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--- vpde_lecture23 [2020/03/26 13:32]
trinh
+++ vpde_lecture23 [2020/03/26 16:42]
trinh
@@ Line 80: / Line 80: @@
 ==== The 1D heat equation with a steady state temperature ====
+We will now examine the methodology for solving the non-homogeneous heat equation with Dirichlet conditions.
+$$
+\begin{gathered}
+u_t = \kappa u_{xx}, \quad x\in[0, L], t \geq 0 \\
+u(0, t) = T_0, \quad u(L, t) = T_1 \\
+u(x, 0) = f(x).
+\end{gathered}
+$$
+The trick is to seek a steady state solution. Seek a solution that does not depend on time. Then $u(x, t) = U(x)$ and we must satisfy:
+$$
+\begin{gathered}
+= \kappa u_{xx}, \quad x\in[0, L], t \geq 0 \\
+U(0) = T_0, \quad U(L) = T_1.
+\end{gathered}
+$$
+The solution is then $U(x) = T_0 + (T_1 - T_0)x/L.
+Next, we set the solution as
+$$
+u(x,t) = U(x) + \hat{u}(x,t).
+$$
+Why do this? Substitute into the system now to see that
+$$
+\begin{gathered}
+\hat{u}_t = \kappa \hat{u}_{xx}, \quad x\in[0, L], t \geq 0 \\
+\hat{u}(0, t) = 0, \quad \hat{u}(L, t) = 0. \\
+\hat{u}(x, 0) = f(x) - U(x).
+\end{gathered}
+$$
+In other words, the effect of the trick of writing the solution using the steady-state $U(x)$ has effectively zero'ed the boundary conditions. So we can simply use the same techniques as developed above for the zero Dirichlet problem.
+The algorithm is summarised in the video, and we will complete the demonstration in Friday's class.

Trinh @ Bath