{ "nbformat": 4, "nbformat_minor": 0, "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": "3.7.9" }, "colab": { "name": "601_Linear Shooting Method.ipynb", "provenance": [], "include_colab_link": true } }, "cells": [ { "cell_type": "markdown", "metadata": { "id": "view-in-github", "colab_type": "text" }, "source": [ "\"Open" ] }, { "cell_type": "markdown", "metadata": { "id": "Z_EdXluqzGCI" }, "source": [ "# Linear Shooting Method\n", "#### John S Butler john.s.butler@tudublin.ie [Course Notes](https://johnsbutler.netlify.com/files/Teaching/Numerical_Analysis_for_Differential_Equations.pdf) [Github](https://github.com/john-s-butler-dit/Numerical-Analysis-Python)" ] }, { "cell_type": "markdown", "metadata": { "id": "TQlbwgeIzGCL" }, "source": [ "## Overview\n", "This notebook illustates the implentation of a linear shooting method to a linear boundary value problem. The video below walks through the code." ] }, { "cell_type": "code", "metadata": { "id": "wCoK3RFEzGCM", "colab": { "base_uri": "https://localhost:8080/", "height": 336 }, "outputId": "2a8772f5-a4ca-4c1b-afe5-2cacc6754eb5" }, "source": [ "from IPython.display import HTML\n", "HTML('')" ], "execution_count": 2, "outputs": [ { "output_type": "execute_result", "data": { "text/html": [ "" ], "text/plain": [ "" ] }, "metadata": {}, "execution_count": 2 } ] }, { "cell_type": "markdown", "metadata": { "id": "SGMwRqi6zGCO" }, "source": [ "## Introduction\n", "To numerically approximate the Boundary Value Problem\n", " \\begin{equation}\n", "y^{''}=p(x)y^{'}+q(x)y+r(x) \\ \\ \\ a < x < b \\end{equation}\n", "with the left and right boundary conditions:\n", " \\begin{equation}y(a)=\\alpha, \\end{equation}\n", " \\begin{equation}y(b) =\\beta. \\end{equation}\n", "\n", "The Boundary Value Problem is divided into two\n", "Initial Value Problems:\n", "1. The first 2nd order Initial Value Problem is the same as the original Boundary Value Problem with an extra initial condtion $y_1^{'}(a)=0$. \n", "\\begin{equation}\n", " y^{''}_1=p(x)y^{'}_1+q(x)y_1+r(x), \\ \\ y_1(a)=\\alpha, \\ \\ \\color{green}{y^{'}_1(a)=0},\\\\\n", "\\end{equation}\n", "2. The second 2nd order Initial Value Problem is the homogenous form of the original Boundary Value Problem, by removing $r(x)$, with the initial condtions $y_2(a)=0$ and $y_2^{'}(a)=1$.\n", "\n", "\\begin{equation}\n", "y^{''}_2=p(x)y^{'}_2+q(x)y_2, \\ \\ \\color{green}{y_2(a)=0, \\ \\ y^{'}_2(a)=1}.\n", "\\end{equation}\n", "\n", "combining these intial values problems together to get the unique solution \n", "\\begin{equation}\n", "y(x)=y_1(x)+\\frac{\\beta-y_1(b)}{y_2(b)}y_2(x),\n", "\\end{equation}\n", "provided that $y_2(b)\\not=0$.\n", "\n", "The truncation error for the shooting method is \n", "$$ |y_i - y(x_i)| \\leq K h^n\\left|1+\\frac{y_{2 i}}{y_{2 i}}\\right| $$\n", "$O(h^n)$ is the order of the numerical method used to approximate the solution of the Initial Value Problems.\n" ] }, { "cell_type": "code", "metadata": { "id": "xEz-lWFwzGCP" }, "source": [ "import numpy as np\n", "import math\n", "import matplotlib.pyplot as plt\n", "import pandas as pd\n", "import warnings\n", "warnings.filterwarnings(\"ignore\")\n", "\n" ], "execution_count": 3, "outputs": [] }, { "cell_type": "markdown", "metadata": { "id": "hPp1gJRtzGCP" }, "source": [ "## Example Boundary Value Problem\n", "To illustrate the shooting method we shall apply it to the Boundary Value Problem:\n", " \\begin{equation}y^{''}=2y^{'}+3y-6, \\end{equation}\n", "with boundary conditions\n", " \\begin{equation}y(0) = 3, \\end{equation}\n", " \\begin{equation}y(1) = e^3+2, \\end{equation}\n", "with the exact solution is \n", " \\begin{equation}y=e^{3x}+2. \\end{equation}\n", "The __boundary value problem__ is broken into two second order __Initial Value Problems:__\n", "1. The first 2nd order Intial Value Problem is the same as the original Boundary Value Problem with an extra initial condtion $u^{'}(0)=0$.