{ "cells": [ { "cell_type": "markdown", "id": "81b1d2f0", "metadata": {}, "source": [ "---\n", "title: \"Euler's Method\"\n", "subtitle: \"Lecture 3\"\n", "date: 2026-01-28\n", "abstract-title: Overview\n", "abstract: | \n", " (1) Basics of Euler's method, \n", " (2) Examples,\n", " (3) Error estimates\n", "format:\n", " html:\n", " other-links:\n", " - text: This notebook\n", " href: L3.ipynb\n", "---" ] }, { "cell_type": "markdown", "id": "63cf0867", "metadata": {}, "source": [ "::: {.callout-note}\n", "\n", "I encourage you to play around with the juptyer notebook for this lecture - you can copy the code with the ```this notebook``` button on the side of this page.\n", "\n", ":::\n", "\n", "::: {.callout-warning}\n", "\n", "These notes are mainly a record of what we discussed and are not a substitute for attending the lectures and reading books! If anything is unclear/wrong, let me know and I will update the notes.\n", "\n", ":::\n", "\n", "::: {.callout-tip}\n", "\n", "This lecture is mostly based on @Burden section 5.2\n", "\n", ":::" ] }, { "cell_type": "code", "execution_count": 8, "id": "db6fe346", "metadata": {}, "outputs": [], "source": [ "#| echo: false\n", "\n", "using Plots, LaTeXStrings, OrdinaryDiffEq" ] }, { "cell_type": "markdown", "id": "5ea01014", "metadata": {}, "source": [ "Recall that we are considering IVPs: seek $u :[0,T] \\to \\mathbb R$ such that\n", "\n", "\\begin{align}\n", " &u(0) = u_0 \\nonumber\\\\\n", " &u'(t) = f\\big( t, u(t) \\big). \\tag{IVP}\n", "\\end{align}\n", "\n", "We will suppose that this equation is well-posed. The numerical schemes we introduce in this section will approximate the solution $u$ to $(\\text{IVP})$ on a *mesh* $\\{t_j\\}_{j=0}^n \\subset [0,T]$. The approximation at time $t_j$ will be written $u_j$ so that $u_j \\approx u(t_j)$. We will consider the error on the mesh: $\\max_j |u(t_j) - u_j |$.\n", "\n", "## Euler's method\n", "\n", "We now fix an equispaced mesh $t_j = j\\frac{T}{n}$. We will sometimes call $h := \\frac{T}n$ the *step size*. \n", "\n", "Replacing the derivative in $(\\text{IVP})$ with the *forward difference* \n", "\n", "\\begin{align}\n", " u'(t_j) \\approx \\frac{u_{j+1} - u_j}{h} \n", "\\end{align}\n", "\n", "yields Euler's method\n", "\n", "\\begin{align}\n", " u_{j+1} = u_j + h f(t, u_j)\n", " \\tag{Euler}\n", "\\end{align}\n", "\n", "::: {.callout-note}\n", "\n", "Last time, we saw that by integrating $(\\text{IVP})$ we obtain\n", "\n", "\\begin{align}\n", " u(t_{j+1}) = u(t_{j}) + \\int_{t_j}^{t_{j+1}} f\\big( s, u(s) \\big) \\mathrm{d}s.\n", "\\end{align}\n", "\n", "Approximating this integral with the *rectangular rule* and replacing $u(t_j)$ with $u_j$ also yields Euler's method.\n", "\n", "::: \n", "\n", "That is, we approximate the differential equation with a *difference equation*.\n", "\n", "We implement Euler's method here:" ] }, { "cell_type": "code", "execution_count": 9, "id": "1f847c0c", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Euler (generic function with 1 method)" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "function Euler( u0, f, T, n )\n", " h = T/n \n", " t = 0:h:T \n", " u = zeros(n+1)\n", " u[1] = u0\n", " for j = 1:n\n", " u[j+1] = u[j] + h * f( t[j], u[j] )\n", " end\n", " return u\n", "end " ] }, { "cell_type": "markdown", "id": "0d920692", "metadata": {}, "source": [ "## Examples\n", "\n", "We now apply Euler's method to each of the examples from Lecture 2:\n", "\n", "::: {#exm-1}\n", "# Exponential growth\n", "\n", "We consider $u' = \\mu u$ with $u(0) = u_0 > 0$. Recall that this problem is well-posed with unique solution $u(t) = u_0 e^{\\mu t}$. We approximate this solution using Euler's method:" ] }, { "cell_type": "code", "execution_count": 10, "id": "84b17ae2", "metadata": {}, "outputs": [ { "data": { "image/png": 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"execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "μ = 2\n", "f(t, u) = μ * u\n", "u0, T, n = 1, 2, 4\n", "\n", "u = Euler( u0, f, T, n )\n", "scatter( 0:T/n:T, u, \n", " label=\"Euler (n=$n)\",\n", " title=\"Exponential (u' = u)\")\n", "\n", "n=100\n", "u2 = Euler( u0, f, T, n )\n", "scatter!