diff --git a/5 Derive Numerique.ipynb b/5 Derive Numerique.ipynb
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+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# 2 Dérivée numérique"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Soit $f:[a,b]\\rightarrow \\mathbf R$, de classe $C^1$ et $x_0\\in[a,b]$.\n",
+ "La dérivée $f'(x_o)$ est donnée par\n",
+ "\n",
+ "$$\\begin{array}{lcl}\n",
+ "f'(x_0)&=&\\displaystyle\\lim_{h\\rightarrow 0^+}\\frac{f(x_0+h)-f(x_0)}{h},\\\\[2mm]\n",
+ "&=&\\displaystyle\\lim_{h\\rightarrow 0^+}\\frac{f(x_0)-f(x_0-h)}{h},\\\\[2mm]\n",
+ "&=&\\displaystyle\\lim_{h\\rightarrow 0}\\frac{f(x_0+h)-f(x_0-h)}{2h}.\n",
+ "\\end{array}$$\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "\n",
+ "Soient $x_0\\in[a,b]$, $(Dy)$ une approximation de $f'(x_0)$\n",
+ "et $(D^2y)$ une approximation de $f\"(x_0)$.\n",
+ "\n",
+ "On appelle\n",
+ "\n",
+ "* **différence finie progressive** l'approximation\n",
+ " $$(Dy)^{P} = \\frac{f(x_0+h) - f(x_0)}{h}$$\n",
+ "\n",
+ "* **différence finie rétrograde** l'approximation\n",
+ " $$(Dy)^{R} = \\frac{f(x_0) - f(x_0-h)}{h}$$\n",
+ "\n",
+ "* **différence finie centrée** l'approximation\n",
+ " $$(Dy)^{C} = \\frac{f(x_0+\\frac12h) - f(x_0-\\frac12h)}{h}$$\n",
+ "\n",
+ "* **différence finie centrée d'ordre 2** l'approximation\n",
+ " $$(D^2y)^{C} = \\frac{f(x_0+h) - 2f(x_0) + f(x_0-h)}{h^2}$$\n"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "# importing libraries used in this book\n",
+ "import numpy as np\n",
+ "import matplotlib.pyplot as plt"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Les différences finies progressive, rétrograde et centrée approchent la dérivée de la fonction en le point $x_0$."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "# Define a function and a point x0\n",
+ "f = lambda x : 0.25*(x-4)**3 - 4*(x-4)**2 +6\n",
+ "x0 = 5 \n",
+ "\n",
+ "# Choosing finite difference size\n",
+ "h = 4\n",
+ "\n",
+ "\n",
+ "# Defininig first order finite differences\n",
+ "DP = ( f(x0 + h) - f(x0) ) / h\n",
+ "DR = ( f(x0) - f(x0-h) ) / h\n",
+ "DC = ( f(x0 + h/2) - f(x0-h/2) ) / h\n",
+ "\n",
+ "# defining the straight lines with slope equal to the FDs\n",
+ "dfp = lambda x : f(x0) + DP*(x-x0)\n",
+ "dfr = lambda x : f(x0) + DR*(x-x0)\n",
+ "dfc = lambda x : f(x0-h/2) + DC*(x-x0+h/2)\n",
+ "\n",
+ "# Drowing points\n",
+ "z = np.linspace(-5, 22, 100)\n",
+ "\n",
+ "plt.plot(z, dfp(z), 'g:', z, dfr(z), 'b:', z, dfc(z), 'r:' )\n",
+ "plt.plot(z, f(z), 'k', x0,f(x0),'ro')\n",
+ "\n",
+ "plt.xlabel('x'); plt.ylabel('$f(x)$');\n",
+ "plt.legend(['$D^P$', '$D^R$', '$D^C$', '$f$'])\n",
+ "plt.show()\n",
+ "\n",
+ "\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "La différence finie d'ordre 2 approche la dérivée deuxième de la fonction en le point $x_0$. \n",
+ "\n",
+ "Pour le voir graphiquement, on peut dessiner une fonction quadratique qui a la forme\n",
+ "\n",
+ "$$ p_2(x) = f(x_0) + (Dy)^{C}\\, (x-x_0) + \\frac12 (D^2y)\\, (x-x_0)^2$$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "# Define a function and a point x0\n",
