Spectral1.ipynb

{
 "metadata": {
  "name": "Spectral1"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Spectral solution of 2-point boundary value problems using Chebfun"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Developed by Randy LeVeque for a course on Approximation Theory and Spectral Methods at the University of Washington.\n",
      "\n",
      "See <http://faculty.washington.edu/rjl/classes/am590a2013/codes.html> for more IPython Notebook examples."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This notebook demonstrates using the Matlab Chebfun package to solve 2-point boundary value problems via a Chebyshev spectral method, following the examples in Chapter 21 of \n",
      "\n",
      "* [ATAP] L.N. Trefethen, \"Approximation Theory and Approximation Practice\", <http://www.ec-securehost.com/SIAM/OT128.html>\n",
      "\n",
      "For more about how to do this, see \n",
      "\n",
      "* <http://www.mathworks.com/matlabcentral/fileexchange/23972-chebfun/content/chebfun/guide/html/guide7.html>"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Using Matlab and Chebfun from IPython..."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "First we need to set things up to use Matlab and set the path to find chebfun.  \n",
      "\n",
      "You must have pymatbridge installed and chebfun available for this to work."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import pymatbridge\n",
      "ip = get_ipython()\n",
      "pymatbridge.load_ipython_extension(ip)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 111
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%matlab\n",
      "path(path,'/Users/rjl/chebfun/chebfun_v4.2.2889/chebfun')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 112
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Setting up the problem"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Define the chebfun x and the linear operator $\\partial_x^2 + I$ with Dirichlet boundary conditions:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%matlab\n",
      "x = chebfun('x');\n",
      "L = chebop(@(u) diff(u,2) +  u)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "display_data",
       "text": [
        "L = chebop\n",
        "   Linear operator operating on chebfuns defined on:          \n",
        "     interval [-1,1]                                          \n",
        "   representing the operator:                                 \n",
        "     @(u)diff(u,2)+u                                          \n",
        "   with n = 6 realization:                                    \n",
        "     42.6000  -68.3607   40.8276  -23.6393   17.5724   -8.0000\n",
        "     21.2859  -30.5331   12.6833   -3.6944    2.2111   -0.9528\n",
        "     -1.8472    7.3167   -9.0669    5.7889   -1.9056    0.7141\n",
        "      0.7141   -1.9056    5.7889   -9.0669    7.3167   -1.8472\n",
        "     -0.9528    2.2111   -3.6944   12.6833  -30.5331   21.2859\n",
        "     -8.0000   17.5724  -23.6393   40.8276  -68.3607   42.6000\n"
       ]
      }
     ],
     "prompt_number": 113
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now let's specify some boundary conditions:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%matlab\n",
      "uleft = 3.;\n",
      "uright = 4;\n",
      "L.lbc = uleft;\n",
      "L.rbc = uright;"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 114
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Note that this changes the $6\\times 6$ matrix shown: the first 4 rows now map function values at 6 Chebyshev points to values of $Lu$ at 4 Chebyshev points..."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%matlab\n",
      "L"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "display_data",
       "text": [
        "L = chebop\n",
        "   Linear operator operating on chebfuns defined on:          \n",
        "     interval [-1,1]                                          \n",
        "   representing the operator:                                 \n",
        "     @(u)diff(u,2)+u                                          \n",
        "   left boundary condition:                                   \n",
        "     3                                                        \n",
        "   right boundary condition:                                  \n",
        "     4                                                        \n",
        "   with n = 6 realization:                                    \n",
        "     33.0353  -51.2657   27.8620  -14.2070   10.1514   -4.5760\n",
        "     -0.6075    5.7509   -9.1443    6.9332   -3.2805    1.3482\n",
        "      1.3482   -3.2805    6.9332   -9.1443    5.7509   -0.6075\n",
        "     -4.5760   10.1514  -14.2070   27.8620  -51.2657   33.0353\n",
        "      1.0000         0         0         0         0         0\n",
        "           0         0         0         0         0    1.0000\n"
       ]
      }
     ],
     "prompt_number": 115
