{ "cells": [ { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "## Math 157: Intro to Mathematical Software\n", "## UC San Diego, winter 2018" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "## Numerical Solutions to ODE" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "### Preview of Ordinary Differential Equations" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "An ordinary differential equation (frequently called an \"ODE\") is an equality involving a function and its derivatives. An ODE of order n is an equation of the form $F(x,y,y',...,y^{(n)})=0$ where y is a function of x, $y^{n}$ is the $n^{th}$ derivative of x." ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "### Sovling Differential Equations with Sage" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "**Example:** Solve $x'+x = 5, x(0) = 1$" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/html": [ "" ] }, "execution_count": 10, "metadata": { }, "output_type": "execute_result" } ], "source": [ "t = var('t')\n", "x = function('x', t)\n", "de = lambda y: diff(y,t) + y - 5\n", "x0 = 1\n", "soln = desolve(de(x),[x,t],[0,x0])\n", "show(soln)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "**Example:** Solve $x''-x' = 0$" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "_K2*e^(-t) + _K1*e^t" ] }, "execution_count": 13, "metadata": { }, "output_type": "execute_result" } ], "source": [ "t = var('t')\n", "de = lambda x: diff(x,t,t) - x\n", "x0 = 1\n", "x1 = 0\n", "desolve(de(x),[x,t])" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "**Example:** Solve for y in terms of t in $\\frac{1}{2}(ln(y-1) - ln(y+1)) = t+C$" ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "[log(y + 1) == -2*C - 2*t + log(y - 1)]" ] }, "execution_count": 17, "metadata": { }, "output_type": "execute_result" } ], "source": [ "C = var('C')\n", "y = var('y')\n", "solve([log(y - 1)/2 - (log(y + 1)/2) == t + C],y)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Note we did not get an expression of y directly; let's try a different input." ] }, { "cell_type": "code", "execution_count": 18, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "[y == -(e^(2*C + 2*t) + 1)/(e^(2*C + 2*t) - 1)]" ] }, "execution_count": 18, "metadata": { }, "output_type": "execute_result" } ], "source": [ "solve([log((y - 1)/(y + 1)) == 2*t + 2*C],y)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Solve the same equation by adding an initial condition: y(1) = 3" ] }, { "cell_type": "code", "execution_count": 19, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "[C == -1/2*log(2) - 1]" ] }, "execution_count": 19, "metadata": { }, "output_type": "execute_result" } ], "source": [ "solny=lambda t:(-e^(2*C + 2*t)-1)/(e^(2*C + 2*t)-1)\n", "solve([solny(1) == 3],C)" ] }, { "cell_type": "code", "execution_count": 20, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "-(e^(2*t - log(2) - 2) + 1)/(e^(2*t - log(2) - 2) - 1)" ] }, "execution_count": 