( 0:T/n:T, u2, \n", " label=\"Euler (n=$n)\" )\n", "\n", "plot!(t-> u0 * exp(μ * t),\n", " label=\"exact solution\")" ] }, { "cell_type": "markdown", "id": "03a17bc0", "metadata": {}, "source": [ "In the following, we plot the errors as a function of $n$:" ] }, { "cell_type": "code", "execution_count": 11, "id": "5536e267", "metadata": {}, "outputs": [ { "data": { "image/png": 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WU6/73bBbmpIuuZ80adKpU6emTJkyfPjwgICAGzduAMCjR4+0+WzevHl2dnZHjhwZPXp0586dJ06cmJiY2LRpU09Pz/nz5z9+/Lh///7adrZv3z516lQ/Pz8/P7+SRVIyFXNqtDCCMG7Q1qhuM1Ho8ZRlAZxm3Uxa9yHoDLLDQgihMpKenk7nWimI/xj88GwUMqlIJDI1Nf3VlnNycjZu3JiUlGRubg4A06dP37Fjh3aWtWHDhlOnTs2vGRQU1KNHDwCYN29e+/btAwICAGDnzp02NjYJCQmOjo4AsHDhQh8fn5L08DcYRiIEAO2Nww4DjRu1F17Ykxw8jNv5LyPP1kAQZMeFEEKlzszMTCnKBI2m6B89UTqNzuBwSrI/V2xsrFqtbteunfZQoVBUq1ZN+9nNza1gzfzDuLi4GjVqaD+bm5vb2Nh8+PBBmwgLnVI2DCgRalF5VuYDJue9f5F9ekveh9dmfQLJjgghhEpdnTp1CJUCYu+Di/f33xIvLnk0bEyhlORmmbW1NUEQd+7cYbFYhb6i0b5JMVQqVfvB0tIyNTVV+1mpVGZmZlpZWRWqU5YM4h7h95iu7vyJG7id/cgOBCGEygKLxRodMIp1bCLIJYW/S42hX1k+c/KEkrVsa2vr4+MzdepUhUIBAOnp6U+fPv3xKT169Dhy5EhMTAwArFmzxtHRsWrVqiW7uk4YaCIEACAIivH/Z8M1GunTMI1CTmpACCFUioIXLnDns1nLfeDlFVApAABkIrizm7nMZ+SQwd26dStxywcPHkxPT3dycnJwcGjcuPG7d+8AwMzMTHvXUMvBwSF/yNikSZN//vmnVatWNjY2p0+fPnbsmHYg6OzsTMqI0OCmRoum0Uie3BKe28ntNpxdpynZ0SCEkO6x2eyw0GvBixavWT9EkiumsoyVuUKHyq7BG1cPGDDgd1q2trY+ePAgAKhUqvxMFhQUVLBOSEhIwUN/f39/f3+1Wl1wPjY6mpyH3DARAgAAhWIxZFbe++fZp7aKw87weoyi27mQHRNCCOkYk8lcMH/evLlz3r59KxQKbW1ttUtUdOVXx3Mluyupc5gIv2K61uZP2iB5dD1962xm1XrcrsOppgb3GmiEUIVHoVCqV69OdhTlSLnIxuUIQRh5teHP2E7l8FKW+ovDz5MdEEIIVSi5ublkh1AYJsIiUFjG3O4jrMevJmh0smNBCKEKQqVS3b59e8yYMWQHUhgmwv9Es7Izbtj+6zG+7xchhH4DlUr18fEpJ/cFCyp3AZVbKcsCso6tV4uFZAeCEEJIl3CxTHFZBa4QhRxNXjyC06ybSZu+OGuKEEJaiYmJUqnUzc3tzZs3SqXS3d0dAN6+ffv8+fP8OgRB9OjRg5THBH8KE2FxUdgcbteh2q1KU5aO4nb2Y9dtRnZQCCFDJ3v7OH3zjCK/4nYZYtKmr/azKOSI8PzuIqtZjgpmVauv/ZyxO1j6LOz7OgSVJpizl8r97hVOACqV6u7du6dPn+7Vq1ezZs02bdqUmZnZvHnzqlWrfr9fjEaj0W5AU65gIvw1NCs7C7+ZedFPs09tEYdf4PXwp9tWJjsohJDhYlX1sF9z+afVTNr0zU+KP2AxZOavBvD58+fmzZuvXr26bdu2XC5Xo9FkZ2f/V+UbN260atXq5cuXtWrV+tULlR68R1gSTLe6/Mkb2bUbZ5/cTHYsCCFEJnt7ezqdzuFwuFwuAISGhv7gPUqtWrXy8/MrV1kQcERYchQqp1k3TrOS786HEEIVw61bt7TJLyEhwczMTCgUZmVlOTs7kx1XceGIUGdyLu2XvY4gOwqEECpr4eHhrVu3BoCcnBxXV9fw8HA9yoKAI0IdYjjXzD65mXb7NLeHP52vy+37EEKoPJs7d6721fa1atWaNWtWwZdO6AUcEeoMq6oHf8pmVnWvtPVTsk9uVktEZEeEEEJlQZsFtSwsLAiCIDGYEsBEqEsElcZp7iuYsYOgM1MWjxRdP6pRKckOCiGE0I9gItQ9ihGH23Wo5d9L8t49S10+GnMhQgiVZ3iPsLTQBU6Wo4IVyfEEFf8jI4RQ+YUjwtJFFzjlf1Z8jlNLxCQGgxBC6HuYCMuO7NWDlMUjxHfOg1pFdiwIIYS+wERYdkza9LUas0z26kHyEn984hAhhMoJTIRlisZ3sPRfaNb7b+HZ7WmbpiuS48mOCCGkTywtLefPn08Ynh07dlhYFLHlt04QGjLeN2tjY/PkyROBQFCcyiqVSi6Xs9ns0o6qLGlUSvGt06LrR4292ph2HYoLakqPVCplMBjl8+UvqPwTiUQmJiZkR4FKF44IyUFQaSategumbwMaXZMnJTschBAyXDgQIROFw+N2GUJ2FAghZNBwRFiO5Fw+kLFzgTI9iexAEELIgGAiLEdM2v7JdK2dunp89snNalku2eEghJBBwERYjuRvVQoAyf8ME986DWo12UEhhFAFh4mw3KEYm/J6BliNCpa+CE9eOkoWFUl2RAghVJFhIiyn6PYuVmOWczsOyj6xSfEpluxwEEKowsJVo+Uau24zdt1mZEeBEEIVGY4I9Ule9FNJRAiQsQcCQghVVJgI9QmFay6+eyll5di82Jdkx4IQQhUETo3qEzrf0TpwheTp7cwDyxhO1bhdh9HM+WQHhRBC+g1HhPqGIIzqNRdM38Gwd01dOTb75Ga1TEJ2TAghpMcwEeolgs4wad2HP2WzWipOWTxCnviO7IgQQkhf4dSoHqNyLcwHTFZ8iqFyS+vtJAghVOFhItR7dDuX/M+aPKk6T0Y1NSMxHoQQ0i84NVqhyBOiU5b651zap5HLyI4FIYT0AybCCoVZpQ5/ymaVKDt50fDcuxfxiUOEEPopnBqtaKhcC7M+gfKE6OxTW3PvXeb28Gc61yQ7KIQQKr9wRFgxMRzdrANXcHy6Z+5bnLl/mSoni+yIEEKonMJEWHERhJFXa8GMHTQrW8VHfL4CIYSKhlOjFRzBYJl2GEh2FAghVH7hiNCwyBPfpW2aJk94S3YgCCFUXmAiNCwMe1cjz9YZOxdkHlyhEmaQHQ5CCJEPE6GBIQjjBm0FM3ZQeZYpywJyrh7SKORkx4QQQmTCRGiICCab29nPeuJ6RVJs8uLhkie38IlDhJDBwsUyhotmzrfwm5kX8yL71FZlWpJpu35kR4QQQiTQcSJUqVRPnz6VSCT16tXjcDi6bRyVBqaLO3/ieo1aRXYgCCFEDh0nwq5du1pbW3M4nCFDhly6dKlKlSq6bR+VCoIgqF/+JKilYunTMCPP1gSdQW5QCCFUNnScCM+cOUOn0wEgODj40KFDc+bM0W37qLQRFKrszSNRyBFut2HsOs3IDgchhEqdjhfLaLMgALx8+RKHg/qIYLIths426xeUc/Vw6urx8rgosiNCCKHSVSqrRnfu3Jmdnd23b9/SaByVAaZrbf6kDZymXTJ2Lcg8uBy3KkUIVWA6SIRyuTw3Nzf/8NChQwcPHjxx4gSFgs9m6DOCMPJqw5+xnWpilrLUXxIRQnZACCFUKoqbq2bPnt2oUSMLC4t///23ULmVlZWDg0PHjh1zcnJOnjy5efPmM2fOGBkZlUK0qKxRWMbcbsOtx6+