+ "f = lambda x : 0.25*(x-4)**3 - 4*(x-4)**2 +6\n",
+ "x0 = 5 \n",
+ "\n",
+ "# Choosing finite difference size\n",
+ "h = 4\n",
+ "\n",
+ "\n",
+ "# Defininig first order centered finite differences\n",
+ "DC = ( f(x0 + h/2) - f(x0-h/2) ) / h\n",
+ "\n",
+ "# Defininig second order centered finite differences\n",
+ "D2 = ( f(x0 + h) - 2* f(x0) + f(x0-h) ) / (h**2)\n",
+ "\n",
+ "\n",
+ "# defining the parabola going through (x_0, y_0)\n",
+ "parabola = lambda x : f(x0) + DC*(x-x0) + 0.5 * D2 * (x-x0)**2\n",
+ "\n",
+ "# Drowing points\n",
+ "z = np.linspace(-5, 22, 100)\n",
+ "\n",
+ "plt.plot(z, parabola(z), 'g:' )\n",
+ "plt.plot(z, f(z), 'k', x0,f(x0),'ro')\n",
+ "\n",
+ "plt.xlabel('x'); plt.ylabel('$f(x)$');\n",
+ "plt.legend(['parabola', '$f$'])\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "### Exercice\n",
+ "\n",
+ "Il s'agit de vérifier numériquement que\n",
+ "\n",
+ "$$\\left\\vert f'(x_0)- (Dy)^{P} \\right\\vert=O(h^1).$$\n",
+ "\n",
+ "On pose $x_0=1$, $f(x)=\\sin(x)$ $\\forall x\\in\\mathbb R$\n",
+ "\n",
+ "Calculez l'erreur commise en utilisant la différence finie progressive.\n",
+ "\n",
+ "Quelle est la valeur de l'erreur pour h=0.1?\n",
+ "\n",
+ "Ensuite verifiez numériquement l'ordre pour $(Dy)^{R}$ et $(Dy)^{C}$."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "def DPerror(f,df, x0,h) :\n",
+ " # formule de differences finies \n",
+ " DP = ( f(x0 + h) - f(x0) ) / h\n",
+ " err = abs(DP-df(x0));\n",
+ " return err\n",
+ "\n",
+ "# code :\n",
+ "# def DRerror(f,df, x0,h) :\n",
+ " # formule de differences finies \n",
+ "\n",
+ "# def DCerror(f,df, x0,h) :\n",
+ " # formule de differences finies \n",
+ "\n",
+ "\n",
+ "# This function just provides a line to compare the convergence in a log-log plot.\n",
+ "from FiniteDifferenceLib import sampleConvergence\n",
+ "# sampleConvergence(h,order,ref) :\n",
+ "# computes a pseudo order of convergence on h\n",
+ "# ref is a reference maximum value \n",
+ "\n",
+ "\n",
+ "x0 = 0.2\n",
+ "\n",
+ "h = 2**np.linspace(-10,-1,10)\n",
+ "plt.plot(h, DPerror(np.sin, np.cos, x0, h) , 'o' )\n",
+ "#plt.plot(h, DRerror(np.sin, np.cos, x0, h) , '*' )\n",
+ "#plt.plot(h, DCerror(np.sin, np.cos, x0, h) , '*' )\n",
+ "\n",
+ "\n",
+ "plt.plot(h, sampleConvergence(h, 1, 1e-1) , ':',\n",
+ " h, sampleConvergence(h, 2, 1e-1) , ':')\n",
+ "\n",
+ "plt.xlabel('h'); plt.ylabel('BDF2 error');\n",
+ "#plt.legend(['DP error','DR error', 'DC error', '$O(h^1)$', '$O(h^2)$'])\n",
+ "plt.xscale('log') \n",
+ "plt.yscale('log')\n",
+ "plt.show()\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "### Exercice\n",
+ "\n",
+ "Il s'agit de vérifier numériquement que\n",
+ "\n",
+ "$$\\left\\vert f\"(x_0)- (D^2y) \\right\\vert=O(h^2).$$\n",
+ "\n",
+ "On pose $x_0=1$, $f(x)=\\sin(x)$ $\\forall x\\in\\mathbb R$\n",
+ "\n",
+ "Calculez l'erreur commise par ce schéma.\n",
+ "\n",
+ "Que vaut l'erreur pour h=0.1?\n"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "# Code\n",
+ "# def D2error(f,df, x0,h) :\n",
