problem: $\frac{\partial w}{\partial t}=a_1(x,t) \frac{\partial^2 w}{\partial x^2}+\Phi(x, y, z, t)$
with
$w=f(x, y, z) \quad$ at $\quad t=0$.(initial condition)
$\frac{\partial w}{\partial x}+k(\mathbf{x}, t) w=g_3(\mathbf{x}, t) \quad$ for $\quad x=0$ (Third boundary )
I found the following content in the "Handbook of linear partial differential equations for engineers and scientists": $\begin{aligned} w(\mathbf{x}, t) & =\int_0^t \int_V \Phi(\mathbf{y}, \tau) G(\mathbf{x}, \mathbf{y}, t, \tau) d V_y d \tau+\int_V f(\mathbf{y}) G(\mathbf{x}, \mathbf{y}, t, 0) d V_y \\ & +\int_0^t \int_S g(\mathbf{y}, \tau) H(\mathbf{x}, \mathbf{y}, t, \tau) d S_y d \tau\end{aligned}$
Is it possible to have an analytical solution (i.e., What is the expression for the Green's function?) similar to the following picture?(Sorry, please allow me to be lazy and use a screenshot.)
Thank you very much for any help!