- #1
Dustinsfl
- 2,217
- 5
Solve Laplace's equation $\nabla^2u = 0$ on a $90^{\circ}$ wedge of radius $a$ subject to the boundary conditions
$$
u(r,0) = 0\quad u_{\theta}\left(r,\frac{\pi}{2}\right) = 0\quad u(a,\theta) = f(\theta).
$$
The standard form of the solutions for $R(r)$ and $\Theta(\theta)$ are
$$
\Theta(\theta) = A\cos\lambda\theta + B\sin\lambda\theta\quad\text{and}\quad
R(r) = r^{\pm\lambda}.
$$
Using the first boundary condition, we have $A = 0$, i.e.
$$
\Theta(\theta) = B\sin\lambda\theta.
$$
Using the second boundary condition, we have
$$
\lambda B\cos\frac{\pi\lambda}{2} = 0.
$$
Therefore, $\lambda = n$ where $n\in\mathbb{Z}$.
The general solution is
$$
u(r,\theta) = \sum_{n = 1}^{\infty}B_nr^{n}\sin n\theta.
$$
With the last condition, we have
$$
u(a,\theta) = \sum_{n = 1}^{\infty}B_na^{n}\sin n\theta = f(\theta)
$$
where the Fourier coefficients are $B_n = \frac{1}{a^n\pi}\int_{-\pi}^{\pi}f(\theta)\sin n\theta d\theta$.
Then the general solution with the defined conditions is $u(r,\theta) = \sum\limits_{n = 1}^{\infty}B_nr^{n}\sin n\theta$ where $B_n$ is defined as above.
Let's take the case where $f(\theta) = \theta$.
\begin{alignat*}{3}
B_n & = & \frac{1}{a^n\pi}\int_{-\pi}^{\pi}f(\theta)\sin n\theta d\theta\\
& = & \frac{1}{a^n\pi}\left[\left.-\frac{\theta\cos n\theta}{n}\right|_{-\pi}^{\pi} + \left.\frac{\sin n\theta}{n^2}\right|_{-\pi}^{\pi}\right]\\
& = & \frac{2(-1)^{n + 1}}{a^nn}
\end{alignat*}
The solution to this Laplace equation when $f(\theta) = \theta$ is
$$
u(r,\theta) = 2\sum\limits_{n = 1}^{\infty}\frac{(-1)^{n + 1}r^{n}}{a^nn}\sin n\theta.
$$
$$
u(r,0) = 0\quad u_{\theta}\left(r,\frac{\pi}{2}\right) = 0\quad u(a,\theta) = f(\theta).
$$
The standard form of the solutions for $R(r)$ and $\Theta(\theta)$ are
$$
\Theta(\theta) = A\cos\lambda\theta + B\sin\lambda\theta\quad\text{and}\quad
R(r) = r^{\pm\lambda}.
$$
Using the first boundary condition, we have $A = 0$, i.e.
$$
\Theta(\theta) = B\sin\lambda\theta.
$$
Using the second boundary condition, we have
$$
\lambda B\cos\frac{\pi\lambda}{2} = 0.
$$
Therefore, $\lambda = n$ where $n\in\mathbb{Z}$.
The general solution is
$$
u(r,\theta) = \sum_{n = 1}^{\infty}B_nr^{n}\sin n\theta.
$$
With the last condition, we have
$$
u(a,\theta) = \sum_{n = 1}^{\infty}B_na^{n}\sin n\theta = f(\theta)
$$
where the Fourier coefficients are $B_n = \frac{1}{a^n\pi}\int_{-\pi}^{\pi}f(\theta)\sin n\theta d\theta$.
Then the general solution with the defined conditions is $u(r,\theta) = \sum\limits_{n = 1}^{\infty}B_nr^{n}\sin n\theta$ where $B_n$ is defined as above.
Let's take the case where $f(\theta) = \theta$.
\begin{alignat*}{3}
B_n & = & \frac{1}{a^n\pi}\int_{-\pi}^{\pi}f(\theta)\sin n\theta d\theta\\
& = & \frac{1}{a^n\pi}\left[\left.-\frac{\theta\cos n\theta}{n}\right|_{-\pi}^{\pi} + \left.\frac{\sin n\theta}{n^2}\right|_{-\pi}^{\pi}\right]\\
& = & \frac{2(-1)^{n + 1}}{a^nn}
\end{alignat*}
The solution to this Laplace equation when $f(\theta) = \theta$ is
$$
u(r,\theta) = 2\sum\limits_{n = 1}^{\infty}\frac{(-1)^{n + 1}r^{n}}{a^nn}\sin n\theta.
$$