Laplaces equation help!!

SandorVS · **Joined:** Thu, 15 Dec 2011 18:21:41 UTC **Posts:** 1

1. Laplace equation's solution for (x,y) plane on a circle x>0, y>0 in polar coordinates with the conditions :
u(r=0,μ) = sinμ/sin∏/8 + sin μ/2 + cos μ/2 if 0≤μ≤π/2
u(r,μ=0) = √r if 0≤r≤1
u(r,μ=∏/2) = r^1/4 + √(2r) if 0≤r≤1

The equation is linear in u.
Use the variable separation method.

Sorry for my bad english, I hope it's understandable.

outermeasure · **Posted:** Thu, 15 Dec 2011 23:41:09 UTC

SandorVS wrote:

1. Laplace equation's solution for (x,y) plane on a circle x>0, y>0 in polar coordinates with the conditions :
u(r=0,μ) = sinμ/sin∏/8 + sin μ/2 + cos μ/2 if 0≤μ≤π/2
u(r,μ=0) = √r if 0≤r≤1
u(r,μ=∏/2) = r^1/4 + √(2r) if 0≤r≤1

The equation is linear in u.
Use the variable separation method.

Sorry for my bad english, I hope it's understandable.

Are you sure about the first condition? I suspect you want $u(1,\theta)$ instead. Also, the last condition currently reads $u(r,\frac{\pi}{2})=r^1/4+\sqrt{2r}$ , the ^1 is redundant.

Recall the Laplacian in polar: $\nabla^2 u=r^{-1}(r u_r)_r+r^{-2}u_{\theta\theta}$ . Separate variables you see $r^k\cos(k(\theta-\theta_0))$ , or a sum of these, are what you should try.

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Laplaces equation help!!

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