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Specify the right hand side in the ODE $u''(x) + u(x) = f(x)$:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%matlab\n",
      "f = 0*x"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "display_data",
       "text": [
        "f = \n",
        "   chebfun column (1 smooth piece)\n",
        "       interval       length   endpoint values   \n",
        "[      -1,       1]        1        0        0   \n",
        "vertical scale =   0 \n"
       ]
      }
     ],
     "prompt_number": 116
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Solve the equation using Chebfun:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%matlab\n",
      "g = L\\f\n",
      "plot(g,'r')\n",
      "title('The Chebfun solution')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "display_data",
       "text": [
        "g = \n",
        "   chebfun column (1 smooth piece)\n",
        "       interval       length   endpoint values   \n",
        "[      -1,       1]       29        3        4   \n",
        "vertical scale = 6.5 \n"
       ]
      },
      {
       "output_type": "display_data",
       "png": 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      }
     ],
     "prompt_number": 117
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Compare to the exact solution:"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "In this case we know the exact solution is a linear combination of $\\sin(x)$ and $\\cos(x)$ and the boundary conditions give us a $2\\times 2$ linear system to solve for the coefficients:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%matlab\n",
      "A = [cos(-1) sin(-1); cos(1) sin(1)];\n",
      "b = [uleft; uright];\n",
      "c = A\\b;\n",
      "u = c(1)*cos(x) + c(2)*sin(x);\n",
      "error = max(abs(u-g))\n",
      "plot(u)\n",
      "title('The exact solution')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "display_data",
       "text": [
        "error =\n",
        "     1.232273452396662e-11\n"
       ]
      },
      {
       "output_type": "display_data",
       "png": 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      }
     ],
     "prompt_number": 118
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Airy equation"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Next we solve the problem in (21.4) of [ATAP]: $$ u'' - xu = 0, \\quad u(-30) = 1, ~u(30) = 0.$$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%matlab\n",
      "L = chebop(@(x,u) diff(u,2) - x.*u, [-30,30]);\n",
      "L.lbc = 1; L.rbc = 0; u = L\\0;\n",
      "plot(u)\n",
      "u"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "display_data",
       "text": [
        "u = \n",
        "   chebfun column (1 smooth piece)\n",
        "       interval       length   endpoint values   \n",
        "[     -30,      30]      184        1  2.2e-16   \n",
        "vertical scale = 6.1 \n"
       ]
      },
      {
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAkAAAAGwCAIAAADOgk3lAAAACXBIWXMAAAsSAAALEgHS3X78AAAA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\nAEQSMAAi/R8ZmgK06g3+JwAAAABJRU5ErkJggg==\n"
      }
     ],
     "prompt_number": 119
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "The nonlinear pendulum"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Finally we solve (21.5a) and (21.5b), the nonlinear pendulum problem\n",
      "$$ \\theta'' + \\sin(\\theta) = 0, \\quad \\theta\\in [0,6] $$\n",
      "with boundary conditions\n",
      "$$ u(0) = -\\pi/2, \\quad u(6) = \\pi/2.$$\n"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%matlab\n",
      "N = chebop(0,6);\n",
      "N.op = @(theta) diff(theta,2) + sin(theta);\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 120
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%matlab\n",
      "N.lbc = -pi/2;\n",
      "N.rbc = pi/2;\n",
      "theta = N\\0"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "display_data",
       "text": [
        "theta = \n",
        "   chebfun column (1 smooth piece)\n",
        "       interval       length   endpoint values   \n",
        "[       0,       6]       70     -1.6      1.6   \n",
        "vertical scale = 5.4 \n"
       ]
      }
     ],
     "prompt_number": 121
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%matlab\n",
      "plot(-cos(theta)), grid on, ylim([-1 1])\n",
      "title('Nonlinear pendulum (21.5)','fontsize',20)\n",
      "xlabel('t','fontsize',20), ylabel('height -cos(\\theta)','fontsize',20)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "display_data",
       "png": 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ORK5CYII=\n"
      }
     ],
     "prompt_number": 122
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "With different boundary conditions:\n",
      "\n"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%matlab\n",
      "N.lbc = -pi/2; N.rbc = 5*pi/2; theta = N\\0;\n",
      "plot(-cos(theta)), grid on, ylim([-1 1])\n",
      "title('Nonlinear pendulum (21.5), another solution','fontsize',20)\n",
