3M236_2018_TP1.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# TP 1 : Introduction à Python en passant par les EDO"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "L'objectif du TP est de vous apprendre ou rappeler les rudiments du langage python indispensable pour illustrer le cours de \"méthodes numériques pour les EDO\". Vous pouvez vous référer au matériel pédagogique dans l'UE Moodle 3M100 ou bien directement aux ressources internet, en sélectionnant les sites \"officiels\" plutôt que les forums.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plutôt que de repartir de 0 nous allons nous concentrer sur les outils dont nous avons besoin pour :\n",
    "<ol>\n",
    "<li> Définir une fonction python \n",
    "<li> Définir des tableaux de nombres contenant les solutions des ODE  (librairie numpy)\n",
    "<li> Afficher ces solutions graphiquement (graphes et portraits de phase) (librairie matplotlib)\n",
    "<li> Faire des opérations d'algèbre linéaire sur des matrices et des vecteurs (librairie scipy.linalg)\n",
    "<li> Calculer des approximations numériques de la solution d'une ODE (librairie scipy.integrate)\n",
    "</ol>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np # donne acces a la librairie numpy, et definit l'abbreviation np\n",
    "import scipy        # donne acces aux librairies scipy, scipy.linalg et scipy.integrate\n",
    "import scipy.linalg\n",
    "import scipy.integrate\n",
    "import matplotlib.pyplot as plt # donne acces a la librairie matplotlib.pyplot, et definit l'abbreviation plt\n",
    "import math     # donne acces aux fonctions et constantes mathematiques\n",
    "%matplotlib inline    \n",
    "# ouvre les fenetres graphiques dans le notebook\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercice 1.  L'oscillateur harmonique simple"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Soit $\\omega_0 > 0$. Commençons avec $\\omega_0= 1$. On considère l’équation différentielle de l’oscillateur harmonique :\n",
    "$$x''(t) + \\omega_0^2 x(t) = 0$$\n",
    "muni des conditions initiales\n",
    "$$\\left\\{\\begin{array}{l}\n",
    "x(0) = x_0\\\\\n",
    "x'(0) = v_0\n",
    "\\end{array}\\right.$$\n",
    "Cette équation est fondamentale. Elle est à la base de la modélisation de nombreux phénomènes physiques\n",
    "<ol><Li>\n",
    "en mécanique : $x(t)$ est le déplacement d’une masse m suspendue à un ressort de constante de raideur\n",
    "$k$ tel que $\\omega_0^2=k/m$\n",
    "<li> $x(t)$ peut aussi mesurer les petites oscillations (en radians, autour de la position\n",
    "d’équilibre stable verticale) d’un pendule de longueur $L$, avec $\\omega_0 =\n",
    "\\sqrt{g/L}$\n",
    "avec  $g$ la constante de\n",
    "pesanteur.\n",
    "<li> en électronique : $x(t)$ est la charge aux bornes d’un condensateur de capacité $C$ qui se décharge dans\n",
    "une bobine d’inductance $L$ avec $\\omega_0=1/\\sqrt{LC}$.</ol>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Question 1 Définir une fonction"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Voici ci-dessous une exemple de fonction en Python"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "exemple( 3 )= 4\n"
     ]
    }
   ],
   "source": [
    "def exemple(x):\n",
    "# le caractere # indique que ce qui suit est un commentaire\n",
    "# En python une fonction comprend une premiere ligne declarative avec le nom de la fonction (ici \"exemple\") suivi des arguments entre parenthèses, terminée par \":\"\"\" \n",
    "# à l'interieur d'un bloc fonction, toutes les commandes doivent etre indentees.\n",
    "    y=x+1   # Cette fonction va programmer en python y=f(x)=x+1 \n",
    "    return y  # le mot reserve \"return\" permet de communiquer les resultats de la fonction a l'exterieur du bloc \n",
    "#Maintenant on va utiliser la fonction. On arrete l'indentation\n",
    "t=3\n",
    "z=exemple(t)\n",
    "print (\"exemple(\",t,')=',z)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On a vu en cours et en TD qu'on peut trouver $a$ et $b$, tels que la fonction $x(t)=a \\cos( \\omega_0 t ) + b \\sin(\\omega_0 t)$ soit solution du problème ci-dessus \n",
    "$$ a= x_0\\quad b= v_0/\\omega_0$$\n",
    "Ecrire  une fonction \"oscille\", qui prend en entrée les valeurs initiales, la valeur de $\\omega_0$, et un temps $t$ et qui calcule la solution. Testez-la avec $x_0 = 2$ et $v_0 = 3$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "À vous de jouer ci-dessous pour répondre à la question 1\n"
     ]
    }
   ],
   "source": [
    "print (\"À vous de jouer ci-dessous pour répondre à la question 1\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3.605017566159969\n"
     ]
    }
   ],
   "source": [
    "def oscille(x0,v0,w0,t):\n",
    "    return x0*math.cos(w0*t)+v0*math.sin(w0*t)/w0\n",
    "x0=2\n",
    "v0=3\n",
    "w0=1\n",
    "t=1\n",
    "print(oscille(x0,v0,w0,t))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Question 2 - Discrétiser un intervalle "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##### On veut maintenant calculer les valeurs de la solution sur un intervalle donné, en discrétisant cet intervalle suffisamment finement pour que l'affichage graphique des valeurs de la fonction aux points discrétisé ait un aspect \"continu\". Plusieurs façons de dicrétiser un intervalle sont possibles en python. En voici trois (il y en a d'autres) :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9]\n",
