solving linear PDE on a curve/contour

Hello,

I was wondering if there is a way to solve a given linear PDE:

A(x,y)*Fx+B(x,y)*Fy=C(x,y)

on a given curve G.

To be able to do that I need to parametrize the curve and transfer my
PDE to ODE right? is there a way to do that?

(I know how to solve the PDE for the whole F(x,y) but I need to find
the values of F only on a given curve without knowing what happens on
points that are not part of the curve)


Thank you for your time
kem

.