mmvHJjgUhhEpFcROho6Pj/PnzHRwc8vLy8gtv3ry5Y8eON2/epKWl0Wi0f/75Z8OGDUKhsGXLlp6enosXLy6dmFFZo1nZMV3d8w/VEjGJwSCEkG4Rml/ZUsTb23vkyJFDhgzRHg4bNozH461cuRIAQkND+/fvn5ycXJx2jIyMPDw8GIwvC/QbNWo0Y8aM/6qsUqnkcjmbzS5+nKj0aOR5otV/06p5sdv0I4xNyQ7n56RSKYPBoFKpZAeC9JJYLMZHovUai8Wi0X7yfMRvPT4RExMzYMAA7edq1aqlpKQU8w+NkZFRYGCgubm59tDGxuYHZ2EiLGc4xjN25Fz5V7R+gknrPhyf7vnPIJZPVCoVEyEqMY1Gg4mwwvutnzChUGhsbKz9rP2zkp2dXZw/NHQ63cfHRyAQ/M7VEVkobA7PdySnSWfhhT0piy9wuwxh18UnDhFC+uq3FnZaWVkJhULt5+zsbIIgrKysdBEV0gM0KzsLv5m83n/nXN6ftmm6Ijme7IgQQqgkfisR1qxZ89GjR9rPkZGRVapUYTKZuogK6Q1Wtfr8KZvZ7t654RfIjgUhhEqiuFOjz549S0tLEwqFr1+/DgkJqVu3rqWl5YgRI7y9vXv37u3s7Dx//vyAgIBSjRWVUxQqp1k3soNACKESKm4iPHny5N27d+3s7J4+ffr06dPg4GBLS8saNWrs3bt3wYIFIpGod+/eY8eOLdVYkb7I2LvI2KsNq0YDsgNBCKGf+7XHJ3TFxsbmyZMnxVwsg6tG9Y4sKjL71FaamRXX158ucCQ3GHx8Av0OkUhkYmJCdhSodOEuaEj3WNXqC6ZuZtdukr5pWtbRdWpxNtkRIYTQf8JEiEoHhWrcuBN/6haCSkteMkp8+4xGpSQ7JoQQKgImQlSKKMamvF6jrcYsk71+mLZhKtnhIIRQEcr1niCoYqALHC1HBavFQrIDQQihIuCIEJURCoeb/1n2+iHu3I0QKicwESIS5MW8TFk8QnznHKhVZMeCEDJ0mAgRCbhdh2pvHCYv8Ze9fkh2OAghg4aJEJGDxnewHLnQrPffwrM7cKtShBCJMBEiMjHd6llP3sSqVj9t/WTh2R1kh4MQMkS4ahSRjKDSTFr1Nm7QRvIoFDQaIAiyI0IIGRYcEaJygcLhcVr0/JoFydj5DyFkmHBEiMqjjL2LQKXkdh9Bs7QlOxaEUAWHI0JUHpkPmsp0rZO6enz2yc1qWS7Z4SCEKjJMhKg8Iqg0TnNfwYwdAJD8zzDxrdOgVpMdFEKoYsJEiMovirEpr2eA1ahg6fPwlJVj894/JzsihFAFhIkQlXd0exersctN2/0pPLNdLcWN2RBCOoaLZZB+YNdpxq7TjOwoEEIVEI4IkV7KvXsx9/4VfMoCIfT7MBEivcSoVF3y8GrKyrF5MS/IjgUhpN9wahTpJbptZavAlbJX97P+XUkTOPF6jqJZ2JAdFEJIL+GIEOkxVs1G/GnbGE7VUleNE17Yo8mTkh0RQkj/YCJE+o2gM0zb9eNP2azKTk9eNFyVmUp2RAghPVOup0aVSuXm7btOXLoulUpbeHtNDxrL4/HIDgqVR1SuhfmAScq0TxSuOdmxIIT0TPlNhCKRyLtNl3jHVmLvBUBjPnkbur9Ry+un/q1evTrZoaFyimZll/9ZmfGZoNKpPEsS40EI6YXymwhnBy97V3OAvPEQ7aGi0aDPDh79/QOf3L5GbmBILygS3mUd38Bp2oXWuCswGGSHgxAqv8rvPcKzl67Jvfp9U2RXMylLkpuLWzCjn2PX8+FP3qRMT85eNUaqfdMhQggVpfyOCBVKJdBZhQoJtmlubq6xsTEpIf1YVlbW4lXr70Q8MTUx+aNz2yGDB1Ao5fffGYaAyrM0HzRF9PZZ7oVdkrsXeD38GZVwXh0hVFj5/aV2cnSAz1HfFCnlhPCzpWV5vOvz7Nmzmt6tVyc73mu16or7pKBTLxu07CCTyciOCwHN0c0ycCWnWbeMPcGZ+5eqc3PIjgghVL6U30S4ZNYk8+NjQZT25ViZZ3Jy4uhhg8vnMKvfyMDPgw8qGw4Ac3uwrZ7T9Z9Xdm2Xrl5PdlwIAAAIwsizlWDGDrpNJZUom+xoEELlC6Eh496JjY3NkydPBALBj6udv3hp9JTZeWZOQGfDxxfj/YdOCwokCKJsgiy+9PT0mu36po6+/E2pVFj9QO/XD26SExP6P6lUymAwqFQq2YEgvSQSiUxMTMiOApWu8nuPEAC6dOrYuWOHmJgYkUhUq1YtOp1OdkRFE4lEhLFZ4VKWiUQi0fm1EhISVm7a/iLqfSV724C//vTy8tL5JQyE5PHN3DvnuL4jGY5VyY4FIUSmcp0IAYAgiMqVK8vl8nKbBQHAzs5OnfIO1CqgFBh2xLtpkNsAACAASURBVD+pVtVNtxc6dPT4uPkr0ltM1jQYAJmJZ/+eN6iN5+pF83V7FQNhVK85QaFm7AlmOFbldRtBNbcmOyKEEDnK4/02vcNgMAb94Wt8bjaolV+KhMkWZyYFT5+gw6sIhcLxs4PTRl/S1OsG1i5QrUXGiFP7QiIfPnyow6sYEIJg120mmLadLnBKWTlGeG4XblWKkGHCRKgbyxfOHedpZrWikdVRf/6+/k57eh5cNa9+/fo6vMStW7ekNTsDk/O1iCAyGw0/cOKsDq9iaAgG07TDQP7kTaqcjOTFIyQRIfjEIUKGprxPjeoLCoUSPHfGnGkTo6KiTE1NK1WqpPNFPUKhUMb+biNNjmVaakmWQUZGRi5cvTn63XsHB4eJIwe3a9dWByHqLSrP0nzA5LzYl8LT24FKM/JoQXZECKGyg4lQl5hMZp06dUqpcTc3N9Pd1zK+LaR/fOLlWe1Xm1q8at3KgxcyOswB75pv0j9ELFza89zlHetX6ipUPcV0rmUdtJbsKBBCZQ2nRvVGgwYN7PI+Ul5e+lr0Ocry4a4hgwb8UjtJSUmrdx3K8D8NlRsA0xjsamX9tf90xPuIiAgdR6znFCkJ4lunNAo52YEghEoXjgj1BkEQIaePDBg59unttSo7d1pWoqUi/cjx/WZm3z258UOhoaE5tXyB8s3/+gyPgSfPXynmwxhqtVqhUDCZzF+6rt6hGJnkxbwU3z7D7TaMXacZ2eEghEoLJkJ9YmVldfXU4c+fP797987e3r5y5coluBMpzpUomKaFS9kmWSLxT899+/bt0MApMYlJQKVzaLB83rQe3bv9agD6gmpiZjF0dt67Z9mnt4nDzvJ8R9HtXcgOCiGke5gI9Y+NjY2NjU2JT6/tXsvsyNYMGF6w0DjubuMe7j8+8f379819+6f03gR/eABAiih16OK/UzOy/If+VeJgyj9mlTr8SRskj66nb5vNrFqP23U41fTXhuAIoXIO7xEaHG9v70rqZFrEkfwSIuqG9bvLff/o/eMTpy5Ymtp5MTh5fDk2sc4etGf+0tWk7NJXpgjCyKsNf8Z2qolZylJ/6Yu7ZAeEENIlHBEaHIIgrp85OmrCtJAVqym21SHtg7uz3d5Lp356zy/yyTPN2G+3EWcaq80dk5KS7OzsijwlLi7u3IVLMR8/N6hdvWePHixW4fdq6REKy5jbbbhx484AFT3xI2RgMBEaIi6Xe2jX5ry8vA8fPjg4OBTz/Y5UKqXwNnIAGmUejVb0n6Jlazeu2H4ww9NPzfVkn3g87Z+m5/7dWXqPl5QNmmWBSWm1SpmdTjPnkxcOQkgHMBEaLiaTWa3aLzyD2Lp5s/jn51Qevb4WiVKN5Nl8fhGZ4NGjR0v3nckMDNUmTql7x8T6/boPGBDz7EGFeRGEMistbc0Elrs3t9NgCodHdjgIoRLCe4SouIJnTbG7tZz+6MiXLVXjIi129NqyIrjIylv2Hc5sPuGb4aOVc65tncjIyO8razSauLi4ly9fyuX69NAezULAn7GdwmAlL/EXhR7XqJQ/PwchVP5gIkTFZWVl9Sz8+hDW88obW9qvadz29brbx3a1b9umyMoJn5LB3KFQoYzrkJycXKgw5Hpo5doNGw2c2HrccvtaXjMXLFar1aXSgVJAYXO4viOtx63Ki3mZssRf+uIe2REhhH4ZTo2iX8Dj8bauWVacmm