20, "metadata": { }, "output_type": "execute_result" } ], "source": [ "C = -1/2*log(2) - 1\n", "solny(t)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "## Visualizing Solutions to ODE" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "The idea of **direction field** (or slope field) related to the first order ODE is similar to vector calculus. \n", "\n", "At each point $(x,y)$, we plot a small vector that has slope $f(x,y)$. A collection of all points $(x,y)$ where the vectors have slope m generate direction field." ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "**Example:** Newton's Equation (MATLAB assignment exercise 2.5)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Newton's law of cooling models the temperature change of an object at a certain temperature when placed in a surrounding environment of a different temperature. The law can be stated as follows: **$dy/dt = k(A - y)$** where k is a positive proportionality constant and A represents temperature of the environment." ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Plot a direction field with A = 1, k = 2, and where the minimum value of t is zero.(We start range of t from 0 because we do not need to worry about negative times)" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false, "scrolled": true }, "outputs": [ { "data": { "image/png": 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" }, "execution_count": 5, "metadata": { }, "output_type": "execute_result" } ], "source": [ "t,y=var('t,y')\n", "f(t,y) = 2-2*y\n", "v=plot_slope_field(f(t,y),(t,0,10),(y,-10,10),headaxislength=3, headlength=3)\n", "\n", "show(v)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "We can see that all solutions approach to 1 which is environmental temperature under our setting." ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "## Euler's Method" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "What if the differential equation cannot be solved with a nice formula? We introduce Euler's Method here. Euler's Method is an algorithmic way of plotting an approximate solution to an initial value problem through the direction field. \n", "\n", "\n", "A more detailed explanation to Euler's method could be found [here](https://en.wikipedia.org/wiki/Euler_method#Informal_geometrical_description). \n", "\n", "Recall from Calculus that $f(x,y)\\simeq \\frac{y(x+h)-y(x)}{h}$, solving for $y(x+h)$ we get: $y(x+h)\\simeq y(x)+h\\cdot f(x,y(x))$ where h is the step size.\n", "\n", "\n", "Note: the first order DE must be in the form of $y'=f(x,y), y(a)=c$ for this method to work." ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "**Example:** Use Euler’s method with $h = 1/2$ to approximate $y(1)$, where $y'-y=4x-4, y(0)=1$" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ " x y h*f(x,y)\n", " 1 1 1/3\n", " 4/3 4/3 8/9\n", " 5/3 20/9 44/27\n", " 2 104/27 212/81\n", " x y h*f(x,y)\n", " 0 1 -3/2\n", " 1/2 -1/2 -5/4\n", " 1 -7/4 -7/8\n" ] }, { "name": "stderr", "output_type": "stream", "text": [ "/ext/sage/sage-8.1/local/lib/python2.7/site-packages/sage/repl/ipython_kernel/__main__.py:5: DeprecationWarning: use the option 'algorithm' instead of 'method'\n", "See http://trac.sagemath.org/6094 for details.