OrthogonalCollocation.ipynb

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        "#Orthogonal Collocation on Finite Elements\n",
        "\n",
        "Discretization of a continuous time representation allow large-scale nonlinear programming (NLP) solvers to find solutions at specified intervals in a time horizon. There are many names and related techniques for obtaining mathematical relationships between derivatives and non-derivative values. Some of the terms that are relevant to this discussion include orthogonal collocation on finite elements, direct transcription, Gauss pseudospectral method, Gaussian quadrature, Lobatto quadrature, Radau collocation, Legendre polynomials, Chebyshev polynomials, Jacobi polynomials, Laguerre polynomials, any many more. There are many papers that discuss the details of the derivation and theory behind these methods1-5. The purpose of this section is to give a practical introduction to orthogonal collocation on finite elements with Lobatto quadrature for the numerical solution of differential algebraic equations."
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        "#@markdown ###Orthogonal Collocation 1\n",
        "\n",
        "# Solve tau dx/dt = -x\n",
        "import numpy as np\n",
        "from numpy.linalg import inv\n",
        "\n",
        "## problem statement\n",
        "x0 = 3.0  # initial condition\n",
        "tf = 10.0 # final time\n",
        "tau = 5.0 # model parameter (time constant)\n",
        "\n",
        "## analytic solution\n",
        "time = np.linspace(0,tf,100)\n",
        "x = x0*np.exp(-time/tau)\n",
        "\n",
        "## numeric solutions\n",
        "# 2nd order polynomial (3 points)\n",
        "t2 = tf * np.array([0.0,0.5,1.0])\n",
        "N2 = np.array([[0.75,-0.25], \\\n",
        "               [1.00, 0.00]])\n",
        "M2 = inv(tf * N2)\n",
        "# Solve linear system of equations by matrix manipulation\n",
        "P2 = tau*M2 + np.eye(2)\n",
        "Q2 = np.dot(inv(P2),(tau*M2))\n",
        "v2 = np.dot(Q2, np.ones(2)*x0)\n",
        "x2 = np.insert(v2,0,x0)\n",
        "\n",
        "# 3rd order polynomial (4 points)\n",
        "t3 = tf * np.array([0.0, \\\n",
        "                    1.0/2.0-np.sqrt(5.0)/10.0, \\\n",
        "                    1.0/2.0+np.sqrt(5.0)/10.0, \\\n",
        "                    1.0])\n",
        "N3 = np.array([[0.436,-0.281, 0.121], \\\n",
        "               [0.614, 0.064, 0.046], \\\n",
        "               [0.603, 0.230, 0.167]])\n",
        "M3 = inv(tf * N3)\n",
        "# Solve linear system of equations by matrix manipulation\n",
        "P3 = tau * M3 + np.eye(3)\n",
        "Q3 = np.dot(inv(P3),(tau*M3))\n",
        "v3 = np.dot(Q3, np.ones(3)*x0)\n",
        "x3 = np.insert(v3,0,x0)\n",
        "\n",
        "# 4th order polynomial (5 points)\n",
        "t4 = tf * np.array([0.0, \\\n",
        "                    1.0/2.0-np.sqrt(21.0)/14.0, \\\n",
        "                    1.0/2.0, \\\n",
        "                    1.0/2.0+np.sqrt(21.0)/14.0, \\\n",
        "                    1.0])\n",
        "N4 = np.array([[0.278, -0.202, 0.169, -0.071], \\\n",
        "               [0.398,  0.069, 0.064, -0.031], \\\n",
        "               [0.387,  0.234, 0.278, -0.071], \\\n",
        "               [0.389,  0.222, 0.389,  0.000]])\n",
        "M4 = inv(tf * N4)\n",
        "# Solve linear system of equations by matrix manipulation\n",
        "P4 = tau * M4 + np.eye(4)\n",
        "Q4 = np.dot(inv(P4),(tau*M4))\n",
        "v4 = np.dot(Q4, np.ones(4)*x0)\n",
        "x4 = np.insert(v4,0,x0)\n",
        "\n",
        "# compare results\n",
        "import matplotlib.pyplot as plt\n",