+ " # formule de differences finies BDF2\n",
+ "\n",
+ "x0 = 0.2\n",
+ "\n",
+ "df = lambda x : -np.sin(x)\n",
+ "\n",
+ "h = 2**np.linspace(-10,-1,10)\n",
+ "#plt.plot(h, D2error(np.sin, df, x0, h) , 'o' )\n",
+ "plt.plot(h, sampleConvergence(h, 1, 1e-2) , ':',\n",
+ " h, sampleConvergence(h, 2, 1e-2) , ':')\n",
+ "\n",
+ "plt.xlabel('h'); plt.ylabel('D2 error');\n",
+ "#plt.legend(['D2 error', '$O(h^1)$', '$O(h^2)$'])\n",
+ "plt.xscale('log') \n",
+ "plt.yscale('log')\n",
+ "plt.show()\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "### Exercice\n",
+ "\n",
+ "Il s'agit de vérifier numériquement que\n",
+ "\n",
+ "$$\\left\\vert f'(x_0)-\\dfrac{3f(x_0)-4f(x_0-h)+f(x_0-2h)}{2h}\\right\\vert=O(h^2).$$\n",
+ "\n",
+ "Cette formule de différences finies est à l'origine du schéma \"BDF2'' pour résoudre numériquement des équations différentielles (Chapitre 9 du livre).\n",
+ "\n",
+ "On pose $x_0=1$, $f(x)=\\sin(x)$ $\\forall x\\in\\mathbb R$\n",
+ "\n",
+ "Calculez l'erreur commise par ce schéma.\n",
+ "\n",
+ "Que vaut l'erreur pour h=0.1?
"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "\n"
+ ]
+ }
+ ],
+ "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": "3.7.7"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}
diff --git a/6.1 Equations Differentielles Ordinaires.ipynb b/6.1 Equations Differentielles Ordinaires.ipynb
new file mode 100644
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--- /dev/null
+++ b/6.1 Equations Differentielles Ordinaires.ipynb
@@ -0,0 +1,367 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Equations Différentielles Ordinaires\n",
+ "\n",
+ "## Problème de Cauchy\n",
+ "\n",
+ "$\\newcommand{\\rr}{\\mathbb R}\n",
+ "f:\\rr_+\\times\\rr \\rightarrow\\rr$ continue, $y_0\\in\\rr$ donné.\n",
+ "On cherche $y:t\\in I \\subset \\rr_+\\rightarrow y(t)\\in\\rr$\n",
+ "qui satisfait le problème suivant\n",
+ "$$\\left\\{\n",
+ "\\begin{array}{l}\n",
+ "y'(t)=f(t,y(t))\\qquad \\,\\forall t \\in I \\\\\n",
+ "y(t_0)=y_0\n",
+ "\\end{array}\n",
+ "\\right.$$\n",
+ "où $y'(t)=\\displaystyle\\frac{d y(t)}{d t}$.\n",
+ "\n",
+ "#### Exemple\n",
+ "\n",
+ "Approximer la solution du problème de Cauchy\n",
+ "$$\\left\\{\n",
+ "\\begin{array}{l}\n",
+ "y'(t)=-t \\,y(t)^2\\qquad \\,\\forall t \\in [0,4] \\\\\n",
+ "y(t_0)=2\n",
+ "\\end{array}\n",
+ "\\right.$$\n",
+ "La solution de ce problème est $y(t) = \\frac{2}{1+t^2}$\n",
+ "Avec les méthodes de Euler Proressive et Rétrograde, Heun, Crank-Nicolson, et Euler modifié.\n"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "# importing libraries used in this book\n",
+ "import numpy as np\n",
+ "import matplotlib.pyplot as plt\n",
+ "\n",
+ "from OrdinaryDifferentialEquationsLib import forwardEuler,backwardEuler,Heun,CrankNicolson,modifiedEuler\n"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 2,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "image/png": 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\n",
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+ "