\n" ] }, { "data": { "image/png": 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" }, "execution_count": 4, "metadata": { }, "output_type": "execute_result" } ], "source": [ "x,y=PolynomialRing(QQ,2,\"xy\").gens()\n", "\n", "eulers_method(4*x+y-4,1,1,1/3,2)\n", "eulers_method(4*x+y-4,0,1,1/2,1)\n", "pts = eulers_method(4*x+y-4,0,1,1/2,1,method=\"none\")\n", "\n", "list_pts = list_plot(pts)\n", "line_pts = line(pts)\n", "show(list_pts+line_pts)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "### Further references" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "In order to improve acccuracy in Euler's method, we usually choose small step sizes. However, it is hard to avoid roundoff errors. If the step size is too small, we will build up so many roundoff errors into approximation. To solve the problem, we introduce an improved Euler's method, which is also called [Heun's Method](https://en.wikipedia.org/wiki/Heun%27s_method). The \"improved\" tangent line approximation at $(a,c)$ is: \n", "\n", " $y(a+h)\\simeq c+h\\cdot\\frac{m+m'}{2} = c+h\\cdot\\frac{f(a,c)+f(a+h,c+h\\cdot f(a,c))}{2}$" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Solving improved Euler's method using [sage](https://www.usna.edu/Users/math/wdj/_files/documents/teach/sm212/sm212-eulers_method-sage/sm212-eulers_method-sage.pdf)." ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "### Fourth-Order Runge-Kutta Method" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "With Euler's method, we are able to solve most cases. However, sometimes we encounter difficult cases that require more sophiscated methods. Thus, we introduce fourth-order Runge-Kutta method. This method involves computing four slopes and taking an average of them. A more detailed explanation could be found [here](http://lpsa.swarthmore.edu/NumInt/NumIntFourth.html)." ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "**Example:** $y' = -y + cos(x)$" ] }, { "cell_type": "code", "execution_count": 29, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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" }, "execution_count": 29, "metadata": { }, "output_type": "execute_result" } ], "source": [ "x = var('x')\n", "y = function('y')(x)\n", "DE = diff(y,x) == -y + cos(x)\n", "desolve_rk4(DE,y,ics=[0,1],end_points=[0,n(2*pi)],step=0.1,output='slope_field')" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "Resources: http://doc.sagemath.org/html/en/tutorial/tour_algebra.html\n", "https://wiki.sagemath.org/interact/diffeq#Euler.27s_Method.2C_Improved_Euler.2C_and_4th_order_Runge-Kutta_in_one_variable" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "## Exercises" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "1.Use Euler’s method to approximate $x(1)$, where $x''-3x'+2x=1, x(0)=0, x'(0)=1$" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "**Solution:** Let $x_1 = x, x_2 = x'$, we trasform given equations to standard form: $x_1' = x_2, x_2' = 1-2x_1+3x_2, x_1(0)=0, x_2(0)=1$.