SciPy_SymPy/tutorial_exercises/Poisson equation.ipynb

{
 "metadata": {
  "name": "Poisson equation"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Initialize printing:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%pylab inline\n",
      "from sympy.interactive.printing import init_printing\n",
      "init_printing()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n",
        "Welcome to pylab, a matplotlib-based Python environment [backend: module://IPython.zmq.pylab.backend_inline].\n",
        "For more information, type 'help(pylab)'.\n"
       ]
      }
     ],
     "prompt_number": 1
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# Poisson Equation\n",
      "\n",
      "Radial Poisson equation is\n",
      "$$\n",
      "\\phi''(r)+{2\\over r} \\phi'(r) = -4\\pi\\rho(r).\n",
      "$$\n",
      "Alternatively, this can also be written as:\n",
      "$$\n",
      "{1\\over r}(r \\phi(r))'' = -4\\pi\\rho(r).\n",
      "$$\n",
      "### Example I\n",
      "\n",
      "Positive Gaussian charge density:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sympy import var, exp, pi, sqrt, integrate, refine, Q, oo, Symbol, DiracDelta\n",
      "var(\"r alpha Z\")\n",
      "alpha = Symbol(\"alpha\", positive=True)\n",
      "rho = Z * (alpha / sqrt(pi))**3 * exp(-alpha**2 * r**2)\n",
      "rho"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{Z \\alpha^{3} e^{- \\alpha^{2} r^{2}}}{\\pi^{\\frac{3}{2}}}$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEIAAAAqCAYAAAAH843fAAAABHNCSVQICAgIfAhkiAAAAxhJREFU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\nAABJRU5ErkJggg==\n",
       "prompt_number": 2,
       "text": [
        "        2  2\n",
        "   3  -\u03b1 \u22c5r \n",
        "Z\u22c5\u03b1 \u22c5\u212f      \n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "     3/2    \n",
        "    \u03c0       "
       ]
      }
     ],
     "prompt_number": 2
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The total charge is $Z$:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "integrate(4*pi*rho*r**2, (r, 0, oo))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$Z$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAABEAAAASCAYAAAC9+TVUAAAABHNCSVQICAgIfAhkiAAAAPRJREFU\nOI3d07ErxVEYxvHPJZfcQndgITGaLDJTSibKXZW6+TeMdn+ATAYxyGyx2pQyGtxkEoOBxPD+Bp3O\n7/5uN5NnOfU8nW+n57wvf6BaiT+JRwzhFffF+V3kS2hiG+dl8F28oIXBJNvEFw6rXniG1Yy/jHdc\nYKAboI7TjD+PZ9xgtOoVc1hJvKbo5QFTVYCchnEtOlroB1DDCT7kO+pJB+Jbd/oF7BWA/cQfwVov\ngHV84jiTtbBVBVjEG67ExKa6RKMbYBod3GE8k28UkFKN4RZPmE2yBtpif9rpxd8LeCR2piOWjxjt\niQJaFz3NiMn9r/oB+/YpvN7S38AAAAAASUVORK5CYII=\n",
       "prompt_number": 3,
       "text": [
        "Z"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Solve for $\\phi(r)$ from the Poisson equation:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "phi = integrate(-4*pi*rho*r, r, r)/r\n",
      "phi"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{Z \\operatorname{erf}{\\left (\\alpha r \\right )}}{r}$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADQAAAAeCAYAAABjTz27AAAABHNCSVQICAgIfAhkiAAAAlZJREFU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\nue5IRWe0A8p1Rxp3PMTisd5EiRIl8ucP/aN3cNLkYuEAAAAASUVORK5CYII=\n",
       "prompt_number": 4,
       "text": [
        "Z\u22c5erf(\u03b1\u22c5r)\n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "    r     "
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Tell SymPy that $\\alpha$ is positive:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "phi = refine(phi, Q.positive(alpha))\n",
      "phi"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{Z \\operatorname{erf}{\\left (\\alpha r \\right )}}{r}$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADQAAAAeCAYAAABjTz27AAAABHNCSVQICAgIfAhkiAAAAlZJREFU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\nue5IRWe0A8p1Rxp3PMTisd5EiRIl8ucP/aN3cNLkYuEAAAAASUVORK5CYII=\n",
       "prompt_number": 5,
       "text": [
        "Z\u22c5erf(\u03b1\u22c5r)\n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "    r     "
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Plot the charge:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sympy import plot"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 6
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot(rho.subs({Z: 1, alpha: 0.1}), (r, 0, 100));"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "display_data",
       "png": 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6HDp0KGobj8eD3t7eMfv6fD5oNBoAgEajgc/nAwD09vbiO9/5Tlif3nN/3j/x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\ndQAAAABJRU5ErkJggg==\n"
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Plot the potential"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot(phi.subs({Z: 1, alpha: 0.1}), 1/r, (r, 0, 100), ylim=[0, 0.12]);"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "display_data",
       "png": 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      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 8
    }
   ],
   "metadata": {}
  }
 ]
}