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fluidistic
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Homework Statement
Consider a circle of radius a whose center is in (0,0). Let [itex](r, \phi)[/itex] be the polar coordinates and (x,y) the corresponding rectangular coordinates of the plane. Calculate the solution to Dirichlet problem (interior) for Laplace equation [itex]\nabla ^2 u =0[/itex] with the following boundary conditions:
1)[itex]u(r=a)=A[/itex]
2)[itex]u(r=a)=A \cos \phi[/itex]
3)[itex]u(r=a)=A+By[/itex]
4)[itex]u(r=a)=Axy[/itex]
5)[itex]u(r=a)=A+B \sin \phi[/itex]
6)[itex]u(r=a)=A \sin ^2 \phi +B \cos ^2 \phi[/itex]
where A and B are constants.
Homework Equations
Already given I think.
The Attempt at a Solution
So I've been checking out internet to confirm my result so far but I've a few questions.
I wrote the Laplacian in polar coordinates [itex]\triangle u = \frac{u_r}{r}+u_{rr}+\frac{u_{\phi \phi}}{r^2}=0[/itex].
I use separation of variables, proposing a solution of the form [itex]u(r, \theta )= \varphi (\phi )R(r)[/itex].
Plugging this back into the Laplace equation I reached 2 ODE's.
[itex]r^2 R''+rR'-k^2R=0[/itex] and [itex]\varphi '' +k^2 \varphi =0[/itex].
The second ODE is easy to me to solve, I reached [itex]\varphi (\phi )=c_1e^{ik \phi }+c_2e^{-i k \phi}[/itex]. However on the Internet they prefer to keep all real values if I understood well, though I don't know how it's possible to do this.
I kind of cheated to solve the first ODE and one solution (I checked out and it indeed is a solution) is of the form [itex]R(r)=c_3 r^k+c_4 r^{-k}[/itex]. Now for R(r) remains finite when r tends to 0, [itex]c_4[/itex] must vanish so that [itex]R(r)=c_3r^k[/itex].
So my solution so far is [itex]u(r, \phi )=c_3 r^k(c_1e^{ik \phi }+c_2e^{-i k \phi} )[/itex].
I've also found out that it's possible to get another form of solution for the first ODE, namely [itex]R(r)=c_5+c_6\ln r[/itex] which would really complicate the number of possible solutions to the PDE. I don't understand what it means physically to have 2 different possible solutions.
Also, how do I deal with the solution that contains complex numbers?
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