\n", "\n", "\n", "Take $h = \\frac{(1-0)}{3} = \\frac{1}{3}$" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "/ext/sage/sage-8.1/local/lib/python2.7/site-packages/sage/repl/ipython_kernel/__main__.py:5: DeprecationWarning: use the option 'algorithm' instead of 'method'\n", "See http://trac.sagemath.org/6094 for details.\n" ] }, { "name": "stdout", "output_type": "stream", "text": [ " t x h*f(t,x,y) y h*g(t,x,y)\n", " 0 0 0.35 1 1.4\n", " 1/3 0.35 0.88 2.5 2.8\n", " 2/3 1.3 2.0 5.5 6.5\n", " 1 3.3 4.5 12. 11.\n" ] }, { "data": { "image/png": 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FRjsAAHLjzTelZcvMkEpacc7kdbYTFRWlEiVK+CMWAADCTlKS1Lev1K2bdPfddkeDq/F6DVNSUpLcbrduuukmtW3bVrt37/ZHXAAAhLy0NKlDB6lMGWnoULujwbV4lTDVrl1bU6ZM0YIFCzR+/Hj9+uuvatiwoVJTU/0VHwAAIWvMGGn5cmnyZKlAAbujwbXkag5TcnKyypcvr1GjRqlDhw6XfT19DlNycjJjBQAA+INffpFq1JA6dpRGj7Y7GmQlVyu2o6Oj9Ze//EXbtm275v0qVaokl8slt9stt9stSQyxBACErfRW3A03SK+8Ync0yI5cJUwpKSnavn272rVrd837JSUlUWECAOB/Ro+WVq40QyppxQUHr9Yw9e7dW0uXLtXOnTu1YsUKtWrV6uKRJwAAIGtbt0r9+0s9ekj169sdDbLLqwrTnj171KZNGx09elQlSpRQ/fr1tWrVKhUrVsxf8QEAEDLSW3Fly0pDhtgdDbzhVcKUkJDgrzgAAAh5o0ZJq1aZIZX589sdDbzBWXIAAATAzz9LAwZITz8t1atndzTwVq7GCmSFsQIAAJhWXP360tGj0saNVJeCEQfBAQDgZyNHSqtXS99+S7IUrALSkouLi1NsbCxroAAAYWfLFmngQKlnT6luXbujQU7RkgMAwE/S0sx6pePHpQ0bpOuvtzsi5BQtOQAA/OT116U1a0wrjmQpuLFLDgAAP9i8OaMVV6eO3dEgt2jJAQDgYxcumPVKJ07QigsVtOQAAPCxESOkdeuk5ctJlkIFLTkAAHzop5+k55+X/vMfqXZtu6OBr9CSAwDARy5cMOuVUlOl9eul666zOyL4Ci05AAB85LXXTKK0ciXJUqhhcCUAAD7w44/SCy9IvXtLd95pdzTwNVpyAADk0vnzphV3+rRZ7E11KfTQkgMAIJdefdUcqksrLnSxSw4AgFz44Qdp8GDp2WelO+6wOxr4Cy05AABy6Px5Mzrg7FnTisuXz+6I4C+05AAAyKHhw6VNm6RVq0iWQh0tOQAAcuD776UXX5See066/Xa7o4G/0ZIDAMBL589Ld91l/ly7lupSOAhISy4uLk5RUVGKj49XfHx8IF4SAAC/GTrUVJhWryZZChdUmAAA8MKmTaYF16eP9NJLdkeDQCFhAgAgm86fN1O809JMKy5vXrsjQqCwSw4AgGx65RUzd+m770iWwg275AAAyIaNG6WXX5b69ZNuu83uaBBotOQAAMjCuXOmFWdZ0po1VJfCES05AACyMGSI9NNPtOLCGS05AACuYcMGs3apf3+pVi27o4FdaMkBAHAV586ZA3VdLqpL4Y7BlQAAXMXLL0ubN7NuCVSYAAC4ovXrzULvgQOl55+3OxrYjYQJAIBLnD1rWnGRkaYVlyeP3RHBbrla9D106FBFRESoZ8+evooHAADbvfSStGWLNGUKyRKMHCdMa9as0cSJE1WjRg1fxgMAgK3WrpWGDZMGDZL4FYd0OUqYUlJS1LZtW02aNEmFCxf2dUwAANji7FmpfXuTKPXpY3c0cJIcJUxdunRRy5Ytdc899/g6HgAAbPPii9Ivv9CKw+W8HiuQmJiojRs3au3atf6IBwAAW6xdKw0fLr3wglS9ut3RwGm8Spj27Nmjp59+Wl9++aXykHoDAELEH1txzz1ndzRwIq/GCsyaNUsPPPCAIiMjlf6wtLQ0uVwuRUZG6uzZs3K5XBfvnz5WoGTJknK5XHK73XK73ZLEEEsAgGP06yeNGCGtW0d1CVfmVcKUmpqqnTt3Zrqtffv2qlKlivr06aMqVapk+hpzmAAATvfdd1KdOmb9Uv/+dkcDp/KqJVegQAFVrVr1stuKFSt2WbIEAIDTnTljWnG1atGKw7Xl+iy5P7bgAAAIJi+8IG3fbo5BiQrI6aoIVrl+e3z99de+iAMAgIBavVp67TVzwG61anZHA6fjLDkAQNg5c8a04QoWlFasoLqErPEWAQCEneefl3bskDZsIFlC9vA2AQCElVWrzAiBIUOkS/YxAVcVkJZcixYtFBUVxewlAICtTp82rbjoaGn5cqpLyL6AvFUSExNZwwQAsN2gQdJvv9GKg/d4uwAAwsLKldLrr0vDhkmMDoS32CUHAAh5p09LNWtKRYqYVlxkpN0RIdhQYQIAhLyBA6WdO6VZs0iWkDMkTACAkLZihTRypDR8uFS5st3RIFjRkgMAhKxTp0wrrlgx6dtvqS4h56gwAQBC1oAB0u7d0pw5JEvIHRImAEBI+vZb6Y03zHlxN99sdzQIdgyuBACEnFOnpBo1pJIlpaVLqS4h9xhcCQAIOf37S3v2SF98QbIE36AlBwAIKcuWSaNHm/Pi/vIXu6NBqGCXHAAgZKSmml1xMTHSkiVUl+A7VJgAACGjXz9p715acfA9EiYAQEhYulQaM0YaNYpWHHyPlhwAIOilpkq33iqVKWNacRERdkeEUEOFCQAQ9Pr2lfbvlxYsIFmCfwTkbRUXF6fY2FglJCQE4uUAAGFkyRLpzTeloUOlihXtjgahipYcACBopaSYAZVut7R4MdUl+A8tOQBA0OrTRzpwgFYc/I+ECQAQlL75Rho71uyMoxUHfyNhAgAElcREaeFCafZs6e67pS5d7I4I4YCECQAQNKZNk/7v/zI+r1+fVhwCg7cZACBofPFF5s83brQnDoQfKkwAAMezLGn6dGnOnMy333abPfEg/JAwAQAc7ddfpc6dpfnzpQcfNMeerFsn3X67NHCg3dEhXAQkYYqLi1NUVJTi4+MVHx8fiJcEAAS5CxekN96QBg2Sihc3i7xbtrQ7KoQrBlcCABxnzRqpY0fp+++l7t2lF1+UCha0OyqEMxZ9AwAc4+RJqUcPqXZt8/nq1dKoUSRLsJ9XCdP48eNVo0YNRUdHKzo6WnXr1tX8+fP9FRsAIIzMni1VrSpNmiS9+qqpMt1+u91RAYZXCVPZsmU1fPhwrVu3TuvWrdM999yj+++/X1u2bPFXfACAELd3r1nMff/9UvXq0k8/Sb16SVFsS4KD5HoNU7FixTRixAh16NDhsq+xhgkAcDVpadL48VLfvlL+/NLo0VLr1pLLZXdkwOVyvIbJ4/EoMTFRp06dUp06dXwZEwAgxP3wg5nS3bWrFB8vbdkiPfQQyRKcy+uC548//qg6derozJkzKliwoGbOnKnKlSv7IzYAQIg5fdrseBsxQqpUSVq2zCROgNN53ZK7cOGCdu3apePHj+vTTz/VxIkTtXTp0ismTektuZIlS8rlcsntdsvtdksSM5kAIMx8+aXUqZO0Z480YID07LNSvnx2RwVkT67XMDVt2lQVK1bUuHHjLvsaa5gAAIcPSz17Sh9+KP3tb9KECWZaNxBMcr0HwePx6OzZs76IBQAQQixLmjJF+s9/zOeTJ0vt27NOCcHJq4Spf//+atGihcqWLauTJ09q6tSpWrJkiRYuXOiv+AAAQeiXX6Qnn5QWL5batpVGjpRKlLA7KiDnvEqYDh48qHbt2mn//v2Kjo7WrbfeqoULF+qee+7xV3wAgCBy7pw0fLg0ZIjkdksLF0pNm9odFZB7nCUHAPCJb781578lJZk23MCBZr4SEAo4Sw4AkCvHjpn2W4MGUnS0tH69NHQoyRJCC4PnAQA5YlnSRx+Zw3JPnZLGjjWJU2Sk3ZEBvkeFCQDgtd9+k/7+dykuTqpXz0zq7tyZZAmhKyAJU1xcnGJjY5WQkBCIlwMA+MmFC9Lrr0vVqpnjTWbNkj791CzwBkIZi74BANmydq1Z1L1xo9Stm/Tyy1LBgnZHBQQGLTkAwDWdPCk9/bR0112SxyOtXi2NHk2yhPDCom8AwFXNmSN16SIdPWrmKz39tBTFbw6EISpMAIDL7Nsn/etfUmysWa/0449mthLJEsIVCRMA4CKPRxo3TqpSRVq2TEpIkObOlSpUsDsywF4kTAAASaaKVL++GQ/w0EPSzz+bsQEclguQMAFA2Dt9WurXT6pVy0ztXrpUevttqUgRuyMDnIM5TAAQxr76Sqpe3cxWGjjQjAxo0MDuqADnYQ4TAIShw4elXr2kDz6Q7r5bmjBBuvlmu6MCnIv9DgAQRixLeu89s+PN45HeeUfq0IF1SkBWWMMEAGHil1+kxo1NgtS8uVnU/eijJEtAdpAwAUCIO3fOHGNy663m0NwFC6QPP5RKlrQ7MiB40JIDgBC2fLk5/23rVtOGGzRIyp/f7qiA4EOFCQBC0PHj0pNPmrlKBQtK69dLw4aRLAE5RYUJAEKIZUkffyz16CGlpkpvvik99ZQUGWl3ZEBwo8IEACFi507pH/8wU7rr1JE2b5a6diVZAnyBwZUAEOQuXJBGjpSqVpU2bZJmzpRmzJBuuMHuyIDQweBKAAhi69aZRd0bNphq0ssvS/y4BXyPlhwABKGUFOmZZ6Q775TS0qRVq6QxY0iWAH9h0TcABJnPP5e6dDHHmwwdahKnPHnsjgoIbSRMABAk9u+XuneXPvlEuvdeafFiqUIFu6MCwgMtOQBwOI9HGjdOqlxZWrpUmjZNmjePZAkIJBImAHCwH3+UGjSQOneW/v1vacsWKT6e89+AQCNhAgAHOn1a6t9fqlVLOnpUWrJEmjRJKlrU7siA8MQcJgBwmEWLzEG5I0ZIAwaY2UoNG9odFRDemMMEAA5x5IjUq5f0/vsmQZowwaxbAmA/dskBgM0syyRJvXqZBd6TJkkdOkgRLJoAHMOrf45Dhw7VnXfeqUKFCikmJkatWrXSL7/84q/YACDkJSVJTZpI7dubUQFbtkiPPUayBDiNV/8kly1bpm7dumn16tX66quvdP78eTVr1kynT5/2V3wAEJLOnZOGDJGqV5d+/VWaP1+aOlWKibE7MgBXkqs1TEeOHFHJkiW1dOlS1a9f/7Kvs4YJAC63YoU5/+3nn00