I've been grappling with this problem for a while now and am at a loss. Here is the problem statement for a function $c(x,t)$ as defined by the following parabolic equation:
$\frac{\partial c}{\partial t} = D_V \frac{\partial^2 c}{\partial x^2}$
$\frac{\partial c}{\partial x}|_{x=0} + \beta_1(t) c|_{x=0} = g_1(t)$
$\frac{\partial c}{\partial x}|_{x=L} = 0$
$c|_{t=0} = c_0, constant$
Here there is a non-homogeneous Robin condition and a homogeneous Neumann condition. Originally, I thought to homogenize the boundary conditions and proceed to use the method of Eigenfunction expansion to solve the problem, but I note that $\beta_1$ is a function of $t$. Are there any other alternatives I should explore?