#!/usr/bin/env python
# coding: utf-8

# # Elliptic PDE: Radially Symmetric Singular Solution
# Copyright (C) 2010-2020 Luke Olson<br>
# Copyright (C) 2020 Andreas Kloeckner
# 
# <details>
# <summary>MIT License</summary>
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# </details>
# 
# -----
# 
# Poisson Problem:
# <center> Given open domain $\Omega \subset \mathbb{R}^2$ </center>  
# $$ -\nabla \cdot \nabla u = f(x)\quad \text{in}\, \Omega $$
# $$ u = g(x)\quad \text{on}\, \partial\Omega $$
# 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:
# $$ u(x,y) = -\frac{1}{2\pi}\ln(r) $$
# 
# <center> $r = \sqrt{x^2+y^2}$, the distance to the origin.</center>

# 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))


# ## Plot

# In[23]:


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()


# Given $C\log(r)$ as the *free-space Green's function*, can we construct the solution to the PDE with a more general $f$?