Heat equation with initial boundary problem

Question

If I have been given in heat equation the initial boundary conditions $u_x(2,t) = 1$ then how I will use this it in question as I will proceed with separation of variable method ..since in question with boundary conditions $u_x(2,t)=0$ I can equate $X'(x)=0$ .but how in $u_x(2,t)= 1$ case

Answer 1

I'm guessing that you were also given a condition like $u_x(0,t)=0$, as otherwise you're solving the heat equation on an infinite domain. You also have some initial condition $u(x,0)=f(x)$. Your question boils down to solving a heat equation with inhomogeneous boundary conditions.

The usual approach here is to subtract out an equilibrium solution. For large $t$, you guess that there's a stable equilibrium $v(x,t)$ that's independent of $t$ but satisfies the boundary conditions (and the heat equation). So an easy example here is $v(x,t)=x/2$. Now consider $w(x,t):=u(x,t)-x/2$, which solves a homogenous heat equation. So, just adjust all the boundary conditions to fit $w$, and solve the heat equation $w_{t}=w_{xx}$. Finally $u(x,t)=w(x,t)+x/2$.