How can I solve a Dirichlet problem defined for a parabolic region?

[zebra]

Homework Statement

Hi, I am looking for a hint, how to solve the following Dirichlet problem. All the standard textbooks have only examples for Dirichlet problems in rectangular or polar coordinate systems, but this problem is defined for a parabolic region.

Homework Equations

u_xx+u_yy=2, x>y²
u(x,y)=0, x=y²

The Attempt at a Solution

Transformation into parabolic coordinates and using separation of variables? I am having a problem even with that. The parabolic coordinates with the parabolic Laplacian are here http://eom.springer.de/P/p071170.htm. I don´t even know, how to transform the boundaries. The parabolic coordinates are so unusual. Is this a right method or would you solve it differently? (with Green functions, or Laplace transform etc)

 

[mighty2000]

Hi there! It seems like you are on the right track with using separation of variables and transforming into parabolic coordinates. However, since this problem is defined for a parabolic region, it may be more appropriate to use the parabolic Laplacian and Green's functions instead of the traditional Laplacian.

One approach you can try is to first transform the given region into a rectangular one using the parabolic coordinate transformation. Then, you can use the standard methods for solving Dirichlet problems in rectangular coordinates, such as separation of variables or Green's functions.

Alternatively, you can also try using the Laplace transform method, where you take the Laplace transform of the given PDE and boundary conditions. This will transform the problem into an ODE, which you can then solve using standard techniques.

I hope this helps! Let me know if you have any further questions.