- #1
fahraynk
- 186
- 6
Homework Statement
$$
\bigtriangledown^2=0 for : 0<r<1 \\
BC : u(1,\Theta)= sin(\Theta), 0<\Theta<\pi \\ u(1,\Theta)= 0, pi<\Theta<2\pi \\
$$
Basically its an interior dirichlet problem for a circle. [/B]
Homework Equations
The Attempt at a Solution
The answer is supposed to be $$U(r,\Theta) = \Sigma r^n[a_n cos(n\Theta) + b_n sin(n\Theta)$$
and the a_n, b_n is basically a Fourier expansion of the boundary conditions.
The books answer is :
$$ \frac{r}{2}sin\Theta + \frac{2}{\pi}(\frac{1}{2} - \frac{r^2}{3}cos(2\Theta)-\frac{r^4}{15}cos(4\Theta)-\frac{r^6}{35}cos(6\Theta)...)$$
Now, I can't seem to get hte Fourier expansion right I guess, because I don't get this answer.
Heres my attempt ...
$$a_n = \frac{2}{\pi}\int_{0}^{\pi}sin(n\Theta)cos(n\Theta)d\Theta = 0
\\
b_n = \frac{2}{\pi}\int_{0}^{\pi}sin^2(n\Theta) = \frac{1}{\pi}
\\
a_0 = \frac{1}{\pi}\int_{0}^{\pi}sin\Theta d\Theta = 0
$$
Cant think of how to make this Fourier expansion into the books answer...