Help me to solve the convection diffusion problem on the upper half plane with a time dependent non-homogeneous boundary at $y=0$ and a time dependent coefficient for the first $y$ derivative.
Solve $$\frac{\partial p}{\partial t}+f(t)\frac{\partial p}{\partial t}=a\left(\frac{\partial^2p}{\partial x^2}+\frac{\partial^2p}{\partial y^2}\right)$$$-\infty\le x\le\infty,0\le y\le\infty,0\le y\le\infty$
With boundary and initial conditions: $$p(\pm\infty,y,t)=0\\p(x,\infty,t)=0\\p(x,y>0,0)=0\\p(x,0,t)=Q(t)$$
The functions $f(t)$ and $Q(t)$ are non-zero functions of time only, and $a$ is a constant. $p$ is a dimensionless variable, hence $Q(t)$ is also dimensionless, and $f(t)$ has units $m/s$.
I can eliminate the first $y$ derivative by defining a new variable $z=y-\int f(t)\,\mathrm dt$ which gives the pde: $$\frac{\partial p}{\partial t}=a\left(\frac{\partial^2p}{\partial x^2}+\frac{\partial^2p}{\partial z^2}\right)$$$-\infty\le x\le\infty,-\int f(t)\,\mathrm dt\le z\le\infty,0\le y\le\infty$
With boundary and initial conditions: $$p(\pm\infty,y,t)=0\\p(x,\infty,t)=0\\p(x,z>-\int f(t)\,\mathrm dt,0)=0\\p(x,-\int f(t)\,\mathrm dt,t)=Q(t)$$
Now that the $z$ boundary is not at zero I am unsure how to solve this problem.
I have also tried taking Laplace and Fourier transforms but I found the inversion too difficult.