7OjtdS3oHgm1f9sdPfOTn1KFjy+PHjP8fPyfA7AVwBAIBaufbcHNns+SuD9endUjQrO8sR82RvHwtPbRWHneH1Gk3nO5IdFEKouHBEiErFmGGDLUKXQd7Xh/SJDw8txQm1a9cuWG3m4tUZ3Zd/yYIAQKHldl144PiZ7+dI1Wq1WPzzR/5JxKrqwZ+yiV27qfz9C7JjQQj9AhwRolJRrVq1jfOnjJ/VKrdml1xjAe/zY1txzPnjBwpthRMVHQ3tPL45k0Il+G6JiYkuLl+2cUlMTBw6dvKL6BgN05gmy5k4evj4v0dRKOXy33AUKqdpF7KDQAj9GkyEqLT07d2jY7vWt2/f/vw5uWbNYd7e3t9vCMdmsUGaA0bfLLnUSLJMTEy0nzMzM73bd0/qslzTxQcAQCGde272+7iZm1YuLpNO/Ba1RJy+bbZp276smo3IjgUh9J/K5T+rUUVhamrapUuXESOGN27cuMhtUfv6dmbd3/NNUWqMOSXP2tpae7R649aURgEaN58v39LZ4h7Lj126npmZWfAksVhcqKQ8oBhxTDsMzD67K33LTEVyPNnhIISKhokQkWnqhLE1P10zOT8bkqMh+zPt4SHB3j8PbVuXX+HWg8dKtxbfnEMQKpcmL158uQ9399696g18XJp2rtH+T4ca9Q8ePlqG4f8cq1p9wdTNrBoN0jZMzT6+UZ2bQ3ZECKHCcGoUkYnFYj28cXnX3gOHzwaLc3ObetWbcS/U3Nw8vwKDQQeFrNBZFKWMTqcDwKNHj7qNmJgxaB9YOAEASLLGrBstkcpGDBlchp34GQqV49PdyLO16PrRlCX+nBY9OC16ElT8q4dQeUGQsmOyjY3NkydPBALBz6sCqFQquVzOZrNLOypUDm3etmPS1Y+SDjO/Fimk1qubxT9/wGKxfDr3CmswC+xqff02T2y3ofXHN4+1RykpKREREVwut169ehwOp2xjL4IiOUF4eivFyMR88DSyY0HFIhKJ8u9Yo4oKp0ZRuTbMb7Br0i2jkBVfnsT49Mp8W4+F0ydq9++OjYv/JgsCAJOjNDITCoUKhWL42InubXoO3HHHd8lRV89mW3ftJaMH36ALHC1HBZv1n0h2IAihr3B+BpVrDAbj0a2rS1ev/3dfD5FY7OJcefm2JV5eXtpvKQDf7wOuzpMwGIzAKbP+TTGTBt4A7SIduWT6loEONvxOHTuUeScKI2j0L580GvHdC0b1WlCMyB+tImSwcGoU6bG/AsYdpDZV1fX9WpT63v3C349vXbWr6Zk66QEQBeY80j40uD7xQci53NzcDVt33Hn0zJzH7du1PZmpUaPJPrlZ8uS2aYcBnMadCmV0VB7g1KghwKlRpMdWLJhlf3Mp8+5ukIlAKae8uGi9t9+eDStTU1MJc4dvsiAAWFX+nJz86tWrql4+c57Aebcx+0y6DVxxuEPPfiqVipwOEASv12jrsctlrx8mL/GXvX5IThgIGTZMhEiPWVlZvXpwa4xtUvUDPV22tRsgD31y46JHvXpcLlctSi9cWyZis1l/DB39qd8eefPRYO8OVZpm9dsernLasnO3topQKMzIyCjjXtD4DpYjF3K7Dc8+tTV9+1xl6scyDgAhA4dTo6hi8m7b9YHnFI3L1y1d2FeXjq2q3HP5Xqr/+W+qZid5XgxYOmuy/6SZImABlcaQZK5YMKNPr55lHLNGpRTfPiO6fpTTtKtph4FlfHVUJJwaNQS4WAZVTId3bGjZ9Y/kat2kbm1Anmv25FAdZtaAP+buCY8pXNXEKuljYp8J8zMGHQKeLQCAOMN/2UiFQjngzz5lGTNBpZm07GXs1SYv7k1ZXhchA4eJEFVMTk5OUY/u7Ny7/+rtnWZck14TenXu1FEsFmuSowpXTXwmzlPkDNn2JQsCAMcie8DOmcGdW/o0XbR6Q8TTlwK+td8f3Xp071YGkVM4XHatrwNZtUSMa0oRKlU4NYoqOKlUymAwqNQvCzIHjPj7pMRZ1jLwy9eSbIsdvQlpVvrUyEIn8hZ50JnMjFZT1ZUbQk6K2b2tjbjSC8cKv0CjtKWuGkflmnO7j6BZ2v68NtI1nBo1BJgIUQVXKBHm5eWNGDf5cnikwrkxTZbNSHy8btHcwOnzkoLuQ8EMp1ZRprmq50QAxzK/zPRE0Ba/Zu41azx4GMExNmrcuLGDg0Npx69RKsS3TotCjxk3bGfSrh+FZVzaV0QFYSI0BJgIUQVXKBFqpaamvnjxgsfj1apVi8lk9ho84pRFd03N9l9rPDpOizym9D/yTVtxkZaHR4CVc7ZrG5oi1+T1+cHd2q74Z24Z9EKdm5Nz5aDk8S3Ttn9ymnWD8vk6xooIE6EhwESIKrgiE2EhSUlJjdp2TfEeLa/jC1Qa9fVV7tnpqlqdhT1XfFNvz0iikoemxagvhxo198jolQNb9Oze5f379wKBoLQHiIqPMdmnt6ilubweo5iutUv1WkgLE6EhwESIKrjiJEIAEIlE85asuBJ6W6lS+3g3GO8/pEW/gLSxIV9rqJUwzwMWPP/mtLRY083dmKYWGofalOxP1iA6tH1DrVq1oDRJn4VJHoVaDCuLkSjCRGgIMBGiCq6YifB73fv5XaHVy/MJ+HLvMP4J5eAY9Yzwbyqt9yUa9NU07PflMOmN4N+/XoaHUCgUjUZT8H1SSE9hIjQE+PgEQkU7vGvzmMkzz61qonGqRxWlmUhTMghZVsEaSa+BxviaBQHAtnqao0+V+k0YFvZAEKw84dpFc7p37VKqcWaf2kq3qWTcsB2U7XJWhCoMTIQIFY3NZu/csEokEkVHR1taWjo5OXXr53fx6emve3x/jiIc6nwzo/Lyiurji6zAEDDlAwCI0obMH7oLoE3LFkqlksfjlUacxl5tsk9uFt85y/MdxXR1L41LIFSx4dQoquBKPDX6vaysrLY9/oxlOGW5tiUUUtOwjVJ7T/mfa77WWN4aRhwAns3XkmfnGEcm8GwrEVQ6S56zauHMnr7dfz+S78le3c8+uZUmcOT1HEWzsPn5Cah4cGrUEOCIEKHiMjMze3TzyrVr127df8Rhs5oPXePrF5AqEwHr/z+UebnfZMH3d+HKavmkm6nm9gAA4vRhi0ZKpLJGXvW5XK6VlZUOY2PVbMR38xDdPJG6apyxd0fTtn8STPy3I0LFgiNCVMHpcET4vSPHT46duyy91TRNJQ/I/EjZPkC9KPrr12u7Qb9VYO365VCjgUtLqbe2WdRqQkizTZQ5ezasaNK4sW5DUgkzhBf25L19LJi1m6AzdNu4AcIRoSHAESFCJde3d8+Gnh5L1m5+cnmXo71dXM2qke/uaKo0/fK1KPVrFgSA88EgFaqCo1JpDABIyUjwHdZv04LJxsbG1atXr1y5sk5ConItzPtPVIuFmAURKiZMhAj9lkqVKm1ZvVT7OT4+vkkH31SfIIV7ZyAokJcLGs2XxZxKOTw5DbMefH0NfUZculg6cNUJpm0VVuJ6dwHn6O4tFhYWOomKwuHmf86LeUEz41PNrXXSMkIVDyZChHTGycnp5b0bsxctv35gp0YDcoH5h9dXv+zclpkIgqoFsmA8HJ0CEy7JTflyABHArScnG/q07u7bvWolh25duxTzxkFxKFMSM3Yt5DTpbNKmL8Fg6apZhCoM3LEQIV3i8XjrlwW/vn/zzYObYReO212Zw3hwAPJygUoFUfrXenf2QMcpX56yAICY+6rzS2Ndu68S1fv7tqROyy479x3UVUjGjTvxJ29SZqYkLxouiQgBMpYFIFSe4WIZVMGV6mKZn8rJyZm9aPmV0NsqlSolJUUUeAXMHQAAdgyGLrNA4AYAoJDCYh8IPPt1xWn8Y8bOgbaWZg6OTiMH9B7wZx+dvPtJHh+VfWoLqNW8HqMYlWv8foOGABfLGAJMhKiCIzcRFnT/wYPug0elt52lrtocjkyCxoPBrRkAwMsrEHUTei/+Uu/eQbizG3ouBCcPyEri3l7XgJm2Z9NqPp+vg15oNJLHN4TndjOda5r9OR5nSn8KE6EhwKlRhMpIo4YNn926NIz2oOahPu6qOHbomi+zlDkpX4aJAJAnhmtrYPw5cPEGGhNMLIVKasi9RzW7