      "xlabel('t','fontsize',20), ylabel('height -cos(\\theta)','fontsize',20)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "display_data",
       "png": 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5uxzrm8N9eUQtEADO604muSv+T9P0v1BSDbL9jhBaeQCHhM/r/NXhv9Er8NCm\nMn3b0Pl3wCnplQdwiPO6GdUCJl6ZAAC1rMbAAADvYTgGBgB4A1pgAABJqmNgVywTPbxbmetZl068\nd593uoX9wVlM96DK8WlbwDaT6ZPvhmekPiTwxLzXXHefd7rifny67rXyd8dJHJ90IQpT/OD2dcbR\ninu6kv7CV0/ue3/umgMm34m2LTB4yH8KPeZ9+bMn0UB5TKX7ihYYkpK4Aaxh3ICea/P6v2Zcd5wc\nChjycr1MWF7T1z5/FcedaL/7hDjf4apMpKlk2Uy5+FVhujg4pdnvPpWAnocXAMAeXYgAAEkUMACA\nJAoYAEASBQwAIIkCBgCQRAEDxrD/LiIgGgUMACCJAgYAkMQHmYHe7L9nBOiDFhgAQBItMGCM9c8G\nAniAFhgAQBIFDAAgiQIGAJBEAQMASKKAAQAkUcCAkfg2KeAxChgwBhPogUp8DgwAIIkWGABAEgUM\nACCJAgYAkEQBAwBIooABACRRwAAAkihgAABJFDAAgCQKGABAEgUMACCJAgYAkEQBAwBIooABACRR\nwAAAkihgAABJ/w9EAEj9lmibygAAAABJRU5ErkJggg==\n"
      }
     ],
     "prompt_number": 123
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Note that theta increases past $\\pi$, indicating the pendulum goes past the vertical and then back down:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%matlab\n",
      "plot(theta)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAkAAAAGwCAIAAADOgk3lAAAACXBIWXMAAAsSAAALEgHS3X78AAAA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\nZ4SXJXIXUAAAAABJRU5ErkJggg==\n"
      }
     ],
     "prompt_number": 124
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Imposing boundary conditions"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Solve\n",
      "$$u''(x) + u(x) = exp(x)$$\n",
      "with boundary conditions\n",
      "$$u(0)=3, u'(2) = 1$$\n"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%matlab\n",
      "L = chebop(@(u) diff(u,2) + u,[0,2],@(u) u-3,@(u) diff(u)-1)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "display_data",
       "text": [
        "L = chebop\n",
        "   Linear operator operating on chebfuns defined on:          \n",
        "     interval [0,2]                                           \n",
        "   representing the operator:                                 \n",
        "     @(u)diff(u,2)+u                                          \n",
        "   left boundary condition:                                   \n",
        "     @(u)u-3 = 0                                              \n",
        "   right boundary condition:                                  \n",
        "     @(u)diff(u)-1 = 0                                        \n",
        "   with n = 6 realization:                                    \n",
        "     33.0353  -51.2657   27.8620  -14.2070   10.1514   -4.5760\n",
        "     -0.6075    5.7509   -9.1443    6.9332   -3.2805    1.3482\n",
        "      1.3482   -3.2805    6.9332   -9.1443    5.7509   -0.6075\n",
        "     -4.5760   10.1514  -14.2070   27.8620  -51.2657   33.0353\n",
        "      1.0000         0         0         0         0         0\n",
        "     -0.5000    1.1056   -1.5279    2.8944  -10.4721    8.5000\n"
       ]
      }
     ],
     "prompt_number": 125
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%matlab\n",
      "f = chebfun('exp(x)', [0,2]);\n",
      "u = L\\f"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "display_data",
       "text": [
        "u = \n",
        "   chebfun column (1 smooth piece)\n",
        "       interval       length   endpoint values   \n",
        "[       0,       2]       23        3      3.6   \n",
        "vertical scale = 3.6 \n"
       ]
      }
     ],
     "prompt_number": 126
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%matlab\n",
      "plot(u)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "display_data",
       "png": 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bzHOfjwuAVT9LiD73AhhKV5PKs6sQAehPVwEDYBz9LCECMJSufpB548IVxWffuuYm/fq9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\n"
      }
     ],
     "prompt_number": 127
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Clearly $u(0) = 3$.  Check that $u'(2) = 1$:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%matlab\n",
      "uprime = diff(u);\n",
      "uprime(2)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "display_data",
       "text": [
        "ans =\n",
        "   1.000000000001889\n"
       ]
      }
     ],
     "prompt_number": 128
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Plot residual $u''(x) - u(x) - e^x$:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%matlab\n",
      "residual = diff(uprime) + u - f;\n",
      "max_residual = max(abs(residual))\n",
      "plot(residual)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "display_data",
       "text": [
        "max_residual =\n",
        "     4.465903202799382e-10\n"
       ]
      },
      {
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAkAAAAGwCAIAAADOgk3lAAAACXBIWXMAAAsSAAALEgHS3X78AAAA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\nRBIwACIJGACRBAyASAIGQCQBAyDS/wFuPJuUGr156QAAAABJRU5ErkJggg==\n"
      }
     ],
     "prompt_number": 129
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 129
    }
   ],
   "metadata": {}
  }
 ]
}