Elliptic PDE: Radially Symmetric Singular Solution

Copyright (C) 2010-2020 Luke Olson
Copyright (C) 2020 Andreas Kloeckner

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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.

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

X = np.arange(-10, 10, 0.2)
Y = np.arange(-10, 10, 0.2)
X, Y = np.meshgrid(X, Y)

Solution

r = np.sqrt(X**2 + Y**2)
Z = -np.log(r)/(2*math.pi)

Check Symbolically

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))

$\displaystyle 0$

Plot

fig = plt.figure(figsize=(8, 6))
ax = fig.add_subplot(111, projection='3d')
#ax.plot_surface(X, Y, Z, rstride=1, cstride=1, cmap=cm.coolwarm,
#                       linewidth=0, antialiased=False)
ax.plot_wireframe(X, Y, Z, linewidth=0.2)
#ax.set_zlim(-1.0, 1.0)
#plt.show()

<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 $f$?