1
$\begingroup$

In the event that I'm solving a partial differential equation through separation of variables, if I end up with an eigenvalue of zero, what do I do with the corresponding eigenfunction?

That is to say, if I end up with an eigenvalue of zero, am I going to add it to the sum that I get from solving for the other eigenvalues? If so, am I going to multiply it by my T(t) first? Something like:

$X_0(x)T(t) + \sum a_nT(t)X_n(x)$ or do we just list 0 as one of our eigenvalues, and then we don't make any changes to the way the function is written out?

  • 0
    For an initial condition $u(x,0) = f(x)$, then $u(x,0) = f(x) = a_0 T_0(0) X_0(x) + \sum a_n T_n(0) X_n(x)$ and you'll determine the $a_n$'s the same way as a _regular_ fourier series. i.e $ a_n = \frac{1}{T_n(0) \|X_n\|^2} \langle f(x), X_n(x) \rangle$ where the scalar product is the proper one.2012-11-14

0 Answers 0