I am studying numeric methods for differential equations and I don't quite understand the purpose for the boundary and initial conditions in the heat equation.
I see that we have three boundary condition, one initial condition for time and two for space. I understand that the time initial condition is for the starting time of the heating, and the two boundary conditions are for the boundaries to which we look the heating in space.
In the ODE case I understood that we are given initial condition because there are many solutions and many functions differing with a constant from each other which satisfy the differential equation. Are the initial and boundary conditions in the PDE case (Heat equation) having a similar purpose or are they solely for restricting the space and time dimensions and the uniqueness being a consequence of that?
Thanks in advance!
Best Answer
Your question is very weird.
Why do people solve differential equations? Well, usually differential equations model something: the flow of heat, the vibration of a string or a surface, an electrical current, the motion of planets, something. The equation codifies the rules of evolution of the system you are studying but it should be obvious to you that to know what is actually going to happen with the heat, or the string, or the planet, you are going to know something about the initial condition (where is it now?) That is what the initial condition and the boundary conditions mean.