The steady state solution for the heat equation $$ \frac{\partial u}{\partial t}-\frac{\partial^2u}{\partial x^2}=0, \qquad 0<x<2, \quad t>0, $$ with the initial condition $$ u(x,0)=0, \qquad 0<x<2, $$ and the boundary conditions $$ \begin{aligned} &u(0,t)=1,\\ &u(2,t)=3, \end{aligned} \qquad t>0, $$ at $x=1$ is
(A) $1$
(B) $2$
(C) $3$
(D) $4$
I tried to solve through separation of variables, but it is too difficult to determine the arbitrary constants from the given conditions. Again I have tried by using Laplace transformation, in this way at the final step, inverse Laplace formation is not good.
Any easiest way to solve this problem ?
By definition, the steady-state solution $u_S$ is time-independent, hence $\dot{u}_S = 0$. In consequence, we have $u_S'' = 0$, which leads to $u_S(x) = Ax+B = x+1$, according to the boundary conditions. Finally, we get $u_S(1) = 2$.