\n", "\\begin{equation}\n", "u^{''} =2u'+3u-6, \\ \\ \\ \\ u(0)=3, \\ \\ \\ \\color{green}{u^{'}(0)=0}\n", "\\end{equation}\n", "2. The second 2nd order Intial Value Problem is the homogenous form of the original Boundary Value Problem with the initial condtions $w^{'}(0)=0$ and $w^{'}(0)=1$.\n", "\\begin{equation}\n", "w^{''} =2w^{'}+3w, \\ \\ \\ \\ \\color{green}{w(0)=0}, \\ \\ \\ \\color{green}{w^{'}(0)=1}\n", "\\end{equation}\n", "\n", "combining these results of these two intial value problems as a linear sum \n", "\\begin{equation}\n", "y(x)=u(x)+\\frac{e^{3x}+2-u(1)}{w(1)}w(x)\n", "\\end{equation}\n", "gives the solution of the Boundary Value Problem." ] }, { "cell_type": "markdown", "metadata": { "id": "IEZioK-qzGCQ" }, "source": [ "## Discrete Axis\n", "The stepsize is defined as\n", " \\begin{equation}h=\\frac{b-a}{N}, \\end{equation}\n", "here it is \n", " \\begin{equation}h=\\frac{1-0}{10}, \\end{equation}\n", "giving \n", " \\begin{equation}x_i=0+0.1 i, \\end{equation}\n", "for $i=0,1,...10.$\n", "\n" ] }, { "cell_type": "code", "metadata": { "id": "VDgbCuF3zGCR", "colab": { "base_uri": "https://localhost:8080/", "height": 281 }, "outputId": "af35eeeb-52cd-4041-a61c-dbbe8e6a19a5" }, "source": [ "## BVP\n", "N=10\n", "h=1/N\n", "x=np.linspace(0,1,N+1)\n", "fig = plt.figure(figsize=(10,4))\n", "plt.plot(x,0*x,'o:',color='red')\n", "plt.xlim((0,1))\n", "plt.title('Illustration of discrete time points for h=%s'%(h))\n", "\n", "plt.show()" ], "execution_count": 4, "outputs": [ { "output_type": "display_data", "data": { "image/png": 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"text/plain": [ "
" ] }, "metadata": { "needs_background": "light" } } ] }, { "cell_type": "markdown", "metadata": { "id": "koRQxasszGCS" }, "source": [ "## Initial conditions\n", "The initial conditions for the discrete equations are:\n", "$$ u_1[0]=3$$\n", "$$ \\color{green}{u_2[0]=0}$$\n", "$$ \\color{green}{w_1[0]=0}$$\n", "$$ \\color{green}{w_2[0]=1}$$" ] }, { "cell_type": "code", "metadata": { "id": "-HG-ZDhCzGCT" }, "source": [ "U1=np.zeros(N+1)\n", "U2=np.zeros(N+1)\n", "W1=np.zeros(N+1)\n", "W2=np.zeros(N+1)\n", "\n", "U1[0]=3\n", "U2[0]=0\n", "\n", "W1[0]=0\n", "W2[0]=1" ], "execution_count": 5, "outputs": [] }, { "cell_type": "markdown", "metadata": { "id": "w1IrA-VOzGCT" }, "source": [ "## Numerical method\n", "The Euler method is applied to numerically approximate the solution of the system of the two second order initial value problems they are converted in to two pairs of two first order initial value problems:\n", "\n", "### 1. Inhomogenous Approximation\n", "The plot below shows the numerical approximation for the two first order Intial Value Problems \n", "\\begin{equation}\n", "u_1^{'} =u_2, \\ \\ \\ \\ u_1(0)=3,\n", "\\end{equation}\n", "\\begin{equation}\n", "u_2^{'} =2u_2+3u_1-6, \\ \\ \\ \\color{green}{u_2(0)=0},\n", "\\end{equation}\n", "\n", "that Euler approximate of the inhomogeneous two Initial Value Problems is :\n", " \\begin{equation}u_{1}[i+1]=u_{1}[i] + h