{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "## TP 3 Schéma des trapèzes implicites" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", "from scipy.integrate import solve_ivp # solveur EDO de Python (initial value problem)\n", "from matplotlib import pyplot as plt" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "def fun(t, y): \n", " return 3* y - 4*np.exp(-t)" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "t0, tfinal = 0, 5\n", "y0 = 1.0" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Code du schéma des trapèzes implicites" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "pas=[];\n", "erreur=[];\n", "h=1;#pas de depart\n", "nbiter=10; #on passe DIX fois dans la boucle\n", "#le dernier pas de temps utilise est 1/2^9 \n", "# la mise à jour du pas s'effectue en effet après le calcul de \n", "# la solution approchee\n", "\n", "# boucle sur les pas de temps de plus en plus petits\n", "for k in range(nbiter):\n", " #initialisation\n", " x=np.arange(t0,tfinal+h,h)\n", " N=np.size(x)\n", " yapp=np.zeros(N)\n", " t = t0\n", " y = y0\n", " yapp[0]=y0\n", " # code la methode des trapezes pour le pas h\n", " for j in range(1,N):\n", " y=yapp[j-1]\n", " #initialisation avec Euler\n", " z=y+h* fun(t,y)\n", " # iterations de point fixe\n", " for i in range(3):\n", " z = y + 0.5*h*(fun(t,y) + fun(t+h,z))\n", " yapp[j]=z\n", " #pas de temps suivant\n", " t = t + h\n", " err=np.max(np.abs(yapp-np.exp(-x)));\n", " pas.append(h);\n", " erreur.append(err)\n", " h=h/2 #divise le pas par deux. c'est pour cela que nbiter=10 et pas 9" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "image/png": 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q1IgFCxbQrl07pkyZwl133XXT00/N9lHchPtGjqRKnjy89d57t/SDNc7ITuejSO/8EgBr1qxhz549JCYmMnv2bJo2bQok/YG+vH9k1qxZKY9fLSwsLOUP6ieffJIyXuvWrZk0aRKQtF/i9OnT6eYrUqQI/v7+zJkzB0j6Y7Vx40Yg6RwYqT95lNYf5PRet1td/jvvvJOVK1emvN5nz55l27ZtnDp1iu7duzNjxgx8fX2vm33Xrl00atSIV199lZIlS3LgwIF0M96My/vDVqxYgY+PDz4+PmmOV6RIEfz8/Jg3bx6QtKZz+Vwil1392rVr14533nkn5W/Rb7/9BsDu3bupXLkyTzzxBJ06dbpiDedWWVHcBM98+fhH5878GhvLqsmTnY5jblB2Oh9FeueXAGjcuDHPPvssAQEB+Pv7p2ymKFSoEFFRUYSEhPDjjz/y0ksvpTnt8ePH8+GHH1K3bl1mzpzJuHHjABg3bhxLly4lMDCQkJCQ65686JNPPmHq1KkEBQVRp06dlJ2o48ePJzIykrp161K7dm3ee++9a57bqlUrNm/enLJD9naX39fXl48++oiIiAjq1q1L48aN2bZtG/PmzWPfvn0MGDAgZad2RtmffvppAgMDCQgIoHnz5gQFBWX4GtyoYsWKERYWxqBBg5g6dWqG486cOZPx48dTt25dwsLCOHTo0BXDr37tXnzxReLi4qhbty4BAQG8+OKLQFI5BQQEEBwczNatWzP8VNUNS2/nRXa9uGpn9mVnjxzR4iJ6f9myLp1PTpLWjjNzc5YuXaodOnRIc1jqHe45VUbLn1Wl/lBEVnOzO7NtjeImFfT15bHGjfk6JobtS5Y4HccYY1zOznB3Cw5v3EiF4GD61qnDpD/+cOm8cgI7w50xWYud4c4NSgUF0bNGDaZHRXHEiuKG5LR/SIzJrm7ld9GK4hYNGz+ei8DY/v2djpLleXl5cezYMSsLYxymqhw7duyGvqeRmqObnkSkPTAO8ASmqOobVw2vAHwEFE0e51lVzfBgJ+7Y9HTZgxUr8u3+/ezbvZti/v5umWd2FBcXR3R09A19D8EY41peXl74+fmRN2/eKx7PaNOTY0UhIp7AdqAtEA2sBSJUdXOqcSYDv6nqJBGpDSxU1UoZTdedRbFp7lyCunbllbvu4iXbsW2Mycay6j6KhsBOVd2tqpeAz4D7rhpHgSLJt32AGDfmu666Xbpwb6lSjFu6lDNXfebZGGNyCieLohyQ+uuP0cmPpTYC6CEi0cBCYEhaExKRgSISKSKRl4+c6S7DR47kuCrvDRzo1vkaY4y7OFkUaR305OrtYBHAdFX1A+4BZorINZlVdbKqhqpq6OWv67tLo/79aVOsGP9ZsIDz6RzwyxhjsjMniyIaKJ/qvh/XblrqB3wOoKqrAS+gpFvS3YThw4dzODGRqbZWYYzJgZwsirVANRHxF5F8QHdg/lXj7AdaA4hILZKKwr3blm5Ai7//nSZFivDml19y8dQpp+MYY0ymcqwoVDUeGAx8D2wBPlfVKBF5VUQuHwP6KWCAiGwEPgV6axb8ML54ePDS8OEcSEhgin2vwhiTw9ghPDKJqtKiWDF2njnDrqNHKVC8uNszGGPMrcqqH4/NUUSEka+9xp+JiUzq29fpOMYYk2msKDJR8yFDaFOsGKPmzyf2yBGn4xhjTKawoshkr40axV+qjO/Vy+koxhiTKawoMtmdjzxCR19f3vr+e07u2+d0HGOMuW1WFC7w6rhxnFRlbI8eTkcxxpjbZkXhAvUiIuhSvjxjVqzgSCac2NwYY5xkReEiI6dN4zzwSkSE01GMMea2WFG4SI02bXgkMJD3N29m27ffOh3HGGNumRWFC7306acUAJ7r18/pKMYYc8usKFyoVJ06PHPXXXz155+sfP99p+MYY8wtsaJwsb/PmkUZDw+efvppNDHR6TjGGHPTrChcrFCpUrz20EOsPnOGL//5T6fjGGPMTbODArpBwqVLBBUpwoWEBKKOHiV/0aJORzLGmCvYQQEd5pkvH2Nfe41d8fGM7dbN6TjGGHNTrCjcpO3TT3N/2bKMXLyYg2vXOh3HGGNumBWFG/1n1izigWdsrcIYk41YUbhR5RYteLppUz7Zu5eV773ndBxjjLkhVhRu9uwXX+Dn6cmQp54iIS7O6TjGGHNdVhRuVuiOOxj96KP8du4cU/v0cTqOMcZclxWFA7qNG0cLHx+emzWLo5s3Ox3HGGMyZEXhAPHwYOLMmZxR5el773U6jjHGZMiKwiF17r2XfzZtyke7d/Pj6NFOxzHGmHQ5WhQi0l5EtonIThF5Np1xuonIZhGJEpFZ7s7oSsPnz6dKnjwMev55Lpw86XQcY4xJk2NFISKewETgbqA2ECEita8apxrwHNBEVesAQ90e1IUKFCvGeyNHsiMujn/dd5/TcYwxJk1OrlE0BHaq6m5VvQR8Blz913IAMFFVTwCo6hE3Z3S5Ns88Q49KlXhj+XK2LFzodBxjjLmGk0VRDjiQ6n508mOpVQeqi8hKEflFRNqnNSERGSgikSISefToURfFdZ3/fPMNhUV4JCKCxPh4p+MYY8wVnCwKSeOxqw9lmweoBrQEIoApInLNoVdVdbKqhqpqqK+vb6YHdbU7AgIY07cvP58+zcQHH3Q6jjHGXMHJoogGyqe67wfEpDHO16oap6p7gG0kFUeO02vyZO7x9eXZL79k15IlTscxxpgUThbFWqCaiPiLSD6gOzD/qnHmAa0ARKQkSZuidrs1pZuIhweTFy4kL9DngQdsE5QxJstwrChUNR4YDHwPbAE+V9UoEXlVRDolj/Y9cExENgNLgadV9ZgziV2vXGgo4/r14+fTp3nHjjBrjMki7Ax3WYwmJnJv6dL8ePQoGxcvplqbNk5HMsbkAnaGu2xEPDyY/O235CdpE1TCxYtORzLG