If the position of some charge Q is known, the boundary condition is u=0 on some parabolic surface, and we know the image charge has its electric volume of Q', then how can I determine the position of the image charge?
Same questions goes for the hyperbolic curve boundary. How can I determine the position of Q'?
I think may be there is a way to transform the coordinates to make everything into an easily-handled form, but I am not sure about it. Another solution I thought about is to put this question into a general question based on basic Poisson's equation and Laplace equation, but I do not have a specific idea on how to conduct this.