{
 "metadata": {
  "name": ""
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "code",
     "collapsed": true,
     "input": [
      "from pylab import *\n",
      "%matplotlib inline\n",
      "from scipy.integrate import odeint\n",
      "\n",
      "# nastavitve za izris grafov (http://matplotlib.org/1.3.1/users/customizing.html)\n",
      "rc('text', usetex=True)\n",
      "rc('font', size=12, family='serif', serif=['Computer Modern'])\n",
      "rc('xtick', labelsize='small')\n",
      "rc('ytick', labelsize='small')\n",
      "rc('legend', fontsize='medium')\n",
      "rc('figure', figsize=(5, 3))\n",
      "rc('lines', linewidth=2.0)\n",
      "rc('axes', color_cycle=['k'])\n",
      "rc('contour', negative_linestyle='solid')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Navadne diferencialne ena\u010dbe"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Pri re\u0161evanju diferencialne ena\u010dbe gre za dolo\u010ditev obna\u0161anja neke koli\u010dine, odvisne od ene ali ve\u010d spremenljivk, \n",
      "pri \u010demer v povezavi med iskano koli\u010dino $y$ in neodvisnimi spremenljivkami $x_i$ nastopajo tudi odvodi iskane koli\u010dine po neodvisnih spremenljivkah, npr. $\\frac{\\partial y}{\\partial x_i}$, $\\frac{\\partial^2 y}{\\partial x_i\\partial x_j},\\ldots$ \u010ce je neodvisna spremenljivka ena sama, govorimo o navadnih diferencialnih ena\u010dbah. \u010ce v diferencialni ena\u010dbi nastopajo samo prvi odvodi, govorimo o navadni diferencialni ena\u010dbi prveg reda. \u010ce poleg njih nastopijo tudi drugi odvodi, je to navadna diferencialna ena\u010dba drugega reda.\n",
      "\n",
      "Navadno diferencialno ena\u010dbo prvega reda lahko numeri\u010dno re\u0161imo tako, da odvod nadomestimo z diferen\u010dnim pribli\u017ekom: \n",
      "\n",
      "$$\\frac{\\mathrm{d}y(x)}{\\mathrm{d}x}=f\\left(y(x),x\\right)$$\n",
      "\n",
      "$$\\frac{y(x+\\Delta x)-y(x)}{\\Delta x}=f\\left(y(x),x\\right)$$\n",
      "\n",
      "$$y(x+\\Delta x)=y(x)+f\\left(y(x),x\\right)\\Delta x$$\n",
      "\n",
      "Na ta na\u010din lahko s poznavanjem za\u010detnega pogoja $y(0)$ z zaporedno uporabo zgornje zveze poi\u0161\u010demo vrednosti koli\u010dine $y$ pri poznej\u0161ih \u010dasih: \n",
      "\n",
      "$$y(0)\\rightarrow y(\\Delta x)\\rightarrow y(2\\Delta x)\\ldots$$\n",
      "\n",
      "Navadno diferencialno ena\u010dbo vi\u0161jega reda lahko z vpeljavo novih spremenljivk spremenimo v sistem navadnih diferencialnih ena\u010db prvega reda:\n",
      "\n",
      "$$\\frac{\\mathrm{d}^2y(x)}{\\mathrm{d}x^2}=f\\left(y(x),\\frac{\\mathrm{d}y(x)}{\\mathrm{d}x}, x\\right),$$\n",
      "\n",
      "$$\\frac{\\mathrm{d}y(x)}{\\mathrm{d}x}=u(x),$$\n",
      "\n",
      "$$\\frac{\\mathrm{d}u(x)}{\\mathrm{d}x}=f\\left(y(x),u(x),x\\right),$$\n",
      "\n",
      "ki ga nato numeri\u010dno re\u0161imo tako, kot smo se nau\u010dili pri navadni diferencialni ena\u010dbi prvega reda:\n",
      "\n",
      "$$y(x+\\Delta x)=y(x)+u(x)\\Delta x$$\n",
      "\n",
      "$$u(x+\\Delta x)=u(x)+f(y(x), u(x), x)\\Delta x$$\n",
      "\n",
      "Tu potrebujemo dva za\u010detna pogoja, vrednost iskane funkcije $y(0)$ in njenega odvoda $u(0)=\\left.\\frac{\\mathrm{d}y}{\\mathrm{d}x}\\right|_{x=0}$:\n",
      "\n",
      "$$\\left(y(0), u(0)\\right)\\rightarrow\\left(y(\\Delta x), u(\\Delta x)\\right)\\rightarrow\\left(y(2\\Delta x), u(2\\Delta x)\\right)\\ldots$$\n",