u_{2}[i] \\end{equation}\n", " \\begin{equation}u_{2}[i+1]=u_{2}[i] + h (2u_{2}[i]+3u_{1}[i] -6) \\end{equation}\n", "with $u_1[0]=3$ and $\\color{green}{u_2[0]=0}$." ] }, { "cell_type": "code", "metadata": { "id": "PWmcmft2zGCT" }, "source": [ "for i in range (0,N):\n", " U1[i+1]=U1[i]+h*(U2[i])\n", " U2[i+1]=U2[i]+h*(2*U2[i]+3*U1[i]-6)\n" ], "execution_count": 6, "outputs": [] }, { "cell_type": "markdown", "metadata": { "id": "Wvn3PlEgzGCU" }, "source": [ "### Plots\n", "The plot below shows the Euler approximation of the two intial value problems $u_1$ on the left and $u2$ on the right." ] }, { "cell_type": "code", "metadata": { "id": "X3xC3lzTzGCU", "colab": { "base_uri": "https://localhost:8080/", "height": 265 }, "outputId": "2532071d-8c06-46b2-f817-488533ebcab1" }, "source": [ "fig = plt.figure(figsize=(12,4))\n", "ax = fig.add_subplot(1,2,1)\n", "plt.plot(x,U1,'^')\n", "#plt.title(r\"$u_1'=u_2, \\ \\ u_1(0)=3$\",fontsize=16)\n", "plt.grid(True)\n", "\n", "ax = fig.add_subplot(1,2,2)\n", "plt.plot(x,U2,'v')\n", "#plt.title(\"U2\", fontsize=16)\n", "\n", "plt.grid(True)\n", "plt.show()" ], "execution_count": 7, "outputs": [ { "output_type": "display_data", "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" } } ] }, { "cell_type": "markdown", "metadata": { "id": "AtAXJZd7zGCV" }, "source": [ "### 2. Homogenous Approximation\n", "The homogeneous Bounday Value Problem is divided into two first order Intial Value Problems\n", "\\begin{equation}\n", "w_1^{'} =w_2, \\ \\ \\ \\ \\color{green}{w_1(0)=0}\n", "\\end{equation}\n", "\\begin{equation}\n", "w_2^{'} =2w_2+3w_1, \\ \\ \\ \\color{green}{w_2(0)=1}\n", "\\end{equation}\n", "\n", "The Euler approximation of the homogeneous of the two Initial Value Problem is \n", " \\begin{equation}w_{1}[i+1]=w_{1}[i] + h w_{2}[i] \\end{equation}\n", " \\begin{equation}w_{2}[i+1]=w_{2}[i] + h (2w_{2}[i]+3w_{1}[i]) \\end{equation}\n", "with $\\color{green}{w_1[0]=0}$ and $\\color{green}{w_2[0]=1}$." ] }, { "cell_type": "code", "metadata": { "id": "dwDtmXUuzGCV" }, "source": [ "for i in range (0,N):\n", " W1[i+1]=W1[i]+h*(W2[i])\n", " W2[i+1]=W2[i]+h*(2*W2[i]+3*W1[i])" ], "execution_count": 8, "outputs": [] }, { "cell_type": "markdown", "metadata": { "id": "6LLstTCjzGCW" }, "source": [ "### Homogenous Approximation\n", "\n", "### Plots\n", "The plot below shows the Euler approximation of the two intial value problems $u_1$ on the left and $u2$ on the right." ] }, { "cell_type": "code", "metadata": { "id": "2qCGNhs-zGCW", "colab": { "base_uri": "https://localhost:8080/", "height": 283 }, "outputId": "87e9b77f-1e80-4998-9796-71c6543f12b8" }, "source": [ "fig = plt.figure(figsize=(12,4))\n", "ax = fig.add_subplot(1,2,1)\n", "plt.plot(x,W1,'^')\n", "plt.grid(True)\n", "plt.title(\"w_1'=w_2, w_1(0)=0\",fontsize=16)\n", "\n", "ax = fig.add_subplot(1,2,2)\n", "plt.plot(x,W2,'v')\n", "plt.grid(True)\n", "plt.title(\"w_2'=2w_2+3w_1, w_2(0)=1\",fontsize=16)\n", "plt.tight_layout()\n", "plt.subplots_adjust(top=0.85)\n", "\n", "plt.show()\n", "beta=math.exp(3)+2\n", "y=U1+(beta-U1[N])/W1[N]*W1" ], "execution_count": 9, "outputs": [ { "output_type": "display_data", "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" } } ] }, { "cell_type": "markdown", "metadata": { "id": "tvIxL2tuzGCW" }, "source": [ "## Approximate Solution\n", "Combining together the numerical approximation of $u_1$ and $w_1$ as a weighted sum \n", " \\begin{equation}y(x[i])\\approx u_{1}[i] + \\frac{e^3+2-u_{1}[N]}{w_1[N]}w_{1}[i] \\end{equation}\n", "gives the approximate solution of the Boundary Value Problem.