Help with understanding BVP for the Heat equation (PDE)?

[Vitani11]

Homework Statement

Find the steady state (equilibrium) solution for the following boundary value problem:
∂u/∂t = (1/2)∂²u/∂x²
Boundary condition:
u(0,t) = 0 and u(1,t) = -1
Initial condition:
u(x,0) = 0

Homework Equations

u(x,t) = Φ(x)G(t)

The Attempt at a Solution

I have found the solution for Φ(x) and G(t) but when implementing the boundary condition u(1,t) = -1 I have that u(1,t) = Φ(1)G(t)=-1. My question is whether or not I should assume that G(t) = -1 and not Φ(1). The reason I ask is because the steady state solution should not depend on time, therefore G(t) needs to be a constant. However I don't trust this assumption and that leads me to a more general question which is: How am I supposed to know which function to make equal to the right hand side of the equation? I mean in any case where you can split a function (for example) u(x,t) = Φ(x)G(t) and have a boundary condition. In other words, if this problem didn't specify that the solution should be a steady state solution then what would you do when trying to figure out which function is equal to the right hand side?

 

[Orodruin]

How am I supposed to know which function to make equal to the right hand side of the equation?

It doesn't matter. You are anyway going to multiply them together to get the result. That being said, why do you think separation of variables is going to work here? There is no obvious Sturm-Liouville operator with homogeneous boundary condition that you find the eigenfunctions of. Furthermore, you are only looking for the stationary solution. The stationary solution by definition does not depend on time.

 

[Vitani11]

Solved, thank you.