add exercise 2: implement heat equation in notebook and Python script

TP Physique Equation de la chaleur/Exercice 2.py

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+# Eq en régime permanent
+# d2T / dx2 = 0
+# T(x =0) = T0
+# T(x = 1) = T1
+# T(i)i E [0, N+1]
+# x(i+1) - x(i) = deltaX = 1/N
+# d2T/dx2 = (T(i-1) - 2*T(i) + T(i+1))/deltaX**2
+
+# Question 1: Résoudre cette équation pour les conditions aux bord
+# d2T / dx2 => dT / dx = A => T(x) = Ax + b
+# Ici b = T0 d'où T(x) = Ax + T0
+# T1 - T0 / 1 - 0 = A = T1 - T0 donc T(x) = (T1 - T0)x + T0
+# Question 2: Ecrire l'équation de la chaleur discrétisée pour i E [1, N-2], puis pour i = 0 et i = N
+# d2T/dx2 = 0
+# T(i+1) - 2*T(i) + T(i-1) = 0
+# Pour i = 0
+# -2T0 + T(1) = -T0-
+# Pour i = N

TP Physique Equation de la chaleur/TP Physique Equation de la chaleur.ipynb

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+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "d72483d906868e17",
+   "metadata": {},
+   "source": [
+    "# TP Physique - Équation de la chaleur\n",
+    "## Exercice 2\n",
+    "\n",
+    "Eq en régime permanent\n",
+    "\n",
+    "d2T / dx2 = 0\n",
+    "\n",
+    "T(x =0) = T0\n",
+    "\n",
+    "T(x = 1) = T1\n",
+    "\n",
+    "T(i)i E [0, N+1]\n",
+    "\n",
+    "x(i+1) - x(i) = deltaX = 1/N\n",
+    "\n",
+    "d2T/dx2 = (T(i-1) - 2*T(i) + T(i+1))/deltaX**2\n",
+    "\n",
+    "### Question 1: Résoudre cette équation pour les conditions aux bord\n",
+    "d2T / dx2 => dT / dx = A => T(x) = Ax + b\n",
+    "\n",
+    "Ici b = T0 d'où T(x) = Ax + T0\n",
+    "\n",
+    "T1 - T0 / 1 - 0 = A = T1 - T0 donc T(x) = (T1 - T0)x + T0\n",
+    "### Question 2: Ecrire l'équation de la chaleur discrétisée pour i E [1, N-2], puis pour i = 0 et i = N\n",
+    "d2T/dx2 = 0\n",
+    "\n",
+    "T(i+1) - 2*T(i) + T(i-1) = 0\n",
+    "\n",
+    "Pour i = 0: T(i-1) = ? => CB = T0-\n",
+    "\n",
+    "-2T0 + T(1) = -T0-\n",
+    "\n",
+    "Pour i = N: T(N+2) = T1+\n",
+    "\n",
+    "T(N) - 2T(N+1) = -T1+\n",
+    "\n",
+    "Système linéaire\n",
+    "$$\n",
+    "-2T0 + T(1) = -T0\n",
+    "\n",
+    "T0 - 2T1 + T2 = 0\n",
+    "\n",
+    "T1 - 2T2 + T3 = 0\n",
+    "\n",
+    ".\n",
+    ".\n",
+    ".\n",
+    "\n",
+    "TN - 2T(N+1) = -T1+\n",
+    "$$\n",
+    "=> MT = b\n",
+    "\n",
+    "Matrice du système (simplifiée pour N=6, normalement N indéfini)\n",
+    "$$\n",
+    "\\begin{bmatrix}\n",
+    "-2 & 1 & 0 & 0 & 0 & 0 & 0\\\\\n",
+    "1 & -2 & 1 & 0 & 0 & 0 & 0\\\\\n",
+    "0 & 1 & -2 & 1 & 0 & 0 & 0\\\\\n",
+    "0 & 0 & 1 & -2 & 1 & 0 & 0\\\\\n",
+    "0 & 0 & 0 & 1 & -2 & 1 & 0\\\\\n",
+    "0 & 0 & 0 & 0 & 1 & -2 & 1\\\\\n",
+    "0 & 0 & 0 & 0 & 0 & 1 & -2\\\\\n",
+    "\\end{bmatrix}\n",
+    "\\begin{bmatrix}\n",
+    "T0\\\\\n",
+    "T1\\\\\n",
+    "T2\\\\\n",
+    "T3\\\\\n",
+    "T4\\\\\n",
+    "T5\\\\\n",
+    "T\n",
+    "\\end{bmatrix}\n",
+    "=\n",
+    "\\begin{bmatrix}\n",
+    "-T0\\\\\n",
+    "0\\\\\n",
+    "0\\\\\n",
+    "0\\\\\n",
+    "0\\\\\n",
+    "0\\\\\n",
+    "-T1+\n",
+    "\\end{bmatrix}\n",
+    "$$"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "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": "3.13.2"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}