\n", "\n", "\n", "The truncation error for the shooting method using the Euler method is \n", " \\begin{equation} |y_i - y(x[i])| \\leq K h\\left|1+\\frac{w_{1}[i]}{u_{1}[i]}\\right| \\end{equation}\n", "$O(h)$ is the order of the method.\n", "\n", "The plot below shows the approximate solution of the Boundary Value Problem (left), the exact solution (middle) and the error (right) " ] }, { "cell_type": "code", "metadata": { "id": "p3Q5hFqszGCX", "colab": { "base_uri": "https://localhost:8080/", "height": 170 }, "outputId": "ddbef255-6b3b-4723-dbd0-fe4eb0c69d3c" }, "source": [ "Exact=np.exp(3*x)+2\n", "fig = plt.figure(figsize=(12,4))\n", "ax = fig.add_subplot(2,3,1)\n", "plt.plot(x,y,'o')\n", "\n", "plt.grid(True)\n", "plt.title(\"Numerical\",\n", " fontsize=16)\n", "\n", "ax = fig.add_subplot(2,3,2)\n", "plt.plot(x,Exact,'ks-')\n", "\n", "plt.grid(True)\n", "plt.title(\"Exact\",\n", " fontsize=16)\n", "\n", "ax = fig.add_subplot(2,3,3)\n", "plt.plot(x,abs(y-Exact),'ro')\n", "plt.grid(True)\n", "plt.title(\"Error \",fontsize=16)\n", " \n", "plt.tight_layout()\n", "plt.subplots_adjust(top=0.85)\n", "plt.show()" ], "execution_count": 10, "outputs": [ { "output_type": "display_data", "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" } } ] }, { "cell_type": "markdown", "metadata": { "id": "CUmD6pdTzGCX" }, "source": [ "### Data\n", "The Table below shows that output for $x$, the Euler numerical approximations $U1$, $U2$, $W1$ and $W2$ of the system of four Intial Value Problems, the shooting methods approximate solution $y_i=u_{1 i} + \\frac{e^3+2-u_{1}(x_N)}{w_1(x_N)}w_{1 i}$ and the exact solution of the Boundary Value Problem." ] }, { "cell_type": "code", "metadata": { "id": "dn0w4Cn4zGCX", "colab": { "base_uri": "https://localhost:8080/", "height": 359 }, "outputId": "c0de42c8-6fc6-42cc-fca1-9f152de60dd0" }, "source": [ "d = {'time x_i': x[0:10], \n", " 'U1':np.round(U1[0:10],3),\n", " 'U2':np.round(U2[0:10],3),\n", " 'W1':np.round(W1[0:10],3),\n", " 'W2':np.round(W2[0:10],3),\n", " 'y_i':np.round(W2[0:10],3),\n", " 'y(x_i)':np.round(W2[0:10],3)\n", " }\n", "df = pd.DataFrame(data=d)\n", "df" ], "execution_count": 11, "outputs": [ { "output_type": "execute_result", "data": { "text/html": [ "
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time x_iU1U2W1W2y_iy(x_i)
00.03.0000.0000.0001.0001.0001.000
10.13.0000.3000.1001.2001.2001.200
20.23.0300.6600.2201.4701.4701.470
30.33.0961.1010.3671.8301.8301.830
40.43.2061.6500.5502.3062.3062.306
50.53.3712.3420.7812.9322.9322.932
60.63.6053.2221.0743.7533.7533.753
70.73.9274.3471.4494.8264.8264.826
80.84.3625.7951.9326.2266.2266.226
90.94.9427.6632.5548.0508.0508.050
\n", "
" ], "text/plain": [ " time x_i U1 U2 W1 W2 y_i y(x_i)\n", "0 0.0 3.000 0.000 0.000 1.000 1.000 1.000\n", "1 0.1 3.000 0.300 0.100 1.200 1.200 1.200\n", "2 0.2 3.030 0.660 0.220 1.470 1.470 1.470\n", "3 0.3 3.096 1.101 0.367 1.830 1.830 1.830\n", "4 0.4 3.206 1.650 0.550 2.306 2.306 2.306\n", "5 0.5 3.371 2.342 0.781 2.932 2.932 2.932\n", "6 0.6 3.605 3.222 1.074 3.753 3.753 3.753\n", "7 0.7 3.927 4.347 1.449 4.826 4.826 4.826\n", "8 0.8 4.362 5.795 1.932 6.226 6.226 6.226\n", "9 0.9 4.942 7.663 2.554 8.050 8.050 8.050" ] }, "metadata": {}, "execution_count": 11 } ] }, { "cell_type": "code", "metadata": { "id": "PDuD6izhzGCY" }, "source": [ "" ], "execution_count": 11, "outputs": [] } ] }