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| - | // | + | ====== Example 15.4: Solution of the inhomogeneous Dirichlet problem ====== |
| + | |||
| + | This lecture starts off with Example 15.4 from the notes, where we look to solve | ||
| + | $$ | ||
| + | \begin{gathered} | ||
| + | u_t = u_{xx} \\ | ||
| + | u(0, t) = 2, \qquad u(\pi, t) = 1 \\ | ||
| + | u(x, 0) = 0. | ||
| + | \end{gathered} | ||
| + | $$ | ||
| + | |||
| + | We show that the solution is given by | ||
| + | $$ | ||
| + | u(x, t) = U(x) + \sum_{n=1}^\infty B_n \sin(nx) \mathrm{e}^{-n^2 t} | ||
| + | $$ | ||
| + | where we have found the steady-state solution | ||
| + | $$ | ||
| + | U(x) = 2 - \frac{x}{\pi}, | ||
| + | $$ | ||
| + | as well as the coefficients | ||
| + | $$ | ||
| + | B_n = \frac{2}{n\pi}[(-1)^n - 2]. | ||