5HJWFFlQ2ZAQ3hk0iJVnzvBGhw5OxzHG5HJWFFnUQ+++S0TFiry8ZAm/TpnidBxjTC5mRZFFiQiTli/Hz9OT/3v0UU5HRzsdyRiTS1lRZGE+FSrwyTvvsDc+niEtWjgdxxiTS1lRZHFNHn2UF5s3Z8bu3Xw6ZIjTcYwxuZAVRTbwwvffE+btzaAJE9i9dKnTcYwxuYwVRTaQx8uLTxYuxFOELh06cOHECacjGWNyESuKbKJS06bMePFFfjt/nifDwpyOY4zJRawospGOr7zCc2FhTN66lRkDBzodxxiTS1hRZDOvLllCq6JFGfTBB/z+xRdOxzHG5AJWFNlMHi8vZv30E0U9PPhbRIR9v8IY43JWFNlQ6bp1mf322+yOi+OhkBASLl1yOpIxJgezosimmg0ZwvgHH+S/R47wYsuWTscxxuRgVhTZ2KOffsojtWszavVqPh082Ok4xpgcyooiGxMRxv/6K819fOg7cSKRH33kdCRjTA5kRZHN5StcmLmrV1PK05P7+/blzw0bnI5kjMlhrChyAN9atZj/2WecTEykY1gYsYcOOR3JGJODWFHkEHW7dOHzESPYeP483erWJf7CBacjGWNyCCuKHOSel19m0sMP8+3RozwaHIwmJjodyRiTA1hR5DADZsxgeJMmTNm2jZFt2zodxxiTA1hR5ECvLV/Ow5Ur89KPPzK9f3+n4xhjsjkrihxIPDyYsnEjbYoXp//UqXz1zDNORzLGZGOOFoWItBeRbSKyU0SezWC8LiKiIhLqznzZWb7ChfkqKooGhQvT/c03WfzGG05HMsZkU44VhYh4AhOBu4HaQISI1E5jPG/gCeBX9ybM/gqXLs3CTZuo6eXF/c89x8pJk5yOZIzJhpxco2gI7FTV3ap6CfgMuC+N8V4D3gTs8563oJi/P4vWrqVc3rzc89hj/DZrltORjDHZjJNFUQ44kOp+dPJjKUSkHlBeVf+b0YREZKCIRIpI5NGjRzM/aTZXKiCAH376CR9PT8J79GDT3LlORzLGZCNOFoWk8ZimDBTxAMYCT11vQqo6WVVDVTXU19c3EyPmHBUaN2bJd9+RX4S7unVjw+zZTkcyxmQTThZFNFA+1X0/ICbVfW8gAFgmInuBO4H5tkP71lVr04afFi2ioIcHrSMiWP/JJ05HMsZkA9ctChEZLCLFXDDvtUA1EfEXkXxAd2D+5YGqekpVS6pqJVWtBPwCdFLVSBdkyTWqtG7NT0uW4O3hQeuHHyZy5kynIxljsrgbWaMoDawVkc+TP86a1iajm6aq8cBg4HtgC/C5qkaJyKsi0ikz5mHS5t+iBcuWLaOopydtevbklw8+cDqSMSYLE1W9/khJ5RAO9AFCgc+Bqaq6y7Xxbl5oaKhGRtpKx43Yv2oVrVu2JCYuji9ee432L7zgdCRjjENEZJ2qprlp/4b2UWhSmxxKvsQDxYC5IvJmpqU0blchLIwV69dTvUAB7n3xRT4dMsTpSMaYLOhG9lE8ISLrSPouw0ogUFUfBUKAv7k4n3GxUgEBLNu6lSY+PvzfhAm807Wr05GMMVnMjaxRlAQeUNV2qjpHVeMAVDUR6OjSdMYtfCpU4Ls9e7i/dGmemDuXF5s2tUOUG2NSXLcoVPUlVd2XzrAtmR/JOMGrWDHm7NlDv+rVGblyJT0qV+bCyZNOxzLGZAF29FiTIo+XFx9s2cLr7doxa98+WleowNEt9r+AMbmdFYW5gnh48Px33zH7739n/ZkzNKpbly0LFjgdyxjjICsKk6ZuY8awbMoUziYk0LhjRxaNGuV0JGOMQ6woTLoa9evHr8uXUz5/fu5+/nneaN/ednIbkwtZUZgMVWralNX79tG1QgWe+/57upQvz+noaKdjGWPcyIrCXFfhUqX4dM8e/nPffXwdE0OjKlXYavstjMk1rCjMDREPD/4xbx6Lx47lWFwcDTp25NPBg52OZYxxAysKc1NaDR3Kul9+oa63N/83cSJ9q1Uj9tAhp2MZY1zIisLctPING/LTkSO80LQp03fuJLRCBTZ89pnTsYwxLmJFYW5JHi8vXvv5Z34cO5YzCQk0iohgXOfOJMbHOx3NGJPJrCjMbWk5dCgbN2+mXalSDJ03jza+vuxdscLpWMaYTGRFYW5byRo1+Domhik9exJ58iSBzZoxuUcP+86FMTmEFYXJFOLhQb+PPuKPlSu5s3hxHvnkE9r5+nLgl1+cjmaMuU1WFCZTVQgLY9HRo7zbvTurjh+nduPGjHvgARIuXXI6mjHmFllRmEwnHh48+umnbFq2jKYlSzL0q69oWKwYkTNnOh3NGHMLrCiMy1Ru0YKFhw8z+8kniblwgUY9e/JEcLAdAsSYbMaKwriUeHjQ7e232bp7N48GBDBh40ZqVKzI9P797aO0xmQTVhTGLXwqVmTC77/zy7RpVCxQgD5Tp9LQx4cVEyc6Hc0Ycx2OFoWItBeRbSKyU0SeTWP4P0Rks4hsEpElIlLRiZwm8zTs04dVJ0/y8WOPcejiRZoNHsyD5cuz9+efnY5mjEmHY0UhIp7AROBuoDYQISK1rxrtNyBUVesCc4E33ZvSuIJHnjw8NHEi2/78kxEtW/JNdDQ1mzfnHyEhdupVY7IgJ9coGgI7VXW3ql4CPgPuSz2Cqi5V1XPJd38B/Nyc0bhQIV9fXl66lO1r1hBRtSrj1q+ncu3avNSsGaf27XM6njEmmZNFUQ44kOp+dPJj6ekHfJvWABEZKCKRIhJ59OjRTIxo3MGvQQM+3LGDqAULuNvPj9dWrMDf3583776bs0eOOB3PmFzPyaKQNB7TNEcU6QGEAm+lNVxVJ6tqqKqG+vr6ZmJE404177mHzw8cYN0nn3BnyZI88913VCpdmn+1a8dJW8MwxjFOFkU0UD7VfT8g5uqRRKQNMBzopKoX3ZTNOKj+//0fC48cYcWkSTQoWZLhixZRsVIlhjdpwtHNm52OZ0yu42RRrAWqiYi/iOQDugPzU48gIvWA90kqCdsGkcs0GTSIhUeOsH7WLML9/Bi1ahUV69ThyeBgdi9b5nQ8Y3INx4pCVeOBwcD3wBbgc1WNEpFXRaRT8mhvAYWBOSKyQUTmpzM5k4PVi4hgzoEDbF64kAerVuXdjRup2qoVncuUYdnYsXaUWmNcTFTT3C2QbYWGhmpkZKTTMQEQbP0AABHnSURBVIwLHYyM5N0nn+T91as5pkqQlxdPdu9OxJgxeBUr5nQ8Y7IlEVmnqqFpDbNvZptsp1xoKK+vXMmBv/5iSu/eJAB9p0/Hr0QJngoNZevChU5HNCZHsaIw2VaB4sXp9+GHbDp7liX/+Q+typZl/Lp11OrQgWY+PswYOJDzx445HdOYbM+KwmR74uHBXf/4B3Oio4netIl/3303h86fp9cHH1CmZEkeDwjg16lTbV+GMbfIisLkKKUCA/nnwoVsv3CBpW+/zT2VKjE1Koo7+/enmpcXL7dowfbvv3c6pjHZihWFyZHEw4OWTz7JrD17OLxvH9P69KFi4cK8tnw5Ndq3p2GhQozr3JmD9sEHY67LisLkeD4VKtBn2jSWHD/OgchIRt97L3GqDJ03D78GDWjs7c1bHTqwa8kSp6MakyVZUZhcpVxICE/Nn89v586xZeFCXg8P51JiIv9cuJCqbdoQVKAAr7RowcbZs22fhjHJ7HsUxgB7V6zgq//8hy9//JGVp0+jQDkPD+6pVo27O3WizRNP4O1nBy82OVdG36OwojDmKof++IMFo0ezcPFiFsfEcAbICzQtWpR7wsJo37cvde6/H/H0dDqqMZnGisKYWxR37hyrpkxh4axZfLtxI79fuADAHSK0LFeOu5o2pVXPnlRr1w7xsC25JvuyojAmkxxYu5bF773H0mXLWLpvHwcTEoCkzVStKlSgVfPmNH3wwaTisDUOk41YURjjApqYyI4ff2TpRx+xdPlylkZHcyR5B3hJERqXLElYUBCN27enQY8eFCxVyuHExqTPisIYN9DERLZ89x2r5sxh1apVrNq7l22XLgGQBwguUIDGVaoQ2qABIXffTc2OHfEsUMDZ0MYks6IwxiHH9uzhl48/ZtWiRayKimLNiRNcPgl8QSCoUCFCKlYkJDSUkHbtqHXvveTx9nYyssmlrCiMySIS4uLY9sMPrFuwgHW//ML6Xbv47eRJYpOHFwDqeHkRULo0ATVqENiwIQFt21KmcWMkTx4no5sczorCmCwsIT6eHT/9xLpvvmH96tX8vns3v584waHkHeUAxYCAQoUILFuWgFq1qNWgAdWbNaNMo0aIl5dz4U2OYUVhTDb01969RH3/PX+sWMHvmzbxx/79/HHqFKdS/c4WBqrny0f14sWp7udHjVq1qB4SQvXmzSkSGAi2FmJukBWFMTmEqhIdFcW2pUvZvm4d27ZsYfuBA2w7doy9ly6R+re5NFDZywt/Hx/8S5fG398f/5o18Q8Oxq9BA/JUqGBFYlJYURiTC1w4f57da9eyfcUKtv32G9u3b2f3oUPsOXmSA5cukfrIVZ5ABcDfywv/okXxL12aCn5++Pn7U75GDfwCA/Hy94fSpSFvXoeWyLiTFYUxuVxcXBwHdu5kz5o17Nm4kT1bt7Jn/372HD7MnlOnOBwXd81zSgLlAb98+ShfpAjlS5bEr0wZylesiF+VKvhVq0b+cuXgjjvA1xeKFgURty+byRwZFYWtdxqTC+TNm5fKtWpRuVatNIefO3eO6P37OfDHHxzYvJnoHTs4sG8f0YcOsffYMVacOsWJv/6CrVuveJ4v4Jd8KefhgV/BgpQrVgw/X1/8ypShXMWKeJcrl1QkJUpAsWJJl+LFk64LF7ZyyQasKIwxFCxYkOo1a1K9Zs10x4mNjeXgwYMc2L2bA1u3JpXJ3r0cjIlh35EjrDp5kmOxsRAbCwcOpDyvCMlFctW1H1DO0xO/okUpUbw4klaRFCsGPj7g7Q1Fivzv+vJtb2/bz+IGtunJGJNpzp8/T0xMDNHR0URHR3Pw4EGi9+7l4J49SY/9+SeHTpwg8apzfeT38KBc/vz4eXpSThW/+Hj8Ll68olxKk85/tgUKXFkeV18XKgQFCyZdUt+++v7Vt/PmzVVrO1l205OItAfGkbRvbYqqvnHV8PzADCAEOAY8qKp73Z3TGHNjChQoQJUqVahSpUq648THx3Po0KGkEkldKMnXv0ZH80V0NJeuep6HhwdlihenXNGi+BUpgl+hQpTz8sIvXz7KiVABqJCYiOeZM7B/P5w5A6dPw7lzSZeb5en5v9IoUADy5wcvr/9dp76d0WPpDcuXL+mSN2/SJa3bqR9zsLgcKwoR8QQmAm2BaGCtiMxX1c2pRusHnFDVqiLSHfg38KD70xpjMkuePHnw8/PDz8+PRo0apTmOqnLs2LE0iyQ6Opot0dH8sHMnp0+fvuJ5+fLlo2rVqlSvXv3KS7Vq3FGkCHK5NC5fzp69/u3Ll4sX4cKF/12fPJl0nfqxy9cXLoArttbkyZN+qeTNC/PmQfXqmT/bTJ/ijWsI7FTV3QAi8hlwH5C6KO4DRiTfngtMEBHRnLa9zBhzBRGhZMmSlCxZkuDg4HTHO3PmTEp57N27lx07drB9+3a2b9/OwoULuXTpf+slxYsXJygo6IpL7YYNyZ8/f+YvgCrEx6dfIhcuQFxc0uXSpSuvb/Z26sdcdJBJJ4uiHHAg1f1o4Op/L1LGUdV4ETkFlAD+Sj2SiAwEBgJUqFDBVXmNMVmMt7c3NWvWpGYaO+ETEhLYv38/27dvZ9u2bfz+++9s2rSJ999/n/PnzwPg6elJQEAAYWFhhIWF0bhxYypXrozc7iYekf/9l58DDvLoZFGk9ZO4ek3hRsZBVScDkyFpZ/btRzPGZHeenp5J30b396ddu3YpjyckJLBz5042btzIxo0bWbNmDR9//DGTJk0C4I477qBp06a0bduW8PBwKleu7NQiZBlOFkU0Sd/nucwPiElnnGgRyQP4AMfdE88YkxN5enpSo0YNatSoQbdu3YCk8oiKimLVqlWsXr2aZcuW8eWXXwJQpUoVwsPD6dChA23atHHNpqoszrGPxyb/4d8OtAYOAmuB/1PVqFTjPA4Equqg5J3ZD6hqt4ymax+PNcbcLlVlx44dLFq0iEWLFrF06VJiY2Px8fGhU6dOdOnShfDwcLxy0JF7s+whPETkHuBtkj4eO01VXxeRV4FIVZ0vIl7ATKAeSWsS3S/v/E6PFYUxJrNdunSJJUuWMHfuXObNm8fx48cpWrQoPXr0oH///gQFBTkd8bZl2aJwBSsKY4wrxcXFsXTpUmbMmMHcuXO5ePEiISEhPPbYYzz00EPZdtNURkXh4e4wxhiTneXNm5fw8HA+/vhjYmJiGD9+PBcvXqRfv374+/vz73//m5MnTzodM1NZURhjzC0qXrw4Q4YMYdOmTSxatIg6derw7LPPUrFiRUaOHElsbOz1J5INWFEYY8xtEhHatm3L4sWLWb9+Pa1ateLFF1+katWqTJw4kbg0DuOenVhRGGNMJqpXrx7z5s1j1apV1KhRg8GDBxMaGsrq1audjnbLrCiMMcYFGjdunPJ9jOPHjxMWFsbAgQM5deqU09FumhWFMca4iIjQuXNntmzZwlNPPcW0adMICgri559/djraTbGiMMYYFytcuDCjR49mxYoV5MmThxYtWvD8888THx/vdLQbYkVhjDFucuedd7Jhwwb69u3LqFGjaN++PceOHXM61nVZURhjjBsVLlyYKVOm8OGHH7JixQpCQ0PZuHGj07EyZEVhjDEO6N27N8uXLycuLo6mTZuyZMkSpyOly4rCGGMc0rBhQ9asWUOlSpW45557mDNnjtOR0mRFYYwxDipbtizLly+nQYMGPPjgg0ydOtXpSNewojDGGIcVK1aMRYsWER4ezoABA/j444+djnQFKwpjjMkCChYsyFdffUXLli3p1asXn3/+udORUlhRGGNMFlGgQAG++eYbwsLC6NGjB8uWLXM6EmBFYYwxWUqhQoX45ptvqFq1Kp07d2br1q1OR7KiMMaYrKZo0aIsWLCAfPny0aFDB44ePepoHisKY4zJgvz9/Zk/fz4xMTFERESQkJDgWBYrCmOMyaIaNWrEu+++y5IlSxgxYoRjOawojDEmC+vTpw99+/Zl5MiRfPvtt45ksKIwxpgsbsKECdStW5eePXty+PBht8/fisIYY7K4AgUKMGvWLM6cOcOgQYNQVbfO35GiEJHiIrJYRHYkXxdLY5xgEVktIlEisklEHnQiqzHGZAV16tRh5MiRzJs3z+3f3HZqjeJZYImqVgOWJN+/2jmgp6rWAdoDb4tIUTdmNMaYLOXvf/87TZo0YciQIRw8eNBt83WqKO4DPkq+/RFw/9UjqOp2Vd2RfDsGOAL4ui2hMcZkMZ6enkyfPp0LFy7w1FNPuW2+ThVFKVX9EyD5+o6MRhaRhkA+YJcbshljTJZVtWpVnn/+eWbPns3ixYvdMk9x1U4REfkBKJ3GoOHAR6paNNW4J1T1mv0UycPKAMuAXqr6SzrjDAQGAlSoUCFk3759t5neGGOyrgsXLhAYGIiI8Pvvv5M/f/7bnqaIrFPV0LSGuWyNQlXbqGpAGpevgcPJBXC5CI6kE7wIsAB4Ib2SSJ7XZFUNVdVQX1/bOmWMydm8vLyYOHEiO3bsYOzYsS6fn1ObnuYDvZJv9wK+vnoEEckHfAXMUNWsedonY4xxSHh4OJ06dWLUqFH89ddfLp2XU0XxBtBWRHYAbZPvIyKhIjIleZxuQHOgt4hsSL4EOxPXGGOynlGjRhEbG8vIkSNdOh+X7aNwSmhoqEZGRjodwxhj3GLgwIFMnz6drVu3Urly5VuejiP7KIwxxrjeiBEjyJs3Ly+88ILL5mFFYYwx2VjZsmV54okn+Oyzz4iKinLJPKwojDEmmxs2bBiFChXi1Vdfdcn087hkqsYYY9ymRIkSDB8+nNjYWFQVEcnU6VtRGGNMDvDss2kdMi9z2KYnY4wxGbKiMMYYkyErCmOMMRmyojDGGJMhKwpjjDEZsqIwxhiTISsKY4wxGbKiMMYYk6Ecd/RYETkK3M4p7koCrj24e9aT25Y5ty0v2DLnFrezzBVVNc0zv+W4orhdIhKZ3qF2c6rctsy5bXnBljm3cNUy26YnY4wxGbKiMMYYkyErimtNdjqAA3LbMue25QVb5tzCJcts+yiMMcZkyNYojDHGZMiKwhhjTIasKJKJSHsR2SYiO0XEdWcAySJEZJqIHBGRP5zO4i4iUl5ElorIFhGJEpEnnc7kaiLiJSJrRGRj8jK/4nQmdxARTxH5TUT+63QWdxGRvSLyu4hsEJHITJ227aNIelMB24G2QDSwFohQ1c2OBnMhEWkOxAIzVDXA6TzuICJlgDKqul5EvIF1wP05/OcsQCFVjRWRvMAK4ElV/cXhaC4lIv8AQoEiqtrR6TzuICJ7gVBVzfQvGdoaRZKGwE5V3a2ql4DPgPsczuRSqrocOO50DndS1T9VdX3y7TPAFqCcs6lcS5PEJt/Nm3zJ0f8diogf0AGY4nSWnMKKIkk54ECq+9Hk8D8guZ2IVALqAb86m8T1kjfDbACOAItVNacv89vAP4FEp4O4mQKLRGSdiAzMzAlbUSSRNB7L0f915WYiUhj4AhiqqqedzuNqqpqgqsGAH9BQRHLspkYR6QgcUdV1TmdxQBNVrQ/cDTyevHk5U1hRJIkGyqe67wfEOJTFuFDydvovgE9U9Uun87iTqp4ElgHtHY7iSk2ATsnb6z8D7hKRj52N5B6qGpN8fQT4iqRN6pnCiiLJWqCaiPiLSD6gOzDf4UwmkyXv2J0KbFHVMU7ncQcR8RWRosm3CwBtgK3OpnIdVX1OVf1UtRJJv8c/qmoPh2O5nIgUSv6ABiJSCAgHMu0TjVYUgKrGA4OB70nawfm5qkY5m8q1RORTYDVQQ0SiRaSf05ncoAnwMEn/ZW5IvtzjdCgXKwMsFZFNJP1DtFhVc81HRnORUsAKEdkIrAEWqOp3mTVx+3isMcaYDNkahTHGmAxZURhjjMmQFYUxxpgMWVEYY4zJkBWFMcaYDFlRGOMGIlJURB5zOocxt8KKwhj3KApYUZhsyYrCGPd4A6iS/CW/t5wOY8zNsC/cGeMGyUer/W9uOfeHyVlsjcIYY0yGrCiMMcZkyIrCGPc4A3g7HcKYW2FFYYwbqOoxYKWI/GE7s012YzuzjTHGZMjWKIwxxmTIisIYY0yGrCiMMcZkyIrCGGNMhqwojDHGZMiKwhhjTIasKIwxxmTo/wG7Uv/w4DeqLQAAAABJRU5ErkJggg==\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "#trace