4 7.20m, §4 Eigenfunctions

To find the eigenfunctions $X_n$, substitute a trial solution $v=T(t) X(x)$ into the homogeneous part of the PDE, $v_t = \kappa v_{xx} + \bar q$. Remember: ignore the inhomogeneous part $\bar q$ when finding the eigenfunctions. Putting $v=T(t) X(x)$ into $v_t = \kappa v_{xx}$ produces:

$$T'X = \kappa T X''$$

Separate variables:

$$\frac{T'}{\kappa T} = \frac{X''}{X} = \hbox{ constant } = - \lambda$$

As always, $\lambda$ cannot depend on $x$ since the left hand side does not. Also, $\lambda$ cannot depend on $t$ since the middle does not. So $\lambda$ must be a constant.

We then get the following Sturm-Liouville problem for any eigenfunctions $X(x)$:

$$- X'' = \lambda X \qquad X'(0) = 0 \qquad X(\ell)=0$$

The last two equations are the boundary conditions on $v$ which we made homogeneous.

This is the exact same eigenvalue problem that we had in problem 7.28b, so I can just take the solution from there. The eigenfunctions are:

$$\fbox{$\displaystyle \lambda_n = \frac{(2n-1)^2 \pi^2}{4\e... ...c{(2n-1) \pi x}{2\ell}\right) \qquad (n = 1, 2, 3, \ldots) $}$$