      "[1.  1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9]\n",
      "[1.         1.11111111 1.22222222 1.33333333 1.44444444 1.55555556\n",
      " 1.66666667 1.77777778 1.88888889 2.        ]\n"
     ]
    }
   ],
   "source": [
    "absc0 = [1+ float(k)/10  for k in  range (10)]    \n",
    "print (absc0)\n",
    "absc1 = np.arange (1,2,0.1)\n",
    "print (absc1)\n",
    "absc2 = np.linspace (1,2,10)\n",
    "print (absc2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Assurez-vous que vous comprenez bien les trois méthodes. Que font les instructions \"range\", \"np.arange\" , \"np.linspace\" ? Que se passe-t-il si on n'utilise pas \"float\" pour définir absc0 ?\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Votre réponse : "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ecrire une fonction $oscillateur(t_,x_0,v_0,\\omega_0)$ qui calcule deux tableaux $x(t)$ et\n",
    "$v(t)$, où $v(t)=x'(t)$, à partir des conditions initiales et d’un tableau de valeurs de temps $t = (t_i)_{i=1,\\ldots,n}$. Tester pour\n",
    "$t_k=6k\\pi/(\\omega_0 n)$, avec $n=10$ puis $n=100$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "a vous de jouer ci-dessous pour repondre a la question 2\n"
     ]
    }
   ],
   "source": [
    "print (\"a vous de jouer ci-dessous pour repondre a la question 2\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Question 3 - Tracer une courbe\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\"Tracer une courbe\" sur un ordinateur, c'est-à-dire par exemple le graphe d'une fonction $y=x(t)$, c'est en fait une succession d'opérations\n",
    "<UL> <li> Définir l'intervalle $[t_0,T]$ sur lequel on veut calculer la fonction\n",
    "<li> Choisir une discrétisation suffisamment fine de cet intervalle et calculer les valeurs de la fonction en ces points (voir exercice 2)\n",
    "<lI> Tracer la ligne brisée reliant les points $(t_i,x(t_i))_{i=1,\\ldots,N}$ avec la fonction plot de la librairie graphique matplotlib\n",
    "<li> Rajouter une légende, le nom des axes, etc, tout ce qui peut rendre la figure compréhensible donc intéressante\n",
    "</UL>\n",
    "Un exemple où on trace la fonction $y=sin(t)$ pour $t\\in[0,\\pi]$ en utilisant la discrétisation absc2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
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XM1MpKK3k/duGMqhre6sjeaTz48J469YhHM4r5VczUzmUW2J1JOUitCgol5F2tJDrZ6ZS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\nVUOLglJKqRr/H5UoDa/RBeKbAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "absc2 = np.linspace (0,math.pi,20)\n",
    "ord2=np.sin(absc2)\n",
    "plt.plot(absc2,ord2)               #trace la courbe\n",
    "plt.title('graphe de y=sin(t)')    # affiche un titre\n",
    "plt.legend(['sin'])                # affiche une legende (indispensable s'il y a plusieurs courbes sur le même graphique)\n",
    "plt.xlabel('t')                    # affiche les labels d'axes\n",
    "plt.ylabel('sin(t)')\n",
    "plt.show()                         # fait apparaitre la figure\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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vS7p0pevBkAWQLxI+7gkbvva6IhNEyhfNy+f3XUOLqsV4cvoa/jNvI0lJdgZY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\nTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# on veut rajouter une autre courbe:\n",
    "ord3=np.cos(absc2)\n",
    "plt.plot(absc2,ord2,absc2,ord3)               #trace la courbe\n",
    "plt.title('graphe de y=sin(t) et cos(t)')    # affiche un titre\n",
    "plt.legend(['sin','cos'])                # affiche une legende (indispensable s'il y a plusieurs courbes sur le même graphique)\n",
    "plt.xlabel('t')                    # affiche les labels d'axes\n",
    "plt.ylabel('f(t)')\n",
    "plt.show()                         # fait apparaitre la figure\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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vS7p0pevBkAWQLxI+7gkbvva6IhNEyhfNy+f3XUOLqsV4cvoa/jNvI0lJdgZY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\nTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(absc2,ord2,label=\"sin\")               #trace la courbe\n",
    "# c'est possible de ne pas tracer les deux courbes dans la meme commande\n",
    "plt.plot(absc2,ord3,label=\"cos\")               #trace la courbe\n",
    "plt.title('graphe de y=sin(t) et cos(t)')    # affiche un titre\n",
    "plt.legend()                # affiche une legende (indispensable s'il y a plusieurs courbes sur le même graphique)\n",
    "plt.xlabel('t')                    # affiche les labels d'axes\n",
    "plt.ylabel('f(t)')\n",
    "plt.show()                         # fait apparaitre la figure\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Tracer sur le même graphique $x(t)$ et $v(t)$ en fonction de $t\\in[0,T]$ et sur un autre graphique la trajectoire $(x(t),v(t))$ pour $t\\in[0,T]$.\n",
    "On choisira $n$ suffisamment grand pour que les courbes  aient un aspect \"continu\".\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "#A vous de jouer ci-dessous :"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Testez en affichant les trajectoires pour différentes valeurs de x0, et v0 sur le même graphique.\n"
   ]
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.5.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}