The second question in my book and I've not the foggiest idea where to begin:

Suppose the rod has a constant internal heat source, so that the basic equation describing the heat flow within the rod is

$$u_t = \alpha ^2 u_{xx} + 1 \qquad for \qquad 0 \lt x \lt 1$$

Suppose we fix the boundaries' temperatures by $u(0, t) = 0$ and $u(1, t) = 1$. What is the steady-state temperature of the rod? In other words, does the temperature $u(x, t)$ converge to a constant temperature $U(x)$ independent of time?

What I know: It seems that if our equation $u(x, t)$ satisfies the boundary conditions, then we can assume that it is not dependent on time. For example, a structure of $u(x, t)$ satisfying the boundary conditions would be something of the form $u(x, t) = x$, so naturally $u_t = 0$. However, I've not any idea how this aids me in solving the problem, and I am not necessarily sure how to graph the PDE. Any help here? They got some wonky answer: $U(x) = - \frac{1}{2\alpha^2} (x^2 - x) +x$.

  • $\begingroup$ It has been really long since I have done this but I remember that Farlow's Partial Differential Equations for Scientists and Engineers has an excellent section on how to solve inhomogeneous boundry value problems like this by using the eigenfunction method. Maybe you can get it from your local/university library. $\endgroup$ – qmd Aug 30 '16 at 14:12
  • $\begingroup$ @qmd this problem comes from his book - particularly lesson 2 problem #2. $\endgroup$ – user312437 Aug 30 '16 at 14:14

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