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Potential Equation¶

Laplace's Equation

\begin{aligned} \nabla^{2} U = 0 \\ \frac{\partial^{2} U}{\partial x^{2}} + \frac{\partial^{2} U}{\partial y^{2}} = 0 & 2D \\ \frac{\partial^{2} U}{\partial x^{2}} + \frac{\partial^{2} U}{\partial y^{2}} + \frac{\partial^{2} U}{\partial z^{2}} = 0 & 3D \end{aligned}

2D Laplaces's Equation¶

Polar Coordinate¶

$z (x, y) = z (r \cos θ, r \sin θ) = U (r, θ)$
$\frac{d (\tan^{- 1} (u))}{d u} = \frac{1}{1 + u^{2}}$
$\begin{matrix} \nabla^{2} z = U_{r r} + \frac{1}{r} U_{r} + \frac{1}{r^{2}} U_{θ θ} \end{matrix}$

Cylindrical Coordinate¶

$V (x, y, z) = V (r \cos θ, r \sin θ, z) = U (r, θ, z)$
$\begin{matrix} \nabla^{2} = U_{r r} + \frac{1}{r} U_{r} + \frac{1}{r^{2}} U_{θ θ} + U_{z z} \end{matrix}$

Spherical Coordinate¶

$\begin{matrix} x = ρ \cos (θ) \sin (ϕ) \\ y = ρ \sin (θ) \sin (ϕ) \\ z = ρ \cos (ϕ) \end{matrix}$
$\begin{matrix} \nabla^{2} V = U_{ρ ρ} + \frac{2}{ρ} U_{ρ} + \frac{1}{ρ^{2} \sin ϕ} U_{θ θ} + \frac{1}{ρ^{2}} U_{ϕ ϕ} + \frac{\cot ϕ}{ρ^{2}} U_{ϕ} \end{matrix}$

Dirichlet Problem¶

Rectangle¶

\begin{matrix} \nabla^{2} U (x, y) = 0 \end{matrix}

一次只能解一個邊不為 0 的 case
若有兩個邊以上不為 0 ，分成若干子問題來解
$u (x, y) = u_{1} (x, y) + u_{2} (x, t) + \dots$

Disk¶

\begin{matrix} \nabla^{2} U (r \cos θ, r \sin θ) = 0 \end{matrix}

Cauchy-Euler Equation

let

u (r, θ) = P (r) Q (θ)

then

\begin{aligned} Q^{″} + λ Q = 0 & # 1 O D E \\ r^{2} P^{″} + r P^{'} - P λ = 0 & # 2 O D E \end{aligned}

characteristic eq. of ODE #2:

\begin{matrix} m^{2} + (1 - 1) m - λ = 0 \\ ⟹ m = \pm \sqrt{λ} \end{matrix}

Upper Half-Plane¶

\begin{aligned} \nabla^{2} u (x, y) = 0 & for - \infty < x < \infty, y > 0 \\ u (x, 0) = f (x) & for - \infty < x < \infty \end{aligned}

Quarter-Plane¶

\begin{aligned} \nabla^{2} u (x, y) = 0 & for 0 < x < \infty, y > 0 \\ u (x, 0) = f (x) & for 0 \leq x < \infty \\ u (0, y) = 0 & for y \geq 0 \end{aligned}

convert to Upper Half-Plane problem,

let

g (x) = {\begin{aligned} f (x) & for x \geq 0 \\ anything & for x < 0 \end{aligned}

then solve

\begin{aligned} \nabla^{2} u (x, y) = 0 & for - \infty < x < \infty, y > 0 \\ u (x, 0) = g (x) & for - \infty < x < \infty \end{aligned}

Sphere in static state¶

assume $u$ is independent of $θ$

\begin{matrix} \frac{\partial U}{\partial t} = \nabla^{2} U = 0 \end{matrix}

let $u (ρ, ϕ) = X (ρ) Φ (ϕ)$

then

\begin{aligned} Φ^{″} + \cot (ϕ) Φ^{'} + λ Φ = 0 & # 1 O D E \\ ρ^{2} X^{″} + 2 ρ X^{'} - λ X = 0 & # 2 O D E \end{aligned}

solve ODE #1¶

ODE # 1 can be converted to

\begin{matrix} [(1 - x^{2}) G^{'} (x)] + λ G (x) = 0 \\ G (x) = Φ (ϕ), (x = \cos (ϕ)) \end{matrix}

which is a Legendre's Equation, then

\begin{matrix} λ_{n} = n (n + 1), n = 0, 1, 2, \dots \\ Φ_{n} (ϕ) = G_{n} (x) = P_{n} (\cos (ϕ)) \end{matrix}

solve ODE #2¶

\begin{matrix} ρ^{2} X^{″} + 2 ρ X^{'} - λ X = 0 \\ m^{2} + (2 - 1) m - n (n + 1) = 0 \\ ⟹ m = - (n + 1), n \end{matrix}

and finally we get,

\begin{matrix} u (ρ, ϕ) = \sum_{n = 0}^{\infty} a_{n} ρ^{n} P_{n} (\cos (ϕ)) \\ u (R, ϕ) = f (ϕ) = \sum_{n = 0}^{\infty} a_{n} R^{n} P_{n} (\cos (ϕ)) \\ ⟹ a_{n} R^{n} = \frac{< f (\cos^{- 1} (x)), P_{n} (x) >}{< P_{n} (x), P_{n} (x) >} \end{matrix}

ref : Fourier-Legendre Expansion

\begin{matrix} a_{n} R^{n} = \frac{2 n + 1}{2} \int_{- 1}^{1} f (\cos^{- 1} (x)) P_{n} (x) d x \end{matrix}

Neumann Problem¶

\begin{aligned} \nabla^{2} u (x, y) = 0 & for (x, y) \in D \\ \frac{\partial u}{\partial n} (x, y) = g (x, y) & for (x, y) \in \partial D \end{aligned}

$D$ 表示定義區域（面）、 $\partial D$ 表示 boundary of $D$
$\frac{\partial u}{\partial n}$ 表示對法向量微分
necessary condition :

\begin{matrix} \oint_{\partial D} \frac{\partial u}{\partial n} d s = 0 \end{matrix}

不滿足則無解（傳說中會考）