      "\n",
      "Algoritmi, ki jih za re\u0161evanje navadnih diferencialnih ena\u010db ponujajo profesionalna ra\u010dunalni\u0161ka orodja, so veliko bolj sofisticirani od zgoraj opisanjega. Z uporabo diferen\u010dnih pribli\u017ekov vi\u0161jega reda, spremenljivega koraka $\\Delta x$, ..., dose\u017eejo ve\u010djo stabilnost in hitrost iskanja re\u0161itve.    "
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Kot primer si oglejmo izra\u010dun poti $s$ in hitrosti $\\frac{\\mathrm{d}s}{\\mathrm{d}t}$ telesa pri padanju le-tega v zemeljskem gravitacijskem polju. Na telo poleg gravitacijske sile deluje zaviralna sila, za katero velja kvadratni zakon upora. Zapi\u0161imo drugi Newtonov zakon:\n",
      "\n",
      "$$m\\frac{\\mathrm{d}^2s}{\\mathrm{d}t^2}=mg-c\\left(\\frac{\\mathrm{d}s}{\\mathrm{d}t}\\right)^2$$\n",
      "\n",
      "Za\u010detna pogoja sta $s(0)=0$ in $\\left.\\frac{\\mathrm{d}s}{\\mathrm{d}t}\\right|_{t=0}=0$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "t=linspace(0, 10, 1000)\n",
      "m=1.0\n",
      "g=9.81\n",
      "c=0.01\n",
      "def deriv(y, t): \n",
      "    return array([y[1], g - c / m * y[1] ** 2])\n",
      "\n",
      "y0=array([0, 0])\n",
      "y=odeint(deriv, y0, t)\n",
      "s=y[:, 0]\n",
      "v=y[:, 1]\n",
      "\n",
      "fig, (axa, axb) = subplots(2, 1, sharex='col', sharey='row', figsize=(5, 6))\n",
      "\n",
      "axa.plot(t, s);\n",
      "axa.set_ylabel(r'Pot (m)')\n",
      "axa.grid(alpha=0.5)\n",
      "axa.annotate('(a)', xy=(0.03, 0.92), xycoords='axes fraction')\n",
      "\n",
      "axb.plot(t, v);\n",
      "axb.set_ylabel(r'Hitrost (m/s)')\n",
      "axb.set_xlabel(r'\\v Cas (s)')\n",
      "axb.grid(alpha=0.5)\n",
      "axb.annotate('(b)', xy=(0.03, 0.92), xycoords='axes fraction')\n",
      "\n",
      "subplots_adjust(hspace=0.1)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAU4AAAGBCAYAAAAJ7VgIAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nIcRB/x+EZJOrI0gmAwAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x1050e8cd0>"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Naloge"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<OL>\n",
      "<LI>Izra\u010dunaj in nari\u0161i, kako sta odklon iz ravnovesne lege in kotna hitrost matemati\u010dnega nihala odvisna od \u010dasa, \u010de ob \u010dasu $t=0$ nihalo miruje, odklon od ravnovesne lege pa je a) $30^\\circ$ in b) $150^\\circ$ (nihalo naj ima togo vrvico). Primerjaj z analiti\u010dnim rezultatom za majhne odklone iz ravnovesne lege. Kro\u017ena frekvenca nihala naj bo $1\\,$s$^{-1}$.  \n",
      "\n",
      "<LI>Izra\u010dunaj in nari\u0161i tira Zemlje in Lune pri gibanju okoli Sonca. Podatki: $M_S=2\\times 10^{30}\\,$kg, $M_Z=6\\times 10^{24}\\,$kg, $M_L=7.35\\times 10^{22}\\,$kg, $d_{ZS}=1.5\\times 10^{8}\\,$km, $d_{LZ}=3.84\\times 10^5\\,$km, $T_Z=365.25\\,$dni in $T_L=29.53\\,$dni. Za\u010detni polo\u017eaj Zemlje naj bo $\\mathbf{r}_Z(0)=(d_{ZS}, 0)$, Lune pa $\\mathbf{r}_L(0)=(d_{ZS}, d_{LZ})$. \n",
      "Za\u010detna hitrost Zemlje naj bo $\\mathbf{v}_Z(0)=(0, 2\\pi d_{ZS}/T_Z)$, Lune pa $\\mathbf{v}_L(0)=(-2\\pi d_{LZ}/T_L, 2\\pi d_{ZS}/T_Z)$. Ker ima Sonce bistveno ve\u010djo maso od Zemlje in Lune, lahko predpostavi\u0161, da miruje v izhodi\u0161\u010du koordinatnega sistema.\n",
      "\n",
      "</OL>"
     ]
    }
   ],
   "metadata": {}
  }
 ]
}