I have started my Differential equation course recently. And My professor was interchangeably using the term boundary condition and initial condition. I'm wondering what are the differences of their meaning in this context when dealing with the partial differential equations. I thought one would not need a boundary condition if a system extends to infinity, but the case of wave on a string would need, since it's a finite system For the initial condition, it always mean the initial state of the system. Can anyone tell me if my understandings are correct. Thank you!
If you think of a differential equation being defined on the coordinates of a certain object, that is mathematically like a number or a time line, and physically like a boundary.
So if $y(0)$ is given, in terms of time, that would be an "initial condition."
But if $y(0)$ and/or $y(L)$ are given for a differential equation that determines the temperature of a rod of length $L$, then $y(0)$ is more appropriately understood to be the "boundary condition" because it gives the temperature of the rod at the boundary of the rod.
Boundary conditions can be understood philosophically as being able to see only outside effects of something, but not directly measure the internal state of an object. The laws of physics governing the internal state of the object are used to formulate differential equations which can be solved to infer the internal state of the object based only on the external state that is known. This is like finding a criminal based on the evidence he leaves behind when committing a crime, and the large field of solving such problems is known as "Inverse Problems."