Random heat evolution HW
I have to admit I'm really stuck on how to approach the Random heat evolution problem.
- What is $g(x)$ supposed to represent? $u(x,0)$? $u_t(x,0)$?
- Are we supposed to actually solve the equation, or just give a qualitative graph showing how the temperature distribution evolves over time?
Comments
edit: nvm, the times are too specific for this to be a qualitative graph, so we might actually need to solve the equation.
Thanks for the answer @flappy_bird. I don’t see how $g(x)$ can tell you $u(x,0)$, because if you plug in $x = 0$ and $x = 1$ into $g(x)$, you won’t get the values the problem gives you for the endpoints. What I’m thinking is that it’s supposed to be the distribution before $t = 0$, that is, before the endpoints are set at the values the problem gives us. So the graph at $t = 0$ would look like $g(x)$ in the middle, but $u(0,0)$ and $u(1,0)$ would have the values given for the endpoints. At each time interval, the graph would smooth out until it became linear, as in the first example graph we did in class. The time intervals do look specific, yes, but they were also specifically used in class as well for a function which was only graphed and never solved (or even precisely defined). It’s also notable that unlike the steady state problem, this one doesn’t ask us for a “description of how you found the solution,” which would be expected if we were actually solving an equation. But you’re right, I think we need @mark to clarify this one.
@AbS I don’t see how $g(x)$ can tell you $u(x,0)$, because if you plug in $x = 0$ and $x = 1$ into $g(x)$, you won’t get the values the problem gives you for the endpoints.
The boundary conditions are $$u(0,t)=a \: \text{ and } \: u(1,t) = b$$ for all $t>0$. The initial condition is $$u(x,0)=g(x) \: \text{ for all } \: 0<x<1.$$ The solution $u$ might or might not be continuous at the endpoints. If $u$ is continuous, then the boundary and initial conditions are said to be consistent. There are plenty of natural situations where the initial and boundary are not consistent, though - like the example from class of the end of a hot metal bar bar being brought into sudden contact with a vat of ice.
So I am still confused over whether this graph is supposed to be just qualitative. I feel like I am missing a diffusivity constant if I wanted a more precise understanding of what my graph looks like at specific times.