The original poster attempts to find a harmonic function U on the disk defined by x² + y² < 6, which satisfies the boundary condition U(x, y) = y + y² on the disk's boundary. The problem involves concepts from potential theory and partial differential equations, specifically the Laplace equation.
Participants are exploring various methods to approach the problem, including integral formulas and separation of variables. There is an ongoing discussion about the correct form of the Laplacian in polar coordinates and the implications of the boundary conditions. No consensus has been reached yet.
Participants note the periodicity of the angle θ, indicating it is 2π periodic, which may be relevant for the solution's formulation. There is also mention of previous experience with similar problems, suggesting a shared understanding of the underlying concepts.
Then is [tex]u(\sqrt{6}, \theta) = \sqrt{6} \sin(\theta) + 6\sin^2(\theta)[/tex] a boundary condition?harsh said: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 said:Then is [tex]u(\sqrt{6}, \theta) = \sqrt{6} \sin(\theta) + 6\sin^2(\theta)[/tex] a boundary condition?
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 [tex]\theta[/tex]? Other than [tex]0 < \theta < 2\pi[/tex] ?harsh said: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