b7vnnpfz57Y4KwLXkquh7/PhxuVwuFWWfKwBk6fhx6amnpHr1pAIFzMG5w4eTLAHBIMeLvi3L0tNPP6369euratWqvowJAEKKZZnjTLp3N4fmjhljBlFGRtodGYDsynHC1LlzZ23evFnLly/3ZTwAEFJ27jQH5X7xhfTPf0pvvindcIPdUQHwVo4Spq5du2ru3LlatmyZSpcuneX9K1WqJJfLJbfbLbfbLUmKj49XfHx8Tl4eABzvwgWTHA0cKBUuLM2YIbVqZXdUAHLK64Spa9eumjVrlpYsWaJy5cpl6zFJSUks+gYQNtavl554QtqwwVSXhgyR+BEIBDevFn137txZU6dO1bRp01SgQAEdPHhQBw8e1JkzZ/wVHwAEjZQUs+vtjjtMhWnlSlNlIlkCgp9XYwUiIiLkusIR2e+++67atWt32e2MFQAQLr74wizkPnxYeuEF6ZlnpDx57I4KgK941ZLzeDz+igMAgtL+/VKPHtLHH0vNmknffCPdeKPdUQHwNYbvA0AOeDzS+PFSlSrS4sVmSvf8+SRLQKgiYQIAL/30k9SggRlC+eCDZmJ3mzbSFVYsAAgRJEwAkE1nzkgDBki1aklHj5rK0jvvSBx2AIS+HA+uBIBw8vXX0pNPSrt2Sf36SX37Svny2R0VgEAJSIUpLi5OsbGxSkhICMTLAYDPHDkitW8vNW4slS4tbdxodsGRLAHhxauxAt5irACAYGVZ0ocfmvEAaWnSa69Jjz4qRbCQAQhL/NMHgEts2yY1bSq1a2f+3LJFevxxkiUgnPHPHwD+59w56ZVXpOrVpe3bpXnzpIQEqVQpuyMDYDcWfQOApBUrzKLuLVuknj2l55+XChSwOyoATkGFCUBYS042R5rUry9dd520dq306qskSwAyo8IEICxZlvTpp1L37tLJk9Ibb0hdukiRkXZHBsCJqDABCDu7dkmxsdK//y3dcYe0ebNJnEiWAFwNc5gAhI20NFNJqlpVWr/eVJg++0wqW9buyAA4HXOYAISFDRukJ54wiVLnztKQIVJ0tN1RAQgWtOQAhLSUFKlXL+n2283YgBUrpLfeIlkC4B0WfQMIWXPnmmrSwYORR0GdAAAQ/0lEQVSmotSrl5Qnj91RAQhGJEwAQs6BA1KPHtJHH5lJ3YsWSTfdZHdUAIIZLTkAIcPjkd5+W6pcWfr6a3MW3IIFJEsAco+ECUBI2LxZatjQTOt+4AHp55+l//s/yeWyOzIAoYCECUBQO3NGGjhQqllTOnxY+uYbafJkqVgxuyMDEEpYwwQgaH3zjako/fab1KeP1K+fOd4EAHyNwZUAgs7Ro1KHDtI990gxMdLGjdKLL5IsAfAfBlcCCBqWZRZy9+wpXbhgDsl97DEpgsUFAPyMHzMAgsL27VKzZlK7dlLjxtKWLWZyN8kSgEDgRw0AR9q1S/r9d+n8eWnoUOmWW6SkJDOMMjFRKlXK7ggBhBMWfQNwFI9HatNGmj5dioyUSpeW9u+XnnlGeuEFqUABuyMEEI5ImAA4yqxZJlmSpLQ0ad8+ae1aqVYte+MCEN5ImADYyrKkH3+U5s0zH8uWZf563rxmxhIA2ImECUDAnTghffWVSZDmz5f27JGuv96MCRg5Upo5U1q82Czofv11pnUDsF9Axgq0aNFCUVFRio+PV3x8vL9eDoBDpVeR5s41SdLy5WYswM03Sy1amI+GDTPmKHk85qiTQoWkcuXsjR0AJOYwAfCTP1aR5s2T9u6V8uc3VaT0JKlCBbujBIDsoSUHwCcsS/rhh4wEKb2KVLmy1Lq1SZAaNGAaN4Dg5PUcpmXLlik2NlZut1sRERGaPXu2P+ICEASSk6VPP5Uef1wqW1aqUcMcUVKwoDR6tLRjhxkwOXKk1LQpyRKA4OV1hSk1NVU1a9bUo48+qgcffNAfMQFwKMuSvv8+o4q0YgVVJADhweuEqXnz5mrevLkkyY/LnwA4RHKyWYs0d67Z0bZvn1mL1LixNGaMSZL+/Ge7owQA/2INE4BMrlZFqlJFiovLqCLly2d3pAAQOCRMAJScLH35ZcZcpH37zBEkjRtLb74pNW9OFQlAeCNhAsKQZUmbNmWuIqWlSVWrSvHxpopUvz5VJABIF5CEqVKlSnK5XHK73XK73ZLEEEsgwI4fz1xF2r8/o4r01lsmSSpf3u4oAcCZApIwJSUlMbgSCDDLkjZuzKgirVyZUUVq04YqEgB4I0djBbZt23Zxh9yOHTu0adMmFS1aVGXLlvV5gACyL72KlL6j7cAB6U9/MlWksWPNWiSqSADgPa+PRlmyZIkaNWok1yWnYT7yyCOaPHlypts4GgXwr6tVkapVyzh+pH59KW9euyMFgODGWXJAkDl2LPNapPQqUpMmJkFq3pwDawHA19glBzicx5O5irRqVUYVqW1bqkgAEAgkTIADHTsmLVyYUUU6eDCjivTf/1JFAoBAI2ECHODSKtLKlea2W26R2rUzVaR69agiAYBdArKGqUWLFoqKimL2EvAHV6oiFSyYeS0SG08BwBlY9A0EiMcjbdiQeS2SxyNVr56xo61uXapIAOBEtOQAP/r994wq0oIFGVWkpk2lCRNMFemGG+yOEgCQFRImwIfSq0hz55okafXqjCrSI49QRQKAYEXCBOTSH6tI8+dLhw5RRQKAUEPCBHjJ45HWr89Yi5ReRbr1VqlDh4wqUp48dkcKAPAVEiYgG44ezbwW6dAhqVAhU0V6+21TRXK77Y4SAOAvJEzAFXg80rp1GVWk777LXEW67z6pTh2qSAAQLkiYgP85etRUj9KrSIcPU0UCABgMrkTYSq8ipe9o++47ybKkGjUy5iJRRQIASAyuRJg5ciTzWqTDh6XoaFNFSp+uXaaM3VECAJyGlhxCmscjrV2beS2SZUk1a0qPP26SpNq1qSIBAK6NhAkh58iRzGuRjhzJqCJ17EgVCQDgPRImBL20tMxVpDVrMqpITzyRsRYpinc7ACCH+BWCoJReRZo716xJSq8iNWsmdepkqkilS9sdJQAgVJAwIShcrYpUq5Zps6WvRaKKBADwB369wLEOH868FunoUalwYVNFeuop6d57qSIBAAKDOUywhccjffqpdOKE9MADUpEipoq0Zk1GFWntWlNFuu22jLlId91FFQkAEHjMYYItHnlEev998/dSpaT69aVvvslcRUqfi1SqlL2xAgDA/9XhN+fOSfv3S3v2SHv3mj/37JF275Y++STjfgcOSBs3msXaVJEAAE7EryXkyKlTmZOgS5OivXulgwdNSy1d/vxS2bLmPLbrrpPOnMn42scfmzEAAAA4EQkTMrEsKTn5yknQHz8/dizz44oUkW64wXzcdpvUsmXG5263+TM6WnK5zP2XLzeTtpOTpb59SZYAAM7GGqYw4vGYnWdXqgb9MSk6dSrjMS6XFBOTOfG5NBFyu031CACAUEWFKUScP2/WC10rEdq3z9wvXVSUSXbSE5+aNS9PikqX5pw1AABImILA6dNXT4TS/37gQOb1Qtdfn5H03Hij1KDB5ZWhkiWliAj7vi8AAIIFCZONLMvMIbrWWqE9e6Tff8/8uMKFMxKfW2+V7rvv8pZZ4cIZ64UAAEDuMLjSTzwec77ZtVpke/dKKSmZHxcTkznxudK6oQIF7PmeAAAIVyz6zoELF0wL7FqJ0N69Zg5RushIqUyZKydC6X8vU0bKm9e+7wsAAFwZLblLnDmTkQBdrTp04ICpIKW77rqMxKd8ealevcuTopIlTdKUUwkJCWFTnfMXrqFvcB1zj2voG1zH3OMaZl9YVZhOnMh62OKRI5kfEx2ddYusaFH/rxeKjY3V7Nmz/fsiIY5r6Btcx9zjGvoG1zH3uIbZFxIVJssyZ5BlNWzx5MnMjytRIiPxqVPnyvOFCha053sCAADO4fhN5WlpJuFZvVqaMUMaM0Zq2TJBbdpId98t3XST2UJfooRUq5aZMN25szRlirRli6kQNWsmDRokTZsmLV0q7dhhWm+HDknr10vx8QkaN07q398cCtukiVS5cs6TpYSEBJ9eA0nau3evT5/PHzH6+jl9/Xy+voaS87/ncHwvBsN72+nX0B/PyXvRmc/p9Gvoj+fM8fNZfnL2rGX17JlsSbJefjn5ivc5c8aytm+3rKVLLWvaNMt69VXL6tHDsh580LLuusuy3G7Lioy0LFNDMh/58llW/vwtrYYNLatNG8vq3duyRo+2rE8/tazVqy1rzx7LOn/eu1hbtmzpg+/Yf89nWZYVExPj0+fzR4xOv46+voaW5fzvORzfi8Hw3nb6NfTHc/JedOZzOv0a+uM5c/p8OWrJWZalk5f2ty7x/PPSG2+ckCQNGHBCK1eaLfN792ZMpD56NPNj/vQns1PM7TaVo4YNzefpH+nrheLjLygx8cRVX/uPR3tkx4ULF3TixNWfz1u+fj7JXHOnx+j06+jrayg5/3sOx/diMLy3nX4N/fGcvBed+ZxOv4b+eM4rPV/BggXlymIxco4Wfacv5gYAAAh22dmclqOEKTsVppEjpcGDT0gqK2m3pkwppFatvH0lAAAA//JbhSk7LEsaNeqEevWK1rvvJqt9e/vHCgAAAOREWM1hAgAAyAnHjxUAAACwGwkTAABAFkiYAAAAskDC5BBjx45VhQoVdP3116t27dpas2bNVe87adIkNWzYUEWLFlXRokXVtGnTa94/XHhzDf8oMTFREREReuCBB/wcYXDw9jomJyerS5cuKlOmjK6//npVrlxZ8+fPD1C0zuTtNXzjjTdUuXJl5c+fX+XKlVPPnj119uzZAEUbXJYtW6bY2Fi53W5FRERwDto1eHutZs6cqWbNmqlkyZKKjo5W3bp1tXDhwgBF63x+TZgKFiyo5ORkFeRAtmuaPn26evXqpcGDB2vDhg2qUaOG7r33Xh259CTg/1myZInatGmjxYsXa9WqVSpbtqyaNWum/fv3Bzhy5/D2GqbbuXOnevfurYYNGwYoUmfz9jqeP39eTZo00a5duzRjxgxt3bpVEydOlNvtDnDkzuHtNZw2bZr69u2rwYMH6+eff9bkyZM1ffp09e/fP8CRB4fU1FTVrFlTY8eOzXIbeLjz9lotXbpUzZo107x587R+/Xo1atRILVu21KZNmwIQbRDI5YRx+MBdd91lde/e/eLnHo/Hcrvd1vDhw7P1+LS0NKtQoULWBx984K8QHS8n1zAtLc2qX7++NXnyZKt9+/ZWq1atAhGqo3l7HceNG2dVrFjRunDhQqBCdDxvr2HXrl2tJk2aZLqtV69eVoMGDfwaZyhwuVzWrFmz7A4jKOT0WlWrVs166aWX/BBR8KElZ7Pz589r3bp1aty48cXbXC6XmjRpopUrV2brOVJTU3X+/HkVLVrUX2E6Wk6v4eDBg1WyZEl16NAhEGE6Xk6u45w5c1SnTh117txZpUqVUvXq1TV06FB5PJ5Ahe0oObmGdevW1bp16y627Xbs2KG5c+fq73//e0BiBq7G+t+Q6nD93XKpHJ0lB985cuSI0tLSFBMTk+n2mJgYbd26NVvP8dxzz8ntdqtJkyb+CNHxcnINly9frnfffZdS8x/k5Dru2LFDX3/9tdq2bat58+YpKSlJnTt3VlpamgYMGBCIsB0lJ9cwPj5eR44cUf369WVZltLS0tSpUyc999xzgQgZuKrXXntNqampat26td2hOAIJk0NZlpWtnvOwYcP00UcfacmSJcqbN28AIgseV7uGKSkpevjhhzVx4kQVKVLEhsiCy7Xeix6PRzExMXr77bflcrlUq1Yt7d27VyNGjAjLhOlqrnUNFy9erFdeeUXjx4/XnXfeqW3btql79+4qXbo01xC2mTZtml566SXNnj1bxYsXtzscRyBhslnx4sUVGRmpgwcPZrr90KFDl/0v9VIjRozQq6++qkWLFqlatWr+DNPRvL2G27dv186dO9WyZUtZ/xt0n95Cyps3r7Zu3aoKFSr4P3CHycl7sXTp0sqbN2+mZKBKlSo6cOCALly4oKio8PoRk5NrOGjQILVr1+5ia7hatWpKSUnRk08+ScIEWyQmJqpjx4765JNP1KhRI7vDcQzWMNksT548+utf/6pFixZdvM2yLC1atEh169a96uNee+01DRkyRAsWLFCtWrUCEapjeXsNq1Spoh9++EEbN27Upk2btGnTJsXGxuqee+7Rpk2bVLZs2UCG7xg5eS/Wq1dP27Zty3Tb1q1bVbp06bBLlqScXcNTp04pIiLzj+KIiAhZlnUxoQcCJSEhQY899pgSEhLUvHlzu8NxFluWmiOT6dOnW9ddd5313nvvWVu2bLE6duxoFS1a1Dp06JBlWZb18MMPW3379r14/+HDh1v58uWzZsyYYR04cODiR0pKil3fgu28vYaXYpec4e113L17t1WoUCGre/fu1i+//GJ9/vnnVkxMjDV06FC7vgXbeXsNX3jhBSs6OtpKTEy0fv31V2vhwoVWxYoVrfj4eLu+BUdLSUmxNm7caG3YsMFyuVzWqFGjrI0bN1q7du2yOzTHyepa9enTx2rXrt3F+0+bNs3KkyePNW7cuEy/W5KTk+36FhyFhMkhxo4da5UvX9667rrrrNq1a1tr1qy5+LVGjRpZHTp0uPj5n//8ZysiIuKyj8GDB9sRumN4cw0vRcKUwdvruGrVKqtOnTrW9ddfb1WsWNEaNmyY5fF4Ah22o3hzDdPS0qwXX3zRqlSpkpU/f36rfPnyVrdu3fgldRWLFy+2XC7XZT//rvXvO1xlda3at29vNWrU6OL9//a3v13xdwvX1nBZFjVfAACAa2ENEwAAQBZImAAAALJAwgQAAJAFEiYAAIAskDABAABkgYQJAAAgCyRMAAAAWSBhAgAAyAIJEwAAQBZImAAAALJAwgQAAJAFEiYAAIAs/D/sibSva6GK7AAAAABJRU5ErkJggg==" }, "execution_count": 3, "metadata": { }, "output_type": "execute_result" } ], "source": [ "RR = RealField(sci_not=0, prec=4, rnd='RNDU')\n", "t, x, y = PolynomialRing(RR,3,\"txy\").gens()\n", "f = y\n", "g = 1-2*x+3*y\n", "L = eulers_method_2x2(f,g,0,0,1,1/3,1,method=\"none\")\n", "\n", "eulers_method_2x2(f,g, 0, 0, 1, 1/3, 1)\n", "\n", "P1 = list_plot([[p[0],p[1]] for p in L])\n", "P2 = line([[p[0],p[1]] for p in L])\n", "show(P1+P2)" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "2.Using the following function definition to complete Runge-Kutta method and Euler method(Note these two methods use different ways to calculate increment)" ] }, { "cell_type": "code", "execution_count": 0, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "def compare_methods(xstart,ystart,xfinish,f,nsteps = 10,tol = 10^(-5.0)):\n", " sol = [ystart]\n", " xvals = [xstart]\n", " h = N((xfinish-xstart)/nsteps)\n", " #your code goes here" ] }, { "cell_type": "markdown", "metadata": { "collapsed": false }, "source": [ "**Solution**" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [ ], "source": [ "def compare_methods(xstart,ystart,xfinish,f,nsteps = 10,tol = 10^(-5.0)):\n", " sol = [ystart]\n", " xvals = [xstart]\n", " h = N((xfinish-xstart)/nsteps)\n", " for step in range(nsteps):\n", " k1 = f(xvals[-1],sol[-1])\n", " k2 = f(xvals[-1] + h/2,sol[-1] + k1*h/2)\n", " k3 = f(xvals[-1] + h/2,sol[-1] + k2*h/2)\n", " k4 = f(xvals[-1] + h,sol[-1] + k3*h)\n", " ek2 = f(xvals[-1]+h,sol[-1]+k1*h)\n", " #Runge-Kutta\n", " sol.append(sol[-1] + h*(k1+2*k2+2*k3+k4)/6)\n", " #Euler's method\n", " sol.append(sol[-1] + h*(k1+ek2)/2)\n", " xvals.append(xvals[-1] + h)\n", " return zip(xvals,sol)" ] } ], "metadata": { "kernelspec": { "display_name": "SageMath 8.1", "name": "sage-8.1" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.14" } }, "nbformat": 4, "nbformat_minor": 0 }