+NnU8Ow7xD8zM/O3giAIo/qtBDN2MCpV16iUv9slhCoEXCyDUNkRCATb1izTfh4zeebh7b4ZjQNAKYfPb77UiH0ANVoDnQ0AoNHAln7QoI+mz/JsggCAEzc33/ZqPHTwIO/6dTp16kQp6VsJCQaT41Ngg5v8pa0IGSQcESJEjg3Lg69u+edv+v0uygheTCgkvwUAkEuByflS490dMLOFxoO/ZKmLS1URx5MbBy5KcRuw5Xp1r2YJCQk6iSTnysHUNePl8VE6aQ0hvYMjQoRI4+Hh4eHhAQCvX7/uPnB4hsAj28gW3oRous4CAPj4HKo0+1L15WVIfAaTQrRJMYdjkfP2oatnc4G1ZbuWPssXzDIzMytxGKbtB9DM+Rm7FjLd6nK7DKVydfMsI0L6AkeECJGvRo0aURFhxyf13NDa3NOWwwxdBxo10Fkgz/1S4/4h6Dz9y9Dw3kG4tAyG71Msik4cF743u3LV2p5r1q6LjIws4eUJwqhBW8GMHTSedcqygJwr/2oUct10DCF9gKtGUQVXflaNFpNMJpsyZ+Hh0+c1XLvMtGT1zPtAELCyPYw+CmwuqJWwsCFMDwOGEQDAtbXw6DjU6UIx4prF36luLDt/ZB+Xy/3ZRf6TKjM1+9wOeVwUt9swo7o+eO8QV40aAhwRIlS+sFisdcuCk15HPjiy4e8+Hc329IfkaDCzh9QYAIDkd2Dv/iULPr8Asfdh6k3oNFXdbFiGTf07L2P41T1d6npv2r5TrVaX4OpUc2uLv2aYD5qSG3ZW8SlWpz1DqJzCESGq4PRuRFhISMj1hWu2vH3zMoNhrRx/CZKj4cpKGLoLAGDTH9BrEfCrgEYDG3qCqze0HQ80BuSksPYNc6CJxw0f3LN7NxsbfD1hyeGI0BDgiBChcq1Nm9a3zh9LjnmzePSfVmubs15dgNgHoMwDAMhOAmsXAIDXIWBmCx2nAI0B7+7A2m6yKq3eNZ0WeFdZt1XXpSvXyGSy34lB9jpCFHIEbxyiigoTIUL6YdLYgOeh5/b0cu3RsiFnZ38QpYKpNWR+BACIuQ/unQAA5BI4NAHGnYX2QeDmo079kJqrmLHzrL1H85ZdeycmJpbs0nTbyvLE9ymLR0ifhumuQwiVF5gIEdIbAoGgb9++Jw/tPzBnpNu/f5pmRFNOzQQAAA1oF7VE3YRa7b/s5b1zCFhVhtkR6qArGV0W3fwgcq7n3axzrwsXL/3qdak8S4shM836T8oJOZy2bpLi43td9gohsmEiREj/dO/a5W1EmDAxemTjSpabOxF5Ynh2HgAgJxXM7QEAkl6DRgXNRwJBwIVFELIe+ixTLnp3p/b4vkH/1G3S6uTJk3L5r011Ml3d+RM3GDVok75tTtah1aqcrFLoGUIkwESIkB7bvGrJrf1rl3aoZJP8kBGyCsxsIfkdAMDHF+DaBAAg7QO8DoWAI2BbAx4cgcNBufV6P6sTMHD3g8q1PH85HRKEcaMO/Bk7KBzTlKX+ebGvSqVXCJUtTIQI6bcaNWpMnjAu9tnDSdVVdlfmUB+fgKQ3wDACmRgAIOoGePYCggJJbyBsJ0y9Aa3HAIUqfX41ybRKn3/2ONTyWrxy7S9dkcIy4nYdxp+8kW7jVCpdQqhsYSJEqCJgsVjBc2d8fB35IjykypkA3ptz8PgkKOWgkAHTGADg4RFoOx5oTHgTCje3wMQrMOqwqsnQVA1n5opNXKdqg0cF/tKrLag8Kwr7y7aoarFQmVLClTgIkU7HiVClUr1+/TokJESlUum2ZYRQcVSvXj0qIuz0HL8+Lb1MNnQCGh3ehQMACD+DpRMAwNU1MGADGJnB9fVw/yCMPqpZ8CKnw+wDt14IqtXrM8T//ftfXgujzExO2zgl68halQhvHCL9o+NEGBwcPG3atO7du+fl5em2ZYRQMVEolObNmx/Zv/vmvtWd5Q+Zb0OIe/vBzB7SYgEARKlgVRnkEgjfByP2AccStvwJsQ81gzYppt8/lmxSq3nn7n8OfvjwYfGvyHCsyp+xg8IySlniLwo9jm86RPpFx4lwzpw5Z8+e5fF4um0WIVQCHh4e5w/vSXn3YgDtifmL45Sz8yEvFwgKKOXw6RU4NwQKDR6dAAsn6LsCNBpY1xXYvLyBW84aNW8zcHQDn1b37t0r5rUoLGNu9xHW41fLY1+mLB4pfXG3VLuGkA7hPUKEKjgul7t/6/qUd89Xzwjkr29F4/Dg/kEAANAAALy8At4DAAD2BYDfdug0FeIfw63tojq9Iyr3aTN6ftW6XmFhYcXcuZRmZWcxfJ5Zn7E5F/elbZqmTE8qtW4hpDOYCBEyCDQaLXC0/5t713dNGcy/s5b94iy8vwsqBchzgWUKWZ+AaQx2teDlFXgXDlNvQv1eEHFcQuNE27dtHbjEpbbXlStXirk1MdOtHn/yRqM6zVSZqaXdL4R+368lQolEkpubW6gwOzv7yZMn35cjhMobMzOzQYMGfYx6tryLaxVrLm1jD7CoBLH3QZwOXAEAQPhe6D4PNGrY2h96/gMjDwKDrcj4GGdSrdP4xY41PE6ePFWsK1Goxk06M93qlmp3ENKJ4ibCrVu3urm5cTic4cOHFyzfuXOns7NzQEBA5cqVr127JpPJsrKy1Gp1dna2RCIphYARQr+LRqP9PXJ49OPw00vGV0m9Szk1C/LEkBwNAJD1Eaxd4M11cGsGlerDmQUgE8Os+9B4kFou+0iY/TF5saCK+579B37pioqPMSlL/GWv7pdKfxD6PcVNhPXq1Ttw4MCUKVMKFmZnZ48bN+7q1av3799fu3ZtQEDA2bNn+/TpU6tWrb/++mv79u2lEDBCSGc6d+oY/fjui9uX64YHM8QpxL0DwBVAZgKkxoJdLVDmwfML0GMBvAuHi8tg9FEYsEFtYp2iYg2bvcqicvX1m7cW80J0exeu78jss7vSt8xUJMeXaqcQ+lW/9j7CefPmvX379tChQ9rDffv2rVu37tGjRwCgVCqtrKwuX77csGHDn7bDYrGsra3zXxHXokWLdevW/VdlfB8h+h36/j7CMvPx48fh46a8iPss5jlraneG7CTw6AGn58DIg7C6EwzaBBQabOwFgzcDzxZOzoKPzwmNxlidO2XMyLGjRhTrv7BaJX94VXbzBKOWN7NVH8JID97zJxaLORwO2VGgkmOxWHQ6/cd1aL9zgYSEBBcXly8N0WhOTk7x8fHFSYRcLvfYsWNWVlb5hz949SUmQvQ7aDQaJsLiqF69evjVc+/fvx8/fU7YnU05IjF49gJJNgBAbiZYVoKjk6HbbOBXgRVtwXc+9FmmOT1PHHNv7qaDS9ZtmTzGf9aUIBrtZz8pbfuom3TMuXxAvD7ItO2fxk27ENTf+hUqA/hi3grvt1aN5ubmMpnM/EM2my0Wi4t1VQrFycnJ+f8sLCx+JwyEkK64urqeP/ZvxNn9gzo0NdrSE4TJkBYLGjVoNJDwFKo2hzu7ofEgqNYC1nWHai0g6LLGuYGEZrJg2xETB7eps+crFIofX4JiZMLrGWA1ZpksKlJ07XDZ9AuhH/itRMjn87Oyvu6olJGRIRAIfjskhBDJ3Nzc9u3c+vbO5WFdmzPXdiRMrODVVaAxQCGD+MdQrRU8PAK1O0KdzrCuO9RoC4FnNQI3GdNs+eGrJnYu4yZP/+lLLegCR0v/haYdBpZNjxD6gd9KhB4eHg8ePNBuK5qSkhIfH1+3Lq6WRqiCsLe337Fp/adXESNauTMOjiaMuBB5HKh0UMkhLvJLOqzTCWq0hvW+0MQPhu7SMI3zzCqtvxRhYusyKnCCVCot5rXUElHOlX/VElGp9gihIhU3Eb5+/Xrbtm2RkZExMTHbtm27e/cuADRr1szBwSEgIODWrVvDhw/v1auXra1taUaLECprFhYWW9etzPjwemxLN8aV5YRKDpGngEIDlRziH0O1lnB3HzQeBIKqsNMP/lgGHSZpxFnyyo22hcXyHFyHjgzIzs7+6VUIOlMtykxeNEIcdhbUuGU/KlPFTYRpaWmRkZG2trb16tWLjIxMTEwEAIIgLl68yGQyFy1aVKdOHXxeAqGKisPhrF0aLEx8N7VdDebD/YQoFR6fBiodlHJIfAZuPnBjM3SeDlIhXF0N486BXU2N8LPcs9+eyGSBa43BQ0dkZGT8oH2CzuD1HmM1Zqns5f3kpQGyqMgy6xpCv/b4hK7Y2Ng8efKkmDcUcdUo+h34+ITOKRSKJavWLVq1Ls/BU2PKB7UKGg+CC4th8Bb4NxC6z4NXVyHzI3SYBHtGgIkl2LsT0WH0z696dWi5YfUKc3PzH7cve3U/+/R2mpUdz3ckzdq+bDr1X0QiEa4arfBwr1GE0K+h0+mzp04UfYpZMrAF+/VFQpgMEceAzoK8XMhIANsacO8g9AqGvf7Qbjx4/gEPjmhqtJE39jv0KEFQw6uf3wihUPiD9lk1G/GnbWVWqZO6bqLo5sky6xcyWJgIEUIlQaPRpowfK/oUs2Joe6MXZwlZLtw/CDQGyHKAxgRROmjUYF8bzsyHoMsQcx8yEqD1GIVzk8PX7lm6ug8cEfCDp60IKs2kZS/BtG1MF/ey7BQyTJgIEUIlR6VSg8aOFiXFrA7wZT88QNBZcO8gKKSQmQjWLvD8IjTqD0/PgrkjNB8JZ+ZD/R7QdIjS2OrgxVs8x6qDh4/6QTqkcLgMhyr5h/hSJ1RKMBEihH4XhUIZ9/cocVLMyoDeRqFrCDoTkl5BZiLkpIC5Azw7B81HwPFp4H8IIk+CJAt6LwE2V2Xnvv9OFM+20pCRo348WQoAGqUifevsjB3zlGmfyqZTyHBgIkQI6QaFQpkQ+Lfo0/vl4wYbX10GyW8BAFKiQSIEpjEoZCCXgCwHPHvBqdkwaBMw2GDEU3kP3nM7yrJKnb+G+//gbW4Ejc6fuoXhXCt1zQThme1qGb73DekMJkKEkC5RKJSJ48YKP75bNtnf6N4OuLMHzO0hLhLobIh9ADXbwdU10GcZnJkP3oPAqR4kPgOvPspKDfZdCuNVrjFq7IS8vLwiWyZodJNWvfnTtqplkpRFw3PvXgS1uox7hyokTIQIId2jUqmTg4KECe+WTPI3jg0jTs0CcToo5aDRwOc3YOEEyZQQRwAAIABJREFUMhEY8SD+CXSdDWE7waMHuDRWsnhbT141tnYIDJr8Xy80pZqYmfUdZ+n/j+TxzZQVf+fFvCjjrqGKBxMhQqi00Gi0qVOmZCW+/+fvAcYKIRF5HJ6dBzobPkcB3w0eHYe24+D4NBh5EK6uhhqtoWoLYHNVtTqtP32L61R1wqTJ/7WFN93OxWrMMtP2A/LePi7jTqGKBxMhQqh00en0GVOnZH+MWTKiByv1DaFRQ3QYZH8CcTrQmMDmwpvrUL8nvLsDZnZQqz1kxkPdbkqbWmsOnDG2qzJr7gL1f0yBsus0Ne30Vxl3B1U8mAgRQmWBRqNNmRQkSY5bPLIn4/5eSH4LLFP4+BKYxhD3CKo0hdgHYO8On15AEz94HQKevYBnq+AIgrcdZJrx/wlerN3f/wfSNk3LvX8ZyNgtC+k1TIQIobJDEMTUSRMkyXEz/QcyXl+C6+sgJRoICmR9Ar4bPDwCrcfC1VXgOx9u74LWf4NGBVVbKF2bzd5yyNjebfXadT9onNdtuORhSMrKMXnv8cYh+gWYCBFCZY1Kpf4zb7Yo6cO4vp3oshxIeQfvwkCcDqI0EKeDmw9cXQ29F8O5YOj5D3x4CJW9wMo5j2YctGAly8Jm+44dRTZLt3e1Clxh2rpv1r8rMnYHKzOSy7hfSE9hIkQIkYPBYKxZuUyY+HZMpwbUxych6xOwuZAWC0xjyEyE5GjwHgDn/oE+KyDiKNTtCixTcG2UZ1dv5NxVJvZVDhz8t8hm2fV8+NO30+2dU1cFCs/v1uQV952IyGBhIkQIkYnNZq9fvUKSHDekS3PKmxB4cBjehYNGA+lxYGoNDCOIPAGdp0PIemg3DpLfQSVP4NmKNYxBYyaZ8B3OnTv3fZsEnWHath9/ymZVTmbqukll3ymkXzARIoTIx2Awdm3dlJP0oXfT2pSMOGAaQW4GpMcBxxziHoFUCA36wqXl0H0uvLgEdboAjQnVWorNXbsND7J0rnHnzp3v26RyLcz7T7SesKbsu4P0CyZChFB5YWxsfOzA7s9vIrvWtqc8PgkRx+FzFAABabFgxAMLR7i1DbrMhIdHoEFfyE4Ch7rAMMqQU5t17mXjXPXu3bvft0nQ6PmfJRHXVZmpZdghpB8wESKEyhdra+uzxw9nxEc3rVGJEKcDgw05aZD5EUysIfszRN2A5v7w9Cy4NAK5BGyrg7kDuDZJVrKadP7DqXrd2NjY/2pZnStMWTkm58JevHGICsJEiBAqj3g8XtjVc3FPwj2tacSzc/DiEnx8AaCBtFgQpoBHd3h1DegscKgDojSwqQp0NgjcEvIYLg1aujdo+ulTES+p4LToyZ+yWZmVkrx4hCQiBJ84RFqYCBFC5Zejo2PE7ZC3j+64WbKIjHigMYDGAGESyCTg4g1RN0CYDE384MUVcO8ISgXY1gQ29+WHJIeqdRo2a/H58+dCDVK5FuYDp1gMmSUOP5+6erz8w2tS+oXKFUJDxr+JbGxsnjx5IhAIilNZpVLJ5XI2m13aUaEKSSqVMhgMKpVKdiDod7148aLrn38lJCVrrFzAwhFoDEiNBXkuVG0O5o7w+BTwbMDEGuIfA88GclIh6yMhzWnVzPvU4QMmJiaFm9NoJI9vCs/tYrq6mw+YDARR5EVFIlER56KKBUeECCH94O7uHvfqcfjFE5bKDHh7C2LugyQTNBpIfgvCJKjVHuRS+BABHr6Q9RGsKoMRT2NX6/rzWK5TtT8G/KVUKr9pjiCM6rcUzNjOqu5FUodQeYGJECGkT7y9vdNiX184sI1HVYJcAnQ2aACEqSBOB9saYFUJ7u4D54bAMgWNBqxdgKBoTAXHQx8yrCuNHTeh0BwYwWAZ1W+ZPxxUS8V449AAYSJECOmfTp06ZSW+O7phCSvrA3x6BVkfIfsz5GYAmwc21eBNKGQmgmdvyEwErg1YOQPHXMM03nDgON3MZuHChf/VrPDU1tS1E+TxUWXZF0Q6vEeIKji8R1jhbdy0acKcYIWaAjxb4AmAzoY8MWR9AjobnBsAnQVvQsHaFXJSQKOGrE8gFdII2Lhs4ciRIwu3pdFIHl0XXtjDrFKH22UolWuB9wgNASZCVMFhIjQQcxcsDF67VUXQgGkMbC4QBChkAADmjmBTDcQZEBcBDnXg81tgsCAzEaQ5LDrl3x2be/ToUagpTZ5UdP2oOPwCx6c74dXBxNyChP6gMoSJEFVwmAgNh0KhGDsuaPuhE2qWKVCoQKUBhQYaDTCMwKoScO0g6yMkvQL7WpAcDQwjyEwEqdCEY3Tx2L9NmzYt1JoqK1V4dqc09pVZ9+FGHi3I6BAqI3iPECFUQdDp9C2b1kuSP/Rq7kGIM0CtBo0aNGpQSCHpDXy4DxQCqrcCjQaUeWBqBSaWIKgqUlCbdejOr1T1zZs3BVujmlmb/zXd6I9xkojraomIrE6hMoAjQlTB4YjQMEkkkvbdeoZHPNFwbYBKA7UKCApoNMBggykfeDYg/197dx4eVXX3Afx77jJr9m0SIICsoUWWIlVAdhQUUAG1alXkpRRR1LdVENC6tAWKWF7bqrhQUZbihijgxiKBgmxq3FBWIYFsJJNt9rn3nvP+MZNArQtCwp1Mfp+HPzLDcPI7z5OHb849mx9FnyI5B4E66CHUlSHo69Sx/bb3387JyWloh+YIWwIaERJC4pDD4fj3pvdKD37eNcMG93EAAIPg0IKoPIrDH+LkN8jphvS2EByyiuyuSMk5fLysVZce/QcN93i+ewjozX/Ds/FloYXPZ19IU6MgJITELZfLtf+TXV/tzs+2aKgphiSB6wDAJIT9KNuPwx+CMWR1hGqHJCMnD/aknV/sT87tPO7a67+9Bx+w97g0fOJI+fwpgU//bUJ/SNOgR6MkztGjURKxY8eOK6+/tS4QgjMdXIPFCXsSVBvAEKhFoA4WO9LawF0IAfgqIRgL+6fcNP7ZxU9/q6nQ4S9q3nxGstiTx0215HY2pTukEdGIkBDSIgwYMKC2+MiaF560Bd0I+RGsg+ck9BBkBamt0KY7sjoi5AM3IEnI6grGRFLmc6+9Lae4Fj7++OlNWTtd6Lr3SccvR