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 [itex]x = 0[/itex] and [itex]x = L(t)[/itex]?

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

EDIT: It might be better to solve for each [itex]\psi_n[/itex] using a Laplace transform in time. The change of variable [itex]\tilde x = x/L_0[/itex] may help to simplify the algebra. Also note that if [itex]\omega[/itex] 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 [itex]\epsilon^{-1}[/itex].