la solutions exacte\n", "xexact=np.arange(t0,tfinal,1e-2)\n", "yexact=np.exp(-xexact);\n", "plt.plot(xexact,yexact,'r');\n", "plt.plot(x,yapp,'-k')#trace la solution approchee\n", "plt.legend(['solution exacte', 'solution approchee trapèzes implicites'])\n", "plt.xlabel('t')\n", "plt.ylabel('y')\n", "plt.show()\n" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "#trace l'erreur en fonction du pas en coordonnées log-log \n", "plt.loglog(pas, erreur,'-og')\n", "plt.loglog(pas,np.power(pas,2),'--og')\n", "plt.legend(['schéma trapèze implicite','pente 2'],loc='best')\n", "plt.xlabel('log pas')\n", "plt.ylabel('log erreur')\n", "plt.title('ordre de convergence du schéma des trapèzes implicites')\n", "plt.show()\n" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [], "source": [ "#solveur basé sur schéma RK45 de Scipy\n", "sol = solve_ivp(fun, [t0, tfinal], [1], 'RK45',dense_output=True)" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "plt.plot(sol.t, sol.y.T, '-ob')\n", "plt.plot(x,yapp,'-k')\n", "t_eval=np.arange(t0,tfinal,1e-2)\n", "plt.plot(t_eval, np.exp(-t_eval),'-r')\n", "plt.xlabel('t')\n", "plt.ylabel('y')\n", "plt.legend(['RK45','Trapèzes implicites', 'sol. exacte'], shadow=True)\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ " ***La solution générale de l'équation différentielle est $ y(t) = C \\, \\exp(3t) + \\exp(-t).$\n", " Pour la solution exacte vérifiant la condition initiale, $y(t) = \\exp(-t)$ i.e. $C=0.$\n", "Les schémas numériques en s'écartant un peu de la solution exacte calculent des approximations avec $C \\ll 1$ mais non rigoureusement nul. Au bout d'un certain temps le terme $C \\, \\exp(3t)$ devient dominant.***" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "data": { "image/png": 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Ho2maBnR/+wfoBNIlFjx2D/098RTFe3jlt3+IdTiapmns3r2b+Ph40tLSum0fOoF0kZknTyZOWdltNbP1k09iHY6maYe5wsJCBg0ahEikm5d3DZ1Ausi4U05hkNePS7x8tvTAbpeiaZp2IBoaGqipqenW9g/QCaRLzX/oHrKaHJQ4mnj+Zl2VpWlabPRE+wfoBNLlzvn1paT5nRQlwYd/+Vusw9E07TCUn5+P1WolMzOzW/ejE0gXS8nM5Ii+CQBsLWugfGd+jCPSNO1wk5+fT3Z2NhaLpVv3oxNIN5hz0w0MqlfUWNy8/fRrsQ5H07TDSH19PZWVlQwdOrTb96UTSDdZ8Pg9ZLniKIpz8cIt98U6HE3TDhMFBQUAh34CEZFTRWSbiOSJyB0Rlj8uIl+HplwRqQ1bFghb9nbPRh6dM689lzS/k92JwjsPPhrrcDRNOwzs3LkTh8NB//79u31fMUsgImIG/gqcBowFLhCRseFllFI3K6UmKaUmAX8BXg9b7N67TCl1Zo8Fvh/6DhvKhOwMzAjb3T7yv1wb65A0TTuEKaXIz89n6NCh3TKAVEuxPAOZDuQppXYqpbzAy8C8dspfALzUI5F1oVlXX8lgV4B6UzMfvvEpHrc71iFpmnaIqqmpoa6urkeqryC2CWQgUBj2vCg0rxURyQaGAh+HzXaISI6IrBGRs7ovzAN30SMLGRy6PmTxXU/EOhxN0w5RO3fuBHqm/QNim0AiXV+v2ig7H1iilAqEzRscGuT9QuAJERkecSci14QSTU5FRcWBRXwALrjnevp54tmV4OHFX/8+ZnFomnboys/PJzExkYyMjB7ZXywTSBEQfplkFrCnjbLzaVF9pZTaE/p/J7ACmBxpRaXUs0qpaUqpaX369DnQmDvNmZTEqWedQHLQwa4Ek25U1zStSwWDQXbu3MnQoUO79f5X4WKZQNYBI0VkqIjYMJJEq95UIjIaSAW+CJuXKiL20OMM4FhgS49EfQCGzpjO+AGpmDGxrdnL1x98EOuQNE07RBQXF+N2uxk1alSP7TNmCUQp5QduBJYCW4FXlVKbReQ+EQnvVXUB8LJSKrx6awyQIyLfAJ8ADymlen0CATjlup8zxBPAJV5WrdqkB6HSNK1L5ObmIiIMHx6xNr9byL7fy4e2adOmqZyc3nGn3EU3309Bsp8B7nguvfdG7N00ZrGmaYeHp556CofDweWXX97l2xaR9aE2533oK9Fj5LLH72JQo5M9zib+c/eTsQ5H07SDWF1dHWVlZYwcObJH96sTSAxd/IebyGyOZ3dCM8/fom//rmla52zfvh2gR9s/QCeQmLI7nZx9zc/I8MVRkBjkP7fq7r2apu2/7du3k5ycTE/3NNUJJMb6DhvKiSdOISlopyDOxH8XPhjrkDRNO4j4fD527tzJqFGjeqz77l46gfQC4045hanZfbEqM7kEeeeBR2IdkqZpB4mCggJ8Pl+b7R/KFyTQ5OuWfesE0kscf9UVHJkYhyhhi8fLh3/+a6xD0jTtILBlyxZsNlubty9p3lZNyQNr8BY1dPm+dQLpRU6/9VeMNpkJiOKbynpWPvd8rEPSNK0XCwQCfPfdd4wePRqr1RqxjPvbSkwOC9bMhC7fv04gvczZ997BCJ/CI37W5ZeQ8983Yh2Spmm9VH5+Pm63m3HjxkVcrvxB3FurcYxJR8xd3z6iE0gvdN6DdzPMFaTJ5GXl19vY8OZbsQ5J07ReaG/1VVtXnzdvq0Z5AsRN7J7eWTqB9FIXPrKQoY2KRpOXT9dvZe1rS2IdkqZpvUggEGDr1q2MGjWqzeor1zcVmOIt2Icnd0sMOoH0Yhc/upDhTUYSWbUxj9WLD7rxtDRN6yYFBQXtVl8FPQGat1bjHN8HMXfPV71OIL3chf+3kGEuRZPJy5qtu/j8xf/EOiRN03qBvdVXI0aMiLi8eWsVyhckbkL3XVyoE8hB4KJHFjLCrXCZvKzJ3c3yp56JdUiapsWQz+dj8+bN7fa+cn1TgTnZhm1IUrfFoRPIQeKChxcywg3N4mdDSZW+2FDTDmO5ubk0NzczadKkiMuDLh/NuTU4J/RBTN13dbpOIAeR+Q/fwxixEETxrdfDq3feH+uQNE2Lga+//pqkpKQ2Lx50fVsJAdVtva/20gnkIPPTe3/LpJRELMpMrkXfgFHTDjcNDQ3k5eUxYcIETKbIX+GudWVY+sVhHdj1Fw+G0wnkIHTqLTdx9LAs4oM2dsShbwWvaYeRjRs3opRqs/rKV9qEt7CB+KP6d/vNFWOaQETkVBHZJiJ5InJHhOWXiUiFiHwdmq4KW3apiGwPTZf2bOSxd9xll3DysVNI8zvJTwrwz9/8CY/bHeuwNE3rRkopvv76a7KyssjIyIhYpmltKZiFuMl9uz2emCUQETEDfwVOA8YCF4jI2AhFX1FKTQpN/witmwYsBGYA04GFIpLaQ6H3GhNOncN5V57LgOZ4ChPc/Ou+/0fR5s2xDkvTtG5SXFxMRUVFm2cfyhek6atynOPSMcdH7p3VlWJ5BjIdyFNK7VRKeYGXgXlRrvtjYJlSqlopVQMsA07tpjh7tb7DhnLpwhsZ3OCg1N7E6y+9q69a17RD1Nq1a7HZbIwfPz7icvfmSpTbT/xR/XsknlgmkIFAYdjzotC8ls4RkY0iskREBu3nuojINSKSIyI5FRUVXRF3r2N3Orni