7iff7h61f8ZddVmdYo0CgpCQkgLcs011wQqTjz1x5lyoBYBL05+A28lQn4EPPDXAAxpbZHVCaoVtkSoDjjTuSNt5qMLLCmuV1555VRDjDkvGeWas0RKSCpfMFWv+I5bn0hzQY9GSZyjR6Pk+8yePeexxUs5k5CQhiQXgj7oIRhhMAkAmARbIqyJqCqErMJTAcEdqrR53epLLrnk9Ha4r05yJH7f/RUk9tGIkBDSQs2fP0+rKr5x9HDmqUDp1wBghCEpSMxEejtkdkRSFoQBxQpuIKsjFItfTuw3alxm206HDx9uaEdyJjWkoFZ6TCv+xpTukLNGQUgIabkkSfrXimWe4iMD+1yI8oMwNAQ9CHrAdQTrUFMCnxuMwZYIxYq0dhA6kl2Vdf7OvS7ueuEvqqu/PTto1Lorn32g+uUnDA9NHDYbFISEkJbO6XRu2/he2ZGvumQlI+RDTSk8FQj7EfJBCDiS4UiBrIIBqg26jvS2sDkPllaltc8bNOyy03dZ2PL6uOYskRwJ5Qtu92x+Teiaif0iZ4iCkBBCAMDlch34bM9Xu7akJ9rhLoKnEoYGQwOToYfhrYS/FmCQJTAgrR2YBGf6v/d+qqZm33LrbQ3tSDZn8lW/yfrf/wsf3Vf+l6mBLz40r0/kjNBiGRLnaLEMOQubNm0ae9P/BDUNsgUJGZBVBD2wJ8GeDEmGEDB0hLzwuSEpkU0XTA89fN/0hx966PR2QgcLatY8KyWkpP36Pjklw6zukB9GI0JCCPm2ESNGBE4WvbBonmyEUH0CIR+0EDiHFkJVEdzH4HPDCEGxwTBgS4AzVcjqI4sWq0kZL7/8ckM71i69XTOeclw0TOh0KlvsoiAkhJDvNmnSpOqiQw/e/VtWVw5DQ3Ux9CC0MOwpcKaCMXAdkgRDhxBIbQ3VqluTbpwy3Z6es3Pnzmgrkuy8+HIlo1X0pRkP4cgPoyAkhJAf8qc//cmoLb9u5EAYGiqPAQIhL4Ie+D1ISENqDhLSoagwNCRmwZYAiyPIrP1HXpORe0FhYeG3Wgt8tr1s7uTgvl1mdIV8NwpCQgj5EYyxV/+1wl9+7KKfd4GvCp4K6BqMMMBQXYqAB7bIylIFqh1J2ZBVWGxur9b+wr55F/by+/0NTdl7DUyZcEftuhcqF8/Ryr4dk8QUFISEEHJG7Hb73h1bT37zdZusdFQVQQj4qqCHYLUjUIfqEggBiw2KgoQ0ONMgq7A4DhSVOV1tL79idMPKRFteH9fMxbbul1Q8eX/N609yX525/SIUhIQQ8hNkZmYeP/jlJ9s2JigCdeVgDF43Ql5Y7OAc1cXQw5AtUKxIzIA9BYoVFufGHR9JiRl333NPtBVJThh4Vfac5yHJZfOneLeuEca3r3wi5w0FISGE/GS9e/f2lBWuXfmCEvYh6AUEArUI1sHiBBhqSwEBxQJJQkIaHCmw2KDa//HiKikx7dlnn400IjkSU8ZPy7xrYfDgZ8Evdv7wdyRNh/YRkjhH+wjJufB4PImJiT/8mYULF97/p4VCkmFPgh6GpMCRBC0MIwxnGowwwgFwQGIIBxH2AVCEvmnd6sGDB5+XTpAfQSNCQgg5JzNmzOB1J2+bMBpeN7QgJAmhAEI+ONPgqYBgsCQAHOEAGGBxwOrUmTxk7HVJma3+e1mpVnK0du0S7veY0peWiYKQEEIawdJ/LtFrygf0+hk8bvjcgIAWgh6G4AhUw54CWYYkQ7GASZCtUCweHe27X9SuSzefz9fQjpzmEuFQ2bwp3m1vgRsm9qjloCAkhJDGIcvy9vzN7qKDua4MBL2oLoahI+yHLQl15XCkQQgwCTYn7EmwJ8HigGopKq9JcLUdPHREZKJKsjlSrr0zc/qC4Fd7yhZMC+7/2OxuxT8KQkIIaUxpaWlFB/d9tWdbgt0KriPsh68aQkfIA4sDhgbDgKxAC0FwSCpUC6zObR99JiWm33HnnZFG1Ox2GbfPTblqcs3qpyuf+4NeftzcTsU3CkJCCGl83bp185QXrV+1VOFhhLzgAtwAA/QQuA5uQLUCDFYnrA6oNljskNXFK16XEtKeeeaZSCO2n1/smvWsrUvvk/+4z79no7k9imO0apTEOVo1Ss7Fmawa/VELFiyYPfdxAQmyDMEhq3CmQQ/D0KFYIDi0AMDAdQgBIaAFFUlsfW99//79Iy1wv0doYTk5/Zw7RL4DjQgJIaRp3X///byu4qaxl0HXIABI0IJQLOAawgEoNghAdcDigGKDYoHFoUMZMPLqpMzs4uJiAJIjsSEFhRYOHz9oZn/iDgUhIYScDytXrtBrynp2aY+QD/4a+KohBPQQfFVwpkLzQ5IhDMgKVGtkHY0nxNt0vbBDl26aduqme+73VL30F/eSR/SKYhO7E08oCAkh5DyRZfnTvbtOHv06I9mJsB9aENwA1+CrhiUBAuAcsgWGDiZBUiCrkC1Hy6osKVmjrrgy2khyumv2c5YO3U8+8bvat57nAa+5nYoDFISEEHJeZWZmVhR9U/DvTRZVhjAgGCCg+QABeyJCPsgKFDWahYoFigJFfX/HR8yZOmvWLABMVhKHXZs9+zke9JfPn+L78B1wbna3mjEKQkIIMUGvXr1ClSXLnlokcQ2hAPQQtBB0DVYnmIKw/9Tue6ZAtkJWIKsLnvqn5Eh55ZVXAEgJKam/uidj6lx/wdbyx+8MH/va7D41VxSEhBBimltuucWoO/m7KTdDCyHkQcgLLQBw2BIhKdBDAINqBWOQVMgWKBYhKzdMmW5NTN2/fz8AtXWHzDsXJI26Wa8oMbs3zRVtnyBxjrZPkHPRKNsnztDQEZfl7y4AY1AskK2QFEgMTIIegqFBsUIPQwDg4AZ0HRAZibZvDn593iqMVzQiJISQmLBl00ZfeVGbzDSEQwh5ofmhhRD2g3MoNjAJkgzGIMmQLbBYoaiVnmBSTrtf9Ol7ejs86CubO9m36z2YMc5pjigICSEkVjgcjuNHDhz5fI9dVaAFoQXBOfQQQl6EvNDDkCRIChjAFEgyVAtkpeDQMeZImTRpUqQRyeZMnzjbv2dT+V+nhw5/bm6PmgUKQkIIiS0dOnTwu0vefmWZBAPhAIQAN6IrSJkMAAIQBiQZTIakRv7qxdfXSo6kxYsXA1DbdMq8+/Gky26oXrXIvfTPurvU5C7FNgpCQgiJRVdeeaVRW/HQvXcyrgECWhCGBq5DD4EbgABjkFUwFt1xqKhCsd0x40HZkbx3714A9p4DXbOes7TpdHLRPbXrl4pQwOw+xShaLEPiHC2WIefifC6W+QGjx45954OdkABJARhkGZICSYoODQWPHlIa+YLrECLBIhcdOZCamgrAqKuqW79U6OG0W2eb3ZVYREFI4hwFITkXMRKEAHRd7/iz7kUlFZAkgEGxQggIA0wCYwCLjhGFAOfgBrgOzjvmZh8+QPsLfwQ9GiWEkGZAUZTCg/tLD++zqwo4hxaEMAAGWQEQvcs+Mi6UZMgyZBWKcqTkJHMkT5gw4T/aEqLuvRV6VbkJ3YhJFISEENJsZGdn+92lm956RWICehhcgx6GHoIsQ5IBASYBIhqQkgJZgWx5Y8NWZk9ctGhRtBXGmKyc/OtddW+/RBOHoCAkhJBmZ/jw4UZd5bwH7oMwoIcBCYYBLQRIYPX/qwsALLrLgimQrfc+PFd2JO7cuRNA4mU3uGYu1mtOls37jX/Pxha+45DmCEmcozlCci5iZ47w+4y96qr1H2wHi2SeBG5AtoABQgACgoEB4OCifh0Nt8k4cfRweno6gHDh/po1z4IbKeNut1zwM7N7Yw4aERJCSDO2bu1avbairSsdXIMRBgBw6Fp04QwTYAxMhiSDSZBVyGqQSxm5Hdtd0BGApV1e1j2LEgaPc780v/q1f5jbF7NQEBJCSPMmy3Lh4YOlh7+yqyoMA1oYgoMbkCRwDsEBQIjoyaXROFSKKmuZPemqq64CY44+Q7PnLHH0GWp2V8xBQUgIIfEgOzvbX1W28a1XpMiGQkODoUNwCEBwgIEDDNHdF5ISScR1m7czW8LChQuZxWrt0L2hNb2ytOVMHFIQEkJI/BgxYoThcz868x4AMDQIA1yHABgHIpvuAYmBAZIMSYEsQ7HOfGSeZE+IrKOJqHn9yZNP/C58bL9pPTmPaLEMiXO0WIaci9hfLPMDLrvssk0ffgRWf2cFAHBICrgAY9FcBIPgAIdhAMIqifIThcnJyRDC/9EHtW8vtXbskTz2f+SUDJM705RoREgIIfFp48aNhsftSk0EN6Br4Dq4gKGBCYADAGOQBCQJTI4c2xbiLCU7t137DmDM0Xd49pwlSkZ2+cI76t5fKcIhszvUVCgICSEkbkmSVHb86PEDnyuyFB35CQ5dh4g8KQUEA1D/pFSCrEBWiyqqmS1x3LhxzGJLuuJW131P6mVFZfOnBAq2md2hJkFBSAghca5NmzZaXcW6l5cxET12BpzD0AEDQgD8tESUIMmRXRZvbshntoS///3vcmpW2sTZabfcHzzwidldaRIUhIQQ0iKMGTOGe92/n3pb9J4KAIYBQwPnkRsOAQYwsMiy0shSGvWeWQ9JNuenn35q7fDz1Bv+1+xONAkKQkIIaUH++te/Cq/74gu7Ri81FAAXEIh+jcjySRbdgM9UyIqQLb37DbInJIdCp6YJq1ctqtu4Smhhc7rRqCgICSGkxdm1c6dWW5FstwKRicPITb8iOjRkIrqJUGZgDJIKWQkKyZac0blz50gLiSN/rRV/Ux6ZOGzmOw4pCAkhpCVSFKWmovTQZ3sl1E8ZCg7DiN7lhPotFpAgsYaLLA6XnGRW50033aSkudJveyD11/fVbX614h/3hY8fMrtDZ4+CkBBCWq5OnToZ3qrlz/4dAAwdEOAcYPX33aP+YWkkESXIKlTLqjffZlbHSy+9ZO14oevefzh+ebn7+YerVy0y6qrM7MzZoiAkhJCW7uabbxbeqpu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u_j$ for all $j$ (equality is only achieved for $j=0$).\n", "\n", ":::\n", "\n", "::: {#exm-logistic}\n", "# Logistic equation\n", "\n", "Recall that the logistic equation takes the form\n", "\n", "\\begin{align}\n", " &u(0) = u_0, \\\\\n", " &u'(t) = \\mu u(t) - \\kappa u(t)^2 \\quad \\text{on } (0,T).\n", "\\end{align}\n", "\n", "Last time, we saw that this problem is well-posed with solution\n", "\n", "\\begin{align}\n", " u(t) = \\frac{\\mu}{\\kappa u_0 (e^{\\mu t}-1) + \\mu } u_0 e^{\\mu t}\n", "\\end{align}" ] }, { "cell_type": "code", "execution_count": 12, "id": "92151263", "metadata": {}, "outputs": [ { "data": { "image/png": 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"output_type": "execute_result" } ], "source": [ "u0, μ, κ, T, n = 1, 1., .05, 15, 5\n", "f2(t,u) = μ * u - κ * u^2\n", "u = Euler( u0, f2, T, n )\n", "u_exact(t) = ( μ/( κ*u0*(exp(μ*t) - 1) + μ) ) * u0 * exp( μ * t )\n", "\n", "plot(u_exact, 0, T, \n", " title=L\"u'=\\mu u(t) - \\kappa u(t)^2\", legend=false,\n", " xlabel=L\"t\", ylabel=L\"u(t)\")\n", "\n", "scatter!( 0:T/n:T, u, \n", " label=\"Euler (n=$n)\")\n", "\n", "n=100\n", "u2 = Euler( u0, f2, T, n )\n", "scatter!( 0:T/n:T, u2, \n", " label=\"Euler (n=$n)\" )" ] }, { "cell_type": "markdown", "id": "8ae3bd28", "metadata": {}, "source": [ "Heres the error as a function of $n$:" ] }, { "cell_type": "code", "execution_count": 13, "id": "1a5f15ab", "metadata": {}, "outputs": [ { "data": { "image/png": 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What do expect to happen in Euler's method when *(i)* $n < 20$, *(ii)* $n = 20$, and *(iii)* $n > 20$?\n", "\n", ":::\n", "\n", "::: {#exm-sin}\n", "\n", "Consider the following (well-posed) linear IVP \n", "\n", "\\begin{align}\n", " &u(0) = 0, \\\\\n", " &u'(t) = t^2 - u \\sin(t) \\quad \\text{on } (0,10)\n", "\\end{align}\n", "\n", "We apply Euler's method to this equation:" ] }, { "cell_type": "code", "execution_count": 14, "id": "a2539ab2", "metadata": {}, "outputs": [ { "data": { "image/png": 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legend=false,\n", " xlabel=L\"t\", ylabel=L\"u(t)\")\n", "\n", "# Euler \n", "n=5\n", "u = Euler( u0, (t,u)->f3(u,0,t), T, n )\n", "scatter!(0:T/n:T, u)\n", "\n", "n=50\n", "u2 = Euler( u0, (t,u)->f3(u,0,t), T, n )\n", "scatter!(0:T/n:T, u2)" ] }, { "cell_type": "markdown", "id": "3945e993", "metadata": {}, "source": [ ":::\n", "\n", "## Error estimates\n", "\n", "We will now prove that, under some assumptions on $u$, Euler's method does indeed converge with rate $O(h)$.\n", "\n", "@Burden chapter 5.2, some of chapter 5.3" ] }, { "cell_type": "markdown", "id": "b94ee96d", "metadata": {}, "source": [ ":::{.callout-tip}\n", "# TO-DO\n", "\n", "* Read: @Burden chapter 5.1, 5.2, 5.3 - please come to the lecture next week with any questions you have,\n", "* Assignment 1: Due today,\n", "* (Sign-up to office hours if you would like)\n", "\n", ":::" ] } ], "metadata": { "kernelspec": { "display_name": "Julia 1.11", "language": "julia", "name": "julia-1.11" }, "language_info": { "file_extension": ".jl", "mimetype": "application/julia", "name": "julia", "version": "1.11.6" } }, "nbformat": 4, "nbformat_minor": 5 }