I've got this problem where it is the diffusion equation with a source term in $ x : [0,L]$.
$\frac{\partial v}{\partial t} = D_V \frac{\partial^2 v}{\partial x^2} + \frac{\partial \bar{c}}{\partial t} $
$\frac{\partial v}{\partial x}|_{x = 0} +\beta_1(t)v|_{x = 0} = 0 $
$\frac{\partial v}{\partial x}|_{x = L} = 0$
$v(x,0) = c_0 - \bar{c}(t=0)$
Here $\bar{c}(x,t) = \frac{g_1(t)}{\beta_1(t)}$, where $\beta_1$ and $g_1$ are functions of time, $t$. What makes it tricky is that one of my boundary conditions, despite being homogenous, has a coefficient that is time-variant. So while trying to use the method of eigenfunction expansion, I found time-variant eigenvalues, $\lambda_n$, of the following form with the corresponding eigenfunctions, $f_n(x,t)$:
$tan(\lambda_nL) = \frac{-\beta_1(t)}{g_1(t)}$, where $\lambda_n > 0$
$f_n(x,t) = cos(x\lambda_n) - \frac{\beta_1(t)}{\lambda_n}sin(x\lambda_n) $
From this I know that the solution will be of the following form:
$v(x,t) = \sum T_n(t)f_n(x,t)$
Where $T_n(t) = exp[D_V \int_0^t\lambda_n(\tau)d\tau]\int_0^t(exp[-D_V \int_0^t\lambda_n(\tau)d\tau]\frac{2\bar{c}(\tau)}{L}\int_0^Lf_n(x,t)dx)d\tau$.
The issue is that I've never encountered time-variant eigenvalues and I'm not sure how to approach this situation since the eigenvalues are time-variant. Is there something I'm missing? I'm still pretty new to this subject.