0TsYVmeizuzh0025vHHvQ7EOS9O0LtTY2MjmzZuZNGkSdrs9YpmmtaWYU+3Yh6f0SEyxTCCRWndUi+fvAEOUUhOA5cDe+5tHs64xU6lnlVLTlFLTenq4x5624PF7GO03Gd18lZfnb74v1iFpmtZFNmzYQCAQ4Kijjoq43FfahGdnHfEzMrv12o9wsUwgRcCgsOdZwJ7wAkqpKqWUJ/T078DUaNc9XJ33wF0cN2oIKQEH+clB/nHbw1QVFcU6LE3TDkAgECAnJ4dhw4a1Oe55w6pixGoiYXrPVF9BbBPIOmCkiAwVERswH3g7vICIZIY9PRPYGnq8FJgjIqmhxvM5oXkacOzFF3HhtReS5YqjKM7F4qcX63toadpBbNu2bdTX1zN9+vSIywONXlxflxM3pS+muO5vPN8rZglEKeUHbsT44t8KvKqU2iwi94nImaFiN4nIZhH5BrgJuCy0bjXwB4wktA64LzRPC0nPyuKqh29jaJ2ZWnMzK3MLWKwvOtS0g45SitWrV5OcnMyoUaMilmlaUwJ+RcJxEZuCu40oFbHp4JA0bdo0lZOTE+swetwbv3+InX5Fg8nDIJeTs355EelZWbEOS9O0KBQUFLBo0SJOP/30iGcgyhek5E9rsQ1MIOPyI7slBhFZr5Sa1nK+vhL9MHD2wjv46RknG9eLxLt58en/8OETf411WJqmRWHlypXEx8czefLkiMubNpQRbPT1+NkH6ARy2Bg6YzrX/OlWhtWbaDR7WVdTzfM3/0Ffva5pvdiePXvYsWMHRx99dMTbtit/kIZPCrENSsQ+ome67obTCeQws+CxezgqI53EgI385ADP3ff/+OLlV2IdlqZpEaxcuRK73c60aa1qjwDj7CNQ6yHxlMHdft+rSHQCOQzN+cX1LLhhAYMb7FTYXKzYup0X9DUjmtarlJaWsnXrVqZPn47D4Wi1XAWMsw/roEQco2JzJyedQA5TKZmZXPHob5lodeAMWNmZHOTp3z6ib4Oiab3E8uXLcTgcHHPMMRGXuzaUE6jxkHRybM4+QCeQw968u27nstDZSLnNxUfffsfzt9yv20Y0LYYKCgrIy8tj5syZOJ3OVsuD3gD1y3dhzUrAMTp295HVCUT7/mxkgsVBfMBKfpKfZx/4ix42V9NiQCnFsmXLSEpKavPCwcZVxQTqvKScPjRmZx+gE4gW5qy7b+fq265jSL2FBrOXr7wu/vmbP7F74zexDk3TDhtbt26luLiYWbNmRex5FWjw0rCiCMe4dOzDer7nVTidQLR9OJOSuOyxuzhxzEj6eeIoTHDz8pL3eEF3+dW0buf1elm6dCl9+/Zl4sSJEcvUL9uF8gdJPi3yeOg9SScQLaKj55/Pzx+6ldEuwapM7EwO8MwDf+bVO++PdWiadsj67LPPqKurY+7cuZjN5lbLvcWNNK0rJeHoTKwZrdtGeppOIFq7Lnh4IVf+8mqG1llwmfxssfp55o5HWPH3f8Y6NE07pFRUVLB69WomTpxIdnZ2q+UqoKh5fTumeCtJJw/er20Hg4GuCnMfOoFoHUrKSOfSx+/irFOOZ1CTk3K7i5VFxfzj1ofZ8OZbsQ5P0w56wWCQ9957D5vNxuzZsyOWaVxdjK+4kZQzh+/XHXdLtm9j0a9voKpod1eF+z2dQLSoHXH8TK585HZmZGTQ1+OkKM7FB19t5Lnf/Imtn3wS6/A07aCVk5NDQUEBp5xyCgkJCa2W+6ubqf9wF44j0nCOz4h6u36fj6VPP4nP00xCWvTrRUsnEG2/zfnFDfz8oVuZbHWS5nWwO8HNmys+57lf/5FdG9bHOjxNO6hUVVXx4YcfMmLECKZOndpquQoqapbkgkDKWcP3q9vuF0sWU1W0mzlX34g9Lq4rwwbA0uVb1A4b8+66HYDX7rqfcmVjd6KLxW99QL//LOOoOVMZ/+PIp+KaphmCwSBvvPEGFouFM888M2JyaPisCM/OOlLPGYklpfUtTdpSuHkja99awpEnzmHo5Mj30jpQOoFoB+zc++8C4OXb7qPKaiSSktVrWLt8A2OnDuHo+efHOEJN650+/fRTioqKOOecc0hKSmq13FvUQP2Hu3AemU7ctH5Rb9fd2MD7f32M1P6ZnHjZ1V0Z8j5iWoUlIqeKyDYRyROROyIsv0VEtojIRhH5SESyw5YFROTr0PR2y3W1njf/4Xu44YHbGB+0ke51UBjnZvnWbfz99od5/5EnYh2epvUqubm5fPrpp0yaNIkjj2w9EFSw2U/1y9swJ1pJ/enIqKuulFIse/YvuGprOP0Xt2JzdF9335idgYiIGfgrMBsoAtaJyNtKqS1hxb4CpimlXCJyHfAwsPfnrFspNalHg9aics59vwPg3YceY09tgBKHi+JGFwW/e5hkr5uf/eE27BHu76Nph4uamhpef/11+vXrx+mnn94qOaigovqVbfir3fS5asJ+9brKeed1tn+5mpkXXkb/4SO7OvR9xLIKazqQp5TaCSAiLwPzgO8TiFIqvGvPGuDiHo1QOyBn3HELAKte+DfbNhZTEe+j3Kb464NPktYU4ITzZzN0RuR7/Wjaocrj8fDKK6+glOK8887DZrO1KlO/fBfNW6tJOXM49mHJUW+74Ov1rFz8PKNmHMtRZ57TlWFHFMsEMhAoDHteBMxop/yVwAdhzx0ikgP4gYeUUm9GWklErgGuARg8eP8uvtG6xnELLuE4oDR3G0uffpWaOKEgyUfx+0vp+/oK0p1BfnpvqxpMTTvkBAIBXnvtNcrKyrjwwgtJT09vVcb1TQUNHxcSN60f8UdnRr3tmpJi3v3zw2QMGsyPr/9Vj9xkMZYJJNKrUxELilwMTANOCJs9WCm1R0SGAR+LyCal1I5WG1TqWeBZgGnTpkXcvtYz+o8azaWP3Y3H7eaN3z9CrTgodbgoRrH7rj+R4vIz4yfHMObEE2MdqqZ1OaUU77//Pnl5eZxxxhmMHNm6eql5ew3Vr27DNiSJ1LNGRJ0EGmuq+e+D9yAmM/Nuvatb2z3CxTKBFAGDwp5nAXtaFhKRU4A7gROUUp6985VSe0L/7xSRFcBkoFUC0Xofu9PJ/IfuAeCrt97i6083UxuvKEjyUbRiJX2W5pAQcHPmHTeSmJ4W42g17cAppVi+fDnr16/nuOOOizhErbewgap/b8Hax0nGgrGIJbo+Ts1Njbz+4D246uo4954HSO7bv6vDb5MoFZsf5SJiAXKBk4FiYB1woVJqc1iZycAS4FSl1Paw+amASynlEZEM4AtgXosG+FamTZumcnJyuv7FaAfM43bz1v3/R63fQYWjGZ8EcAatpDdbSbH7+cnvbtYN79pBSSnFxx9/zMqVK5k6dSpz587FZNo3OXiLG6n85ybEbqbvdRMxJ9mj2rbX7eK/f7yX0rxczr5jIdrFOC8AACAASURBVEMmTO6Ol4CIrFdKtcp6MUsgACJyOvAEYAaeU0o9ICL3ATlKqbdFZDkwHigJrbJbKXWmiBwDPAMEMboiP6GU6vDufjqBHBx2bVjPp/9+l0ZnHJVWF0FRJAbspLiFfulWzvjtr2MdoqZFRSnFRx99xKpVq5gyZQpnnHFGq+Th2V1P5XObMTnM9Ll6PJb06H4oNTc18vofF1K6Yztzb7qN0Ucf1x0vAeilCaSn6QRy8Pnq7bf4ZsUmGuJsVFlcAEYyaTaR7AjoMxOt1woEArzzzjt8/fXXbSaP5h21VL2wBVOClT5XjceSGt2V5q76Ol7/40IqdhVwxq9uY+T0yOOmdxWdQNAJ5GC34tl/kv9dKU1OK1UWN0oUcUEbac0W4qSZU39xBWlZWbEOU9Nobm5myZIl5OXlccIJJzBr1qxWDeJNOaXUvJ6HJcNJnyuPxJwcXbVV9Z4i3njo9zRUV3LmLb9j2JSjuuMl7EMnEHQCOZTk/PcNNq/6liangyqri4AorMpMmteB0+Nh6LgsTrjysliHqR2GysvLeeWVV6iurmbu3LmtGsxVQFH/YQENnxZhH5lC+kVjMDmi689UuHkjbz/6IGI2c9atdzFg1JjueAmt6ASCTiCHqm0rV/Ll6x/TbIuj1u7DZfICkBRwkNRsIo5mZl12DgPGjYtxpNqhbtOmTbzzzjtYrVbOPfdchgwZss/yQL2Hqpe24c2vI35Gf1LOHI6YO+5tpYJB1r3zOqtefoHU/gM4+457SekXXW+r0vw61ry5k9OuHY/d2bmOt20lEH0zRe2gN3rmTEbPnAkYvbnef/hJausVTQ4Te+LcBEWR9+oSUv3vEecJEm/2cdK1F9N3aOzHlNYODS6Xi/fee4/NmzeTlZXFueeeS3LyvleQuzdXUfPGdpQnQOp5o4ifEt3NEZtqa/jwmT+zc8M6Rh09kznX/CKqW7MHg4pvlhey5q0dxKfYaar1dDqBtEWfgWiHtO8++YR173yKxxJHo11Ra3KDgFmZSPU7iPMEcZi9HHP+XIZMaT0Wg6a1RynFtm3bePfdd3G5XMyaNYtjjz12n/HMA00+at/egfubCqz940m7YDTWfvFRbfu7zz/l40XP4nO7OGHBVUyaMzeqiwvrKlx89PxWSvLqGDapDydecgSO+Ojvp9WSrsJCJxAN1r/9FltWfE2zxUmjPUiduRkAUZAUdJDgNWHzeRg4rC/HX3UpNkf04y9oh5eKigr+97//sWPHDvr27cvZZ59NZuYPtx5RQYVrfRl1/ysg2Own6aTBJJ6QFdUFgvUV5Xy86Fl25Kwhc8Rofnzdr0jPGtTheoFAkG9XFLPmrR2YzCZmnj+S0TP6H/BtTXQCQScQrbV1//0v363ehFecuO1CrcWDXwIAOJSVJJ8NuzeAHS+jpo3hqPN/1iP3GNJ6r4aGBlatWsW6deuwWq3MmjWL6dOn73PW0ZxXQ927+fhKm7BlJ5F69gis/Ts+6/C6XXz55musf+9NBOGY8y9m6tx5mEzmDtct3FrNyle3U1PSxOBxacy66AgS07rmB5BOIOgEonWsqngPK55+joYmwWOz0Wj102DyfH/nNkfQSqLfisOrsCkP2WMHc8yCi7BYO189oB0c6uvr+fzzz1m/fj2BQIDJkydz0kknfT+GuVIKT14t9R8X4s2vw5xqJ/m0oTjHZ3T4o8PX3MzGj5ay9q3XcNXVMmbmiRw3fwFJGX06jKt8Vz3r3s2nYFMVSRkOjv3ZSIZO7Hif+0MnEHQC0Trnm6X/Y/PyNXgDNrw2K03WAPUmD0qMz45VmUkI2HD6TVh9PpyWIKN/NImJ805vdeGYdnBRSlFYWMjatWvZsmULSikmTpzI8ccfT1qacZ825Q/i/raShs/34CtswJRkI/H4LBJmZCLW9v/+7sYGNi77gPXvvYm7oZ5B4yYw88JLyRwxusPYyvLryfmggIKNldjjLEyeM5iJJw/CYu34bGV/6QSCTiBa18ld/Tkb3l5Gs8eKz2bFY4EGixeP+L8vY1MWEgJWHD7B6vdjNwcZODyL6RecgzMpMYbRax2pq6vj22+/ZePGjZSVlWG325k8eTLTp0//PnH4Kt00rS3Ftb6UYJMfc7qDxOOziJ/ar912DqUUJdu/45tlH7Dti5UEfD6GTprKjLPPZ+ARY9uNK+APsmNDORs/KaIsvx57nIVJpwxi/ImDuryHVTidQNAJROtezW43qxe9wJ68PfiUHb/VSrNV0WD24g1LLKIgTtmIC1iw+cHi9+OwBhk8ZjiTzppLfHLrsbG17ldTU0Nubi5bt26loKAAgIEDBzJp0iQmTJiA3W7HX+XGtakS96ZKfMWNYALnmHTiZ2RiH5GCmNquNqopKWbbF6vYtvozKgt3YXU4GTvzRCbOPo0+2W13KVdKUVZQT+6aUravL6e50UdKvzjGz8riiKP7Y4vyIsQDoRMIOoFosdFUV8eaxf+hbEcpPr8Vv8WGz2qi2RKkyeTFF2q038sRtOAIWrEHBGtAYQ0GcNpNDBw+hCPnzia5X+tBiLT919TURGFhIbt27WL79u1UVlYCkJ6ezvjx4xk/fjypCcl48uvxbK+hOa8Wf5lxPzbboESc4zOIm9QXc1LrEQUBgsEApXnbKfhmA3k5a6go2AlA5qgjGHf8yYw57gRszsjXcwQDQUrz69m1qZIdGyqoq3BjtpoYOjGDMUdnMmhMWrvJqqvpBIJOIFrvU7FnN+teWUJVYS3BoJWgxYbfYsJrhmazH5d4Cci+n1GLMuEIWrEFTVgDYA0qLErhtFvoO6A/w6ZPod/YYTji9E0m93K73ZSXl1NWVkZJSQm7d++mqqoKALPZTHZ2NiNHjmR432wSGi14CxvxFjbgLWyAoAKLCfvQJByjUnGOz8CS0rp3U8Dvo6Ignz3bv6N462Z2f/sNzU2NIELmiFGMPnomI2ccG7FhXClFbZmLPdtrKc6tZfeWKjxNfkwmYcCoFEZN78fwyX2xdWM1VXt0AkEnEO3g4vP52frFCvI+/YKmag+BoBVlthO0mPGbTfjMQZpNftziIyDBVuublQm7MmMNmrAGBUvQuPWE3WQmPs5Bar/+9BkxhPQhmSSkJeOMd2KxHLw3p3C73dTW1lJbW0tNTQ21tbVUV1dTXl5OfX399+WcTidZ/QcyMLkf/S1ppDfHoyo9+MtdBF2hqkaLCduAeOxDk7GPTMGenbxPg7jX7aKycDeVhbuoKtxFWX4eZTvy8PuM2+gkpGeQPX4SQyZMZvD4ScQl7XtVelOdh8rCRiqLGijf1UBJXi3uBp8RX5KNwWPTGDI+g0Fj07q1bSNaOoGgE4h2aAkEgpQUF7Bt5YdUbt2FtyFAMGhHmRwoiwVlNhMwC36TwmcK4pEAHvF/f51LJCYlWJUZizJhUYJFCVZMWE1m7DYbzrg47EkJOJMTiUuKxx4Xh81hw+q0YXXYjcc2K2azGZPJhIhgMpm+n/Y+FxGCweA+k1Lq+8eBQACv1/v95PF4vn/sdrtxuVw0NTXtM/l8vn1ei81iJdmRSLo9hTRJINUfT6rLgb1BEPVD9Y8pzoKlTxzWvnFYsxKwZSUi6VbcjfU0VFVSX15KXUU59RVl1FWUU1u6h/qK8u/Xt9jt9Bk8hAGjjiBzpDElZfQh4A9SX+mmruKHqbbMRWVRI+567/frJ2U4yByRwoCRKQwYkUJyX2evu9ZIJxB0AtEOP0op6hpdFOV/S+FXq6nZsQtfnQ/ltWLCCSY7WOxgtqBMJpRJCJogYIKAKCP5SBCfBPDhb9VeEwsmBIfYcCobDmw4glacQStxyk6icpConCQoJ3YsCGIMORdvIuiEgC2A3+rDa/bQbHLjkgbc3gbcDfW46utw19fiqqvD42pqtd+45BSSMvqRkNaHpD5ZJKRl4kjMREyJuBv8NNV6aKrz4qr74f/wr1erw0xK3zjSB8aTkZVIxqAEMrISsMf1/muIOp1ARORG4D9KqZruCq6n6ASiaW1r9gWorK2norSA0t2bqN39Hc1lZfjqmsAdwOxzYAk6sRCHSZyIyY6YLGAyhyZBmUxgEhBBTCYEMIkJJQoRQWEsAzBjQjCF/RMEo5eaAGYlmJVgUabQY5BgEBX0E1QB/MpHIOgz/ld+fEEPnqAbb8CNJ+jGE3DjDbrxBT1tvmaT2YrJYsdsicdsjcdsicdkjkNM8SBOxJSIIpFgMAFfs9De16Uz0Upckp34ZBtxKXYSUu2k9HGS3DeO5D5OHAnWXndmEa0DuRtvf2CdiGwAngOWqi46bRGRU4EnMYa0/YdS6qEWy+3AC8BUoAo4XylVEFr2W+BKIADcpJRa2hUxadrhymE1k9Unlaw+qTC+7bG1lVI0evxU1LmoriqjsWoPDdUlNNTtwl1bQKCuBNy1iKcBk9eH2WcGvxXltxP0x6H8dlB2TAEb5oAZkzJjCpoxKRNgImgyEzAJQTERNJlQYsxTYkaJEAwlIQWhhARgBrEgpKAkA8QMFitmTFiMNIZJrIhYMYUmwRKqTgMTQSwEsQQVZhQWwGwCiwgWSwCLtR57GtjtJux2we4wG5PTjCPBSlyiFbPDBlarMdksYDWBNQhWD/iD4ApbfpAmkpaiqsISI23OAS4HpgGvAv9USu3o9I5FzEAuMBsoAtYBFyiltoSVuR6YoJS6VkTmA2crpc4XkbHAS8B0YACwHBillGr3/FqfgWhaz/IFglQ1eilvaKaqpoaGqhLcNWV460tRjeUEmsrx+cohWI3F3ICJJkyBZgj48PjNNPtM+H1CwGci6DMhXsHsFeKaIc4D8c2KOA8kNGPM87YfjwLELFgQI0kEFWYVxOL3Y/b5sXi9mD3NWNzNmP1+LAE/pu6o5jebQ4kmLKnsncxmsFi6/v/f/hb6RXcL+ZYOaDwQpZQSkVKgFPADqcASEVmmlLqtUxEZX/55SqmdoQBfBuYBW8LKzAPuDT1eAvy/UDKbB7yslPIA+SKSF9reF52MRdO0bmA1m+if7KB/sgOyUoDIF8x5/UHK6pspqnGzp9ZNSU0jNVVluKpL8NaXE2woJ11VM1AqyZRKMsxVxFtrCYqLWrOJWpOJQpOJOrONenMyDcqByyt4mgP4XG6c7iDxHohrVsQ3G4kn1Wcl1WMhqUkR1+jD7PGD0wns22NKnE4sKSlYMzKwpKVhTUvDmpqCJTkZa1Iy1sQETBYL4veDz2dMXu8Pj/dnnt9vTIFA2//7fNDc3HG5lv/fcEOnE0hbOkwgInITcClQCfwDuFUp5RMRE7Ad6GwCGQgUhj0vAma0VUYp5ReROiA9NH9Ni3UHthH/NcA1AIMHD+5kqJqmdSebxcSgtDgGpYVfWHfE948CQUVxjZudlY3kVzaxurKJ/Mom9pRXQW0RA6WSLKlktLWc6Y4KstlDmmcPJuUnaINah4mK5Ewq0odSkdSPcmcS35qFYl89RY3FlDaVYvEoklyQ5IJ0t5lhKp3B/mT6eZ2kN5mIr2nGnJuLv7zc+EIOY0pIwDZkSMTJnNDxXXgPVtGcgWQAP1VK7QqfqZQKisgZB7DvSJWALc8V2yoTzbrGTKWeBZ4FowprfwLUNK13MJuEwelxDE6PY1aL+ww2evxsK61ny556tpTU8/qeer4rbSDg95IlFUyOq2JmaiUTLEVMq8rDvuMLZG9ttzMVsqbjH3IyZX1HUZyQRpG7goK6AnbU7WBl7Q6KG/O+35fD7GBMyiSmWoYxJtCXYc2JpNQF8BcV4S3YhXvDBurfe4/w1nZr9mAcR4zBMWYMjjFHYD9iDNZ+fXvisHW7DhOIUuqedpZtPYB9FwHhI6RkAXvaKFMkIhaMc8vqKNfVNO0