Discussion
The steady state solution for the heat equation
\[ \begin{aligned} &\frac{\partial u}{\partial t} - \frac{\partial^{2} u}{\partial x^{2}} = 0, \\ &0 \lt x \lt 2,\ t \gt 0, \end{aligned} \]
with the initial condition u(x, 0) = 0, \(0 \lt x \lt 2\) and the boundary conditions u(0, t) = 1 and u(2, t) = 3, \(t \gt 0\), at x = 1 is
Answer: B
The steady-state value at x = 1 is 2.
At steady state u no longer depends on t, so the equation reduces to \(u'' = 0\), and u is linear in x. The initial condition does not affect the steady state; only the boundary values do:
\[ \begin{aligned} u(x) &= a + bx,\ u(0) = 1,\ u(2) = 3 \\ u(x) &= 1 + x,\ \ u(1) = 2 \end{aligned} \]
The steady temperature is the straight line between the end values, so at the midpoint it is their average, (1 + 3)/2 = 2. 1 and 3 are the end values themselves, and 4 is their sum.
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