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When I'm learning laplace's equation with the case is dirichlet problem for a circle, i know that naturally i have to separate variables in polar coordinate.

And i derive from rectangle to polar coordinat, then i separate the variable until i have two ODE as follows

$\Theta''+\lambda\Theta=0 \\ r^2R''+rR-\lambda R=0$

Solve the ODE and Plug in to the solution $u(r,\theta)=R(r)\cdot \Theta(\theta)$ and i have:

$u=\left(Cr^n+Dr^{-n}\right)(A\cos n\theta+B\sin n\theta)$

But i still don't know, in the textbooks that i've read told me that D is vanish and summing the remaining solution, it become full Fourier Series as follows:

$u=\frac{1}{2} A_0+\displaystyle\sum_{n=1}^{\infty}r^n(A_n\cos n\theta+B_n\sin n\theta)$

Please, explain to me about the last part. Thanks.

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There is a boundary condition in the disk given which makes the other part go away.

$$ | u(0,\theta)| \leq \infty $$

This conditions enforces that

$$ | R(0) | \leq \infty $$

For this to work you note that in

$$ R(r) = Cr^{n} + Dr^{-n} $$

when $ r \to 0$ this will blow up. So for it to work $ D = 0$.

You'll be left with

$$ u(r,\theta) = \sum_{n=0}^{\infty} A_{n}r^{n} \cos(n\theta) + \sum_{n=1}^{\infty} B_{n} r^{n} \sin(n\theta) $$

Then we take out $A_{0}$ since

$$ A_{0} r^{0} \cos(0 \cdot \theta) = A_{0} $$

So we can change $n$ to $n=1$

$$ u(r,\theta) = A_{0} + \sum_{n=1}^{\infty} A_{n} r^{n} \cos(n \theta) + \sum_{n=1}^{\infty} B_{n} r^{n} \sin(n \theta) = A_{0} + \sum_{n=1}^{\infty} r^{n} (A_{n} \cos(n \theta) + B_{n} \sin(n \theta) ) $$

I believe $\frac{A_{0}}{2} $ comes from the interval.

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