wkGC3MDU7janZad/P8weCbC9vJGdXDesLqnm0oIbiWjcAmXFwTnYDJyWXMiaQi7N0HZbtSxkIDDRZmT5oOgw/CcZcDf0n4Ao0k1+XT15tHt9Vf8fmqs28WPkBzQFjMLLUuFSmHT+N6efNZnr/6YyyZ+IrLMRbUIAnLw/Pd9to3rqVhqU/9POxZGYSN2UKcdOm4pwyFfvIEchBeOfmmF0HEkoIucDJQDFGI/qFSqnNYWVuAMaHNaL/VCl1noiMAxbzQyP6R8BI3YiuaVpbSuua+WJnJZ/lVvJZbgVVTV5E4KjsNM4e7eDU1EJSK3JgxydQutFYKS4DRs6BcWfDsFlgMe575Q/62Vm3k28rv2V92XrWla6jpKkEgHRHOscOPJaTBp/EMQOOwWkxbikTaGzEs20bzZu34P76K1w5643qMMCUnEz80UeTcPzxJMw8DkufjscB6Um98kJCETkdeAKjG+9zSqkHROQ+IEcp9baIOIB/A5MxzjzmhzW63wlcgdGo/yul1Acd7U8nEE3TAIJBxZaSej7aWs77m0rYVtZgJJMhaZwzZSBnDLcQX/gZ5C2H3A/BUweOZDjiJ3DkT41kEjZKoFKKooYi1pau5cuSL1m1ZxUN3gYcZgdHDzia2dmzOXnwycRZ4/ZZx1dcjCsnB9e6dTStXPV9QnGMG0fCySeRPHcutuzsHj46rfXKBNLTdALRNC2SvPIG3ttYytvfFLOjookEu4UzJw3gwumDObKfA3augM1vwHfvgacekgfDlAUw+WJIymy1PV/Qx/qy9Xyy+xM+LvyY0qZSnBYnc7LnMG/EPKb2m4pJ9q2yUkrh2baNxhWf0vjpp7i/+goAx/jxJM09neR587CkpvbE4WhFJxB0AtE0rX1KKTbsrmHxl4W8u3EPHn+QY4anc/2sERw7Ih0JeGHb+5DzL8j/1LhgceyZcNzNkDmxzW1+Vf4Vb+14i6UFS2nyNTE0eSiXjL2Enwz7CQ5L5HHLfSUl1L//AfXvvUfzli2IzUbiqT8mdf4FOCdP6tGr2nUCQScQTdOiV+fy8UrObv6xMp/yBg/jByZz40kjmDO2n/HlXbUD1i8yJk89DD8ZZv4ahhzb5jbdfjfLdi3jxS0vsrV6Kyn2FM4bfR6XjLmEFEdKm+t5tm+n5uVXqHvzTYJNTdjHjiHjmp+TOGd2jzS+6wSCTiCapu0/jz/AGxuKeeazneRXNjEtO5Xfnj6Gqdmh6qTmOlj3D/jib+CqNHpwzf4D9D+yzW0qpVhftp5/b/k3nxR+Qrw1ngVjF3DJ2EtIsCW0uV6wqYm6d96letEivAUF2IYPJ+Pan5M0d263JhKdQNAJRNO0zvMHgry2vojHluVS0eBh7oRMFp4xlr5JoSoorwtynoPPHjGSyqSL4OR7ILH9q7/zavL42zd/Y9muZSTbk/nFpF/ws1E/wxzWSN+SCgRoWLqUyqefwZObi33sGPrddjvxP2p5LXbX0AkEnUA0TTtwLq+fZz/byd9W7MBuMXHHaUdwwVGDMe0dYtZdA5/9H6x9FqxOmPOA0djeQZvFlqotPJrzKGtL1zImbQy/m/E7JvWd1O46Khik/v0PqHjsMXx79pBw0kn0v/N3WAdGvDFHp+kEgk4gmqZ1nfzKJu58YxOrd1QxfUgaj8+fxMCUsGGEK7fDO7+EXZ/DkJkw7/9B6pB2t6mUYumupTyy7hHKXeVcPOZifjnll202tO8V9HiofuEFKp96GoC+t9xC6oUXdFm1lk4g6ASiaVrXUkrx2voi7ntnCyaBh382gVOPDOvWGwzChudhWeiGHmf+2bgosQMun4snNzzJ4u8WMyx5GA/OfJBx6eM6XM9XXEzJwntpWrUK55QpDHj4YWxZB342ohMIOoFomtY9dlU18YuXvmJjUR2X/Cibu88Yi80S9uu/ZhcsuQKKc2Dq5XDqH43qrQ6s3rOauz+/m2p3NTdPvZlLxl7SYfddpRR1b71F2f0PgMnEgAcfIPGUUw7o9ekEgk4gmqZ1H68/yCNLv+PvK/OZMTSNpy6eSlq87YcCAR98/Af4/EkYMAUueAkS+3e43TpPHQtXL+Sj3R9x6pBT+f0xv9/nivY249m9m+Kbb6F582ZSF1xCv9/8BrHZOlwvkrYSyMF39y5N07ReyGYxcefcsTxx/iS+Kqxl3l9XkVfe+EMBsxVm3wfzF0PFNvj7SVDyTYfbTbYn8/isx/nVlF/x4a4PWfDBAspd5R3HM3gw2S8tJvXii6lZ/BLN27cfyMuLSJ+BaJqmdbGvdtdw9Qs5BBU8f/l0xmftO0gVJRvhpfngroULX4ahx0e13c+LP+eWFbeQZE/iqZOfYkTqiKjW8+7eje0AxkPSZyCapmk9ZPLgVF679hicVjMX/H0Na3ZW7VsgcwJc9RGkDIL/nAu5SyNvqIVjBx7LolMX4Q/6WfC/BWyq2BTVegeSPNqjE4imaVo3GJoRz5LrjqZ/soPL/7WOnILqfQskZcJl70OfI+DlC2Hru1Ftd0z6GF48/UWSbclcs+waNlZs7Iboo6MTiKZpWjfJTHby0tU/IjOURDYV1e1bID4dLn0bBkyGJZfDjo+j2u7AhIH869R/kepIjWkS0QlE0zStG/VJtPPiVTNIclq55Lkv921YB2OckYteg4xR8PJFsPvLqLbbP74/z/34OdIcaVz/0fXsrNvZDdG3TycQTdO0bjYgxcniq2dgMQlXPr+O6ibvvgWcqXDJG5CYCYvPM+70G4X+8f15ZvYzmMXMdcuui6p3VleKSQIRkTQRWSYi20P/txolRUQmicgXIrJZRDaKyPlhyxaJSL6IfB2a2r9hjKZpWoxlp8fzzCXTKKlr5tp/r8fjbzECd0JfuHgJiAkWn2/00IrCoMRB/O2Uv1HrqeX65dfj8rm6IfrIYnUGcgfwkVJqJMZ45ndEKOMCFiilxgGnAk+ISPgN829VSk0KTV93f8iapmkHZmp2Ko/8bAJrC6q5+81vWxdIGwbnvwg1BfDaZRDwR7XdcenjeHTWo2yv3c49q++hpy7PiFUCmQc8H3r8PHBWywJKqVyl1PbQ4z1AOdC7RprXNE3bT/MmDeTGE0fwak4Rr+UUti4w5Fg443HY+QmseDDq7R438DhumnwTSwuW8vzm5zteoQvEKoH0U0qVAIT+79teYRGZDtiA8IrBB0JVW4+LiL2dda8RkRwRyamoqOiK2DVN0w7IzbNH8aNhadz91rfkljW0LjDlEmPM9ZWPQd5HUW/3iiOvYHb2bB7f8DhflkTXGH8guu1KdBFZDkS60cudwPNKqZSwsjVKqYijxYtIJrACuFQptSZsXilGUnkW2KGUuq+jmPSV6Jqm9Rbl9c2c/udVpMRZeefG43DaWgwg5XXBP06GxnK47vOo7psF0ORrYv6783H5XLw+73WS7ckdr9SBHr8SXSl1ilLqyAjTW0BZKAnsTQYRuw6ISBLwHnDX3uQR2naJMniAfwHTu+t1aJqmdYe+SQ6enD+JvPJGHl76XesCtjj42b/A54I3roUof+zHW+P50/F/otpTze+/+H23tofEqgrrbeDS0ONLgbdaFhARG/AG8IJS6rUWy/YmH8FoP4nQGqVpmta7HTsig0uPzmbR6gLW5le3LtD3CJjzB6M9ZMMLUW93bPpYbpx0I8t2LePtHW93YcT7ilUCeQiYLSLbgdmh54jINBH5R6jMecDxwGURuuv+R0Q2AZuADOD+ng1f0zSta9x+2hEMSo3j1iXf4PJG6HU19QpjRMMP74K64qi3e9m4y5jWbxp/XPtHSptKuzDiH+i78WqapsXYmp1VzH92DVceN5S7zxjbukB1Pjx1DGQfAxct6XB89b2KGor46ds/ZXr/6fzlpL90OBhVW/TdeDVN03qpHw1L54Lpg1m0uoBtpRF6ZaUNhVPuhbzlsOXNqLeblZjFDZNu4NOiT9lY2fX3y9IJRNM0rRe47cejSXRYWPj2t5Ebvo+6CvqNh6V3GT20onTxmIt54bQXmNhnYhdGa9AJRNM0rRdIjbfxmzmjWbOzmnc2lrQuYDLD6Q9DfRF8/kTU2zWbzEzuO7kLIw0LqVu2qmmapu23C6YP5siBSTzw3hbc3kDrAtnHwJE/g1VPGLc7iTGdQDRN03oJs0lY+JNxlNV7WLS6IHKh2fcZZyPLf9+jsUWiE4imaVovctSQNE4c3YenVuRR5/K1LpA8EH50PWx+HUqjG9K2u+gEomma1sv85sejqW/288xnbYwLcswvjIGoPo7tJXA6gWiapvUy4wYkc+bEAfzr8wLKG5pbF3CmwLG/hNz/QeG6ng8wRCcQTdO0XuiW2aPwBoI8vaKNoWpnXAvxfeCj2LWF6ASiaZrWCw3JiGfexAG8vG43NS2HwAWwxcPMX0PByqjHUe9qOoFomqb1Uj8/YTgub4AXvtgVucCUBcZ46p8/2bOBhegEomma1kuN7p/IKWP6smh1fuQbLdriYfo1sO09qMjt8fh0AtE0TevFrps1nBqXj1fWRRj+FowEYnHC6p4/C9EJRNM0rRebmv3/27v7GKvqO4/j788MICMPDvMADs8PqyKiAl5p0cats7BLbbdQY3fVdEtaG5LdbtKN2V1t9q9NulmaTdb+02TDanfZtmltq0ZjaxUtblsV6YBQnlRQUKY8DSArCCIM3/3jHpIpvZdhLnPOmXvn80pu7rnn/s453xMjnzm/3zn318T8qU08/KtdnOk++4cNRrTA3C/Apkfh/b2Z1uYAMTMb4L78iWn87uhJfvF6yclbYcFXIbph3X9mWpcDxMxsgFt47VjarhjOd9eWGUxvmgZXL4bXvgtnStyxlRIHiJnZADekvo5750/mVzsO8XbX8dKNCvfBB12wPb0pbM+XS4BIapK0WtKO5H1MmXbdPaazfarH+mmSXk22fzSZP93MrGb95fxJDK0X31v7bukGM9phzFTo+E5mNeV1BfIg8EJEXAW8kHwu5WREzElen+2x/pvAQ8n27wH3pVuumVm+xo4azuLZbfx4/Z7St/TW1cFNX4J3XoKD2zOpKa8AWQKsSpZXAUsvdkMVJ/VtB35SyfZmZtXqiwumbjfNOAAACn1JREFUcOzDMzy9qcSEU1C8G6t+WGZXIXkFyLiI2AeQvI8t0264pA5JayWdC4lm4GhEnIvgTmBCuQNJWp7so6Orq6u/6jczy1xhyhimtYzgsQ2dpRuMaIFZS4u39J4u8SOM/Sy1AJH0vKQtJV5L+rCbyRFRAO4FviVpBqAS7UpMIJx8EbEyIgoRUWhtbe3jWZiZDRySuHPuBF7ddYQ9R8rMiz7nXjj1f/DmM6nXk1qARMTCiJhd4vUkcEBSG0DyXvLm5ojYm7y/DbwIzAUOAY2ShiTNJgLZPj1jZpaTz80rdrg88drvSjeYdhuMaitehaQsry6sp4BlyfIy4MnzG0gaI+myZLkFuBXYFhEBrAHuutD2Zma1aOKYy1kwvZnHN3RS/OfwPHX1cP3nYedq+OBQqrXkFSArgEWSdgCLks9IKkh6OGlzLdAhaRPFwFgREduS7x4A7pe0k+KYyCOZVm9mlqM7501g9+ETrH/nvdINbrwHzp6BLY+lWodKJliNKhQK0dHRkXcZZmaX5PipM9z8jedZOnc8/3rnDaUb/ccnoG4oLF9zyceTtD4Zj/49fhLdzKzKjLxsCH923Tie2bKf06V+YBHghrth74ZUf+bdAWJmVoXuuL6NoydOs/btw6UbXH8XINj6eGo1OEDMzKrQbVe3MmJYPT/bvL90g1FXwuQFsC29e4wcIGZmVWj40Hrarx3Hc1v3l54nBOC6pXBwW2rdWA4QM7MqdcfsKzn8wUes232kdINr/7z4vj2dqxAHiJlZlfrkNWNpGFrPM+W6sUaPh0kfS60bywFiZlalGobVc/vMVn6+dT/dZ8s8kjFrCezfDIff6vfjO0DMzKrYp2a30XXsFBveLfNQ4aylsHgFNJScdumSOEDMzKrYH1/TypA6lZ8v/YoJ8PG/hsub+v3YDhAzsyo2evhQClPHsKZcgKTIAWJmVuVuv2Ysr+8/xt6jJzM9rgPEzKzKtc8szsn34hvZTprnADEzq3J/NHYkExobyo+DpMQBYmZW5STRPnMsL+08xKkz3Zkd1wFiZlYDbp/ZysnT3azbVeap9BQ4QMzMasCC6S1cNqQu026sXAJEUpOk1ZJ2JO9/8ISLpNslbezx+lDS0uS7/5a0q8d3c7I/CzOzgaNhWD0fm97MSzvTnca2p7yuQB4EXoiIq4AXks+/JyLWRMSciJgDtAMngOd6NPmHc99HxMZMqjYzG8BumdHMmweO03XsVCbHyytAlgCrkuVVwNJe2t8FPBMRJ1Ktysysit0yoxmAV8pNMtXP8gqQcRGxDyB5H9tL+7uBH5y37l8k/VbSQ5IuK7ehpOWSOiR1dHVle4+0mVmWrht/BaOGD+HljLqxUgsQSc9L2lLitaSP+2kDrgee7bH668BM4GagCXig3PYRsTIiChFRaG1treBMzMyqQ32d+Pj0Zl5+K5srkCFp7TgiFpb7TtIBSW0RsS8JiAvdNvAXwBMRcbrHvvcli6ck/Rfw9/1StJlZlbtlRjOrtx1gz5ETTGq6PNVj5dWF9RSwLFleBlxotpN7OK/7KgkdJIni+MmWFGo0M6s6t8xoAbIZB8krQFYAiyTtABYln5FUkPTwuUaSpgKTgP89b/vvS9oMbAZagG9kULOZ2YB39biRtIwcxisZdGOl1oV1IRFxGPiTEus7gK/0+LwbmFCiXXua9ZmZVStJLJjRwstvHSIiKHbUpMNPopuZ1ZgF05s58P4pdh36INXjOEDMzGrMzVOLP+6x/p0y09z2EweImVmNmdE6kisahjpAzMysb+rqxLzJjXQ4QMzMrK8KU5vYefA4R098lNoxHCBmZjXopinFcZAN76Z3FeIAMTOrQTdObGRInVIdB3GAmJnVoIZh9Vw3fjQdux0gZmbWRzdNaWJT51FOd59NZf8OEDOzGlWYOoYPT59l2973U9m/A8TMrEadG0hP63ZeB4iZWY0aN3o4ExobeC2lO7EcIGZmNWzOpEY27jmayr4dIGZmNWzOpEY63zvJoeOn+n3fDhAzsxo2f1oTn76hjROnuvt937nMB2JmZtm4cVIj3753Xir7zuUKRNLnJW2VdFZS4QLtFkt6Q9JOSQ/2WD9N0quSdkh6VNKwbCo3M7Nz8urC2gLcCfyyXANJ9cC3gU8Bs4B7JM1Kvv4m8FBEXAW8B9yXbrlmZna+XAIkIrZHxBu9NJsP7IyItyPiI+CHwBIV52dsB36StFsFLE2vWjMzK2UgD6JPAPb0+NyZrGsGjkbEmfPWm5lZhlIbRJf0PHBlia/+KSKevJhdlFgXF1hfro7lwHKAyZMnX8RhzczsYqQWIBGx8BJ30QlM6vF5IrAXOAQ0ShqSXIWcW1+ujpXASoBCoVA2aMzMrG8GchfWb4CrkjuuhgF3A09FRABrgLuSdsuAi7miMTOzfpTXbbyfk9QJLAB+KunZZP14ST8DSK4u/hZ4FtgO/Cgitia7eAC4X9JOimMij2R9DmZmg52Kf9APDpK6gHcq3LyFYvfZYOJzHhx8zoPDpZzzlIhoPX/loAqQSyGpIyLKPvRYi3zOg4PPeXBI45wH8hiImZkNYA4QMzOriAPk4q3Mu4Ac+JwHB5/z4NDv5+wxEDMzq4ivQMzMrCIOEDMzq4gD5CKUm5ekVkn6jqSDkrbkXUtWJE2StEbS9mSumq/lXVPaJA2XtE7SpuSc/znvmrIgqV7Sa5KezruWrEjaLWmzpI2SOvptvx4DubBkXpI3gUUUf5/rN8A9EbEt18JSJOk24DjwPxExO+96siCpDWiLiA2SRgHrgaU1/t9ZwIiIOC5pKPBr4GsRsTbn0lIl6X6gAIyOiM/kXU8WJO0GChHRrw9P+gqkdyXnJcm5plRFxC+BI3nXkaWI2BcRG5LlYxR/PqempwmIouPJx6HJq6b/opQ0Efg08HDetdQCB0jvys1LYjVK0lRgLvBqvpWkL+nO2QgcBFZHRK2f87eAfwTO5l1IxgJ4TtL6ZIqLfuEA6V2f5h+x6iZpJPAY8HcR8X7e9aQtIrojYg7FaRHmS6rZLktJnwEORsT6vGvJwa0RMY/iFOFfTbqpL5kDpHfl5iWxGpOMAzwGfD8iHs+7nixFxFHgRWBxzqWk6Vbgs8l4wA+Bdknfy7ekbETE3uT9IPAExa75S+YA6V3JeUlyrsn6WTKg/AiwPSL+Pe96siCpVVJjstwALARez7eq9ETE1yNiYkRMpfj/8S8i4gs5l5U6SSOSG0OQNAL4U6Bf7rB0gPSil3lJapKkHwCvANdI6pR0X941ZeBW4K8o/lW6MXndkXdRKWsD1kj6LcU/lFZHxKC5tXUQGQf8WtImYB3w04j4eX/s2LfxmplZRXwFYmZmFXGAmJlZRRwgZmZWEQeImZlVxAFiZmYVcYCY5UxSo6S/ybsOs75ygJjlrxFwgFjVcYCY5W8FMCN5ePHf8i7G7GL5QUKznCW//vv0YJl7xWqHr0DMzKwiDhAzM6uIA8Qsf8eAUXkXYdZXDhCznEXEYeAlSVs8iG7VxIPoZmZWEV+BmJlZRRwgZmZWEQeImZlVxAFiZmYVcYCYmVlFHCBmZlYRB4iZmVXk/wFDFnW9a6QmcQAAAABJRU5ErkJggg==\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# tracé de solutions voisines de la solution qui s'en écartent progressivement.\n", "plt.plot(t_eval, np.exp(-t_eval),'r')\n", "for C in [-3e-7, -2e-7, -1e-7,-0.5e-7,0.5e-7,1e-7, 2e-7, 3e-7 ]:\n", " plt.plot(t_eval, C*np.exp(3*t_eval)+ np.exp(-t_eval))\n", "plt.xlabel('t')\n", "plt.ylabel('y')\n", "plt.title('sol C exp(3t) + exp(-t) pour C entre -3e-7 et 3e-7')\n", "plt.show()" ] } ], "metadata": { "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": "3.7.4" } }, "nbformat": 4, "nbformat_minor": 2 }