Elliptic PDE: Radially Symmetric Singular Solution¶

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Poisson Problem:

Given open domain $\Omega \subset \mathbb{R}^2$

$$-\nabla \cdot \nabla u = f(x)\quad \text{in}\, \Omega$$ $$u = g(x)\quad \text{on}\, \partial\Omega$$ Let: $$f(x) = \delta(x)$$
$\delta(x)$ is the Dirac delta function. This problem describes a unit charge at the origin.

Solution¶

Potential due to point charge: $$u(x,y) = -\frac{1}{2\pi}\ln(r)$$

$r = \sqrt{x^2+y^2}$, the distance to the origin.
In [18]:
import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
from matplotlib import cm
import math

import sympy as sym
sym.init_printing()


Set up Grid¶

In [19]:
X = np.arange(-10, 10, 0.2)
Y = np.arange(-10, 10, 0.2)
X, Y = np.meshgrid(X, Y)


Solution¶

In [20]:
r = np.sqrt(X**2 + Y**2)
Z = -np.log(r)/(2*math.pi)


Check Symbolically¶

In [21]:
sx = sym.Symbol("x")
sy = sym.Symbol("y")
sr = sym.sqrt(sx**2 + sy**2)
ssol = sym.log(sr)

sym.simplify(sym.diff(ssol, sx, 2) + sym.diff(ssol, sy, 2))

Out[21]:
$\displaystyle 0$

Plot¶

In [23]:
fig = plt.figure(figsize=(8, 6))

<mpl_toolkits.mplot3d.art3d.Line3DCollection at 0x7f8af6335dd0>
Given $C\log(r)$ as the free-space Green's function, can we construct the solution to the PDE with a more general $f$?