I have been trying to solve a heat-exchanger problem where two fluids are separated by a conducting wall between them and the fluids flow perpendicular to each other. So i need to consider two dimensional conduction in this separating wall while heat gets exchanged between the two fluids through this wall.
Are there any examples of analytic solution to a Laplace equation coupled with another ODE ? I have failed to find any such example. Most texts deal with variants of the Laplace operator with different bc(s).