fThere is something I do not understand about the Goursat problem:
- For a first order PDE if you prescribe the a value at a point, you can propagate the solution along a charateristic.
- For a second order hyperbolic PDE you need prescribe the function and normal derivative. In this case can you also propage the solution along a charateristic? If so, along which charateristic is it (there are two charateristic curves passing through each point).
- Finally, the Goursat problem involves prescribing the function along two intersecting charactristics. But if (2) makes any sense then if you know the function and normal derivative at a point, then you know the solution along a characteristic, and then you have the Goursat data and can solve. So knowing the function and derivative at one POINT is enough. I know this does not make sense, but where is the error??