I would like to solve the particular problem for $f(x,y)$ and $g(x,y)$ on the half-space $y\ge 0$ below:
\begin{align} \dfrac{\partial^{2}f}{\partial y^{2}}+\dfrac{\partial g}{\partial x} & =0\text{ on }y>0,\\ \dfrac{\partial^{2}g}{\partial y^{2}}+\dfrac{\partial f}{\partial x}+1 & =0\text{ on }y>0,\\ g & =0\text{ on }y=0,\\ \dfrac{\partial f}{\partial y}-cf & =0\text{ on }y=0,\\ f & \sim-x\text{ as }y\to\infty,\\ g & \sim0\text{ as }y\to\infty, \end{align}
where $c > 0$.
I am stumped as to how to solve this problem exactly and I have tried several approaches. I am aware that the PDEs can be decoupled to give \begin{align*} \dfrac{\partial^{4}f}{\partial y^{4}}-\dfrac{\partial^{2}f}{\partial x^{2}}=0,\\ \dfrac{\partial^{4}g}{\partial y^{4}}-\dfrac{\partial^{2}g}{\partial x^{2}}=0, \end{align*} and then this makes them amenable to (a) similarity solutions, or (b) transform methods. I have tried setting, say, $f(x,y)=F(\eta)$ where $\eta = y/\sqrt{x},$ and whilst this turns the decoupled PDEs into (nonlinear) ODEs in $\eta$, the Robin boundary conditions on $f$ doesn't make any sense (since it is not an ODE in $\eta$). I am aware the considering the decoupled PDEs and taking a linear combination of Fourier sine and cosine transforms with respect to $y$ can help (and that the constants can be chosen such that the boundary terms involving the first and second derivatives incorporate the Robin condition on $f$) but then applying this transform to the PDE results in terms involving the second and third derivatives evaluated at $y=0$, which must be determined from the solution. I have tried both of these approaches but cannot make any progress with them.
I will spare the exact details of what I have tried, but my supervisor and I tried looking at this sort of problem a while ago and he suspects that it can only be solved numerically (since it apparently results in a singular integral equation), even though it is possible to make headway in the cases where $c=0$ and $c=\infty$ (where the boundary condition on $f$ at $y=0$ is either Neumann or Dirichlet, respectively).
I would be very interested to hear whether there are any other approaches (e.g. different integral transforms that I may not have considered) that might be useful for attacking a problem like this.