially is the important part behind why we do mathematics.  Also is there some sort of visual representation of these solutions?  Thanks

There are tons of situations where the solution of a PDE discribes the behaviour over time:
1) pendulum
2) (radio active) decay
3) mass-spring-damper systems
4) inductance-capacitance-resistance systems
5) heat distribution over time
6) water level in a bucket with a leak
7) Temperature of a heated house
8) (electromagnetic) waves
