Find Harmonic Function on Disk: U(x,y)=y+y^2

[Tony11235]

I am to find a function U, harmonic on the disk $$x^2 + y^2 < 6$$ and satisfying
$$u(x, y) = y + y^2$$ on the disk's boundary. I am not sure where to start. Hints, help, anything?

 

[matt grime]

Use the integral formula.

 

[LeonhardEuler]

I would think Cauchy's integral formual would be useful here: you have the value of a function on a boudry and want the value in the interior.

 

[harsh]

You are trying to solve the Laplace equation on a disk. Try separation of variables, then break it down to 2 ODE's. Here is a start for you..

You will probably need to solve the PDE in polar coordinates.

- harsh

 

[Tony11235]

You are trying to solve the Laplace equation on a disk. Try separation of variables, then break it down to 2 ODE's. Here is a start for you..
You will probably need to solve the PDE in polar coordinates.
- harsh

Then is $$u(\sqrt{6}, \theta) = \sqrt{6} \sin(\theta) + 6\sin^2(\theta)$$ a boundary condition?

 

[harsh]

Then is $$u(\sqrt{6}, \theta) = \sqrt{6} \sin(\theta) + 6\sin^2(\theta)$$ a boundary condition?

Looks right. Make sure you solve the correct PDE, the laplacian in r,theta is not as simple as U_rr and U_theta*theta

- harsh

 

[Tony11235]

Looks right. Make sure you solve the correct PDE, the laplacian in r,theta is not as simple as U_rr and U_theta*theta
- harsh

I know. In an earlier problem I had to compute the laplacian in polar. Oh and one more thing, is there anything else I need to know about $$\theta$$? Other than $$0 < \theta < 2\pi$$ ?

 

[harsh]

The theta condition that you are going to use, I believe, will be that theta is 2pi periodic.

- harsh