Klein-Gordon Equation with boundary conditions

[dsaun777]

I am trying to find solutions for the Klien-Gordon equations in 1-d particle in a box. The difference here is the box itself oscillating and has boundary conditions that are time dependent, something like this L(t)=L0+ΔLsin(ωt). My initial approach is to use a homogeneous solution and use Eigenfunction expansion to get a solution. But I can't make progress. Are there other methods that are easier? maybe numerical methods...

 

[pasmith]

Do you mean that you have homogenous BCs at x = 0 and x = L(t)?

Set \epsilon = \Delta L/L_0 so that the boundary condition is applied at x = L_0(1 + \epsilon \sin \omega t). Then for small \epsilon, you can seek an asymptotic expansion \psi = \psi_0 + \epsilon \psi_1 + \dots where the boundary conditions on the \psi_n are shifted to x = L_0 by expanding \psi(L_0(1 + \epsilon \sin \omega t),t) in Taylor series in x about x = L_0 and comparing coefficients of powers of \epsilon. This gives you a series of problems which can be solved by eigenfunction expansion.

EDIT: It might be better to solve for each \psi_n using a Laplace transform in time. The change of variable \tilde x = x/L_0 may help to simplify the algebra. Also note that if \omega is a natural frequency of any of these problems then the resulting resonance will cause the asymptotic assumption to break down, since eventually the amplitude will exceed \epsilon^{-1}.