        "plt.figure(1)\n",
        "plt.subplot(2,1,1)\n",
        "plt.plot(time,x,'r-',linewidth=3)\n",
        "plt.plot(t2,x2,'b-.',linewidth=2,markersize=20)\n",
        "plt.plot(t3,x3,'k:.',linewidth=2,markersize=20)\n",
        "plt.plot(t4,x4,'g.-',linewidth=2,markersize=20)\n",
        "plt.xlabel('Time')\n",
        "plt.ylabel('Value')\n",
        "plt.legend(['Exact','3 Points','4 Points','5 Points'])\n",
        "plt.text(2,0.5,'tau dx/dt = -x')\n",
        "\n",
        "plt.subplot(2,1,2)\n",
        "# exact solutions at collocation points\n",
        "y2 = x0*np.exp(-t2/tau)\n",
        "y3 = x0*np.exp(-t3/tau)\n",
        "y4 = x0*np.exp(-t4/tau)\n",
        "plt.plot(t2,x2-y2,'b--.',linewidth=2,markersize=20)\n",
        "plt.plot(t3,x3-y3,'k:.',linewidth=2,markersize=20)\n",
        "plt.plot(t4,x4-y4,'g.-',linewidth=2,markersize=20)\n",
        "plt.xlabel('Time')\n",
        "plt.ylabel('Error')\n",
        "plt.legend(['3 Points','4 Points','5 Points'])\n",
        "plt.show()\n"
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rj+Jzny/ZdmMLEUmP8HIqRLuyHXBRmgalGN3yoRo+ClX/r3FYsRSXeXPoH/yA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hKIiooiPj6esLAwoqKi1OWvvfYaGzZswM/Pj7FjxzJ48GBSU1MrXccWes4czrm2MeUXOEN2z+8h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\nEgAAAABJRU5ErkJggg==\n",
            "text/plain": [
              "<Figure size 576x396 with 2 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "metadata": {
        "id": "1a5Z7nmi-F__",
        "colab_type": "code",
        "cellView": "form",
        "outputId": "8d4155dc-ffbb-4b64-f9a8-58ff883936d6",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 361
        }
      },
      "cell_type": "code",
      "source": [
        "#@markdown ###Orthogonal Collocation 2\n",
        "\n",
        "# Solve tau dx/dt = -x + k u\n",
        "import numpy as np\n",
        "from numpy.linalg import inv\n",
        "\n",
        "## problem statement\n",
        "x0 = 0.0  # initial condition\n",
        "tf = 10.0 # final time\n",
        "tau = 5.0 # model parameter (time constant)\n",
        "k = 2.0   # gain\n",
        "\n",
        "## analytic solution\n",
        "time = np.linspace(0,tf)\n",
        "x = k*(1.0-np.exp(-time/tau))\n",
        "\n",
        "## numeric solutions\n",
        "# 2nd order polynomial (3 points)\n",
        "t2 = tf * np.array([0.0,0.5,1.0])\n",
        "N2 = np.array([[0.75,-0.25], \\\n",
        "               [1.00, 0.00]])\n",
        "M2 = inv(tf * N2)\n",
        "# Solve linear system of equations by matrix manipulation\n",
        "P2 = inv(tau*M2 + np.eye(2))\n",
        "Q2 = np.dot(tau*M2,np.ones(2)*x0) + k*np.ones(2)\n",
        "v2 = np.dot(P2, Q2)\n",
        "x2 = np.insert(v2,0,x0)\n",
        "\n",
        "# 3rd order polynomial (4 points)\n",
        "t3 = tf * np.array([0.0, \\\n",
        "                    1.0/2.0-np.sqrt(5.0)/10.0, \\\n",
        "                    1.0/2.0+np.sqrt(5.0)/10.0, \\\n",
        "                    1.0])\n",
        "N3 = np.array([[0.436,-0.281, 0.121], \\\n",
        "               [0.614, 0.064, 0.046], \\\n",
        "               [0.603, 0.230, 0.167]])\n",
        "M3 = inv(tf * N3)\n",
        "# Solve linear system of equations by matrix manipulation\n",
        "P3 = inv(tau*M3 + np.eye(3))\n",
        "Q3 = np.dot(tau*M3,np.ones(3)*x0) + k*np.ones(3)\n",
        "v3 = np.dot(P3, Q3)\n",
        "x3 = np.insert(v3,0,x0)\n",
        "\n",
        "# 4th order polynomial (5 points)\n",
        "t4 = tf * np.array([0.0, \\\n",
        "                    1.0/2.0-np.sqrt(21.0)/14.0, \\\n",
        "                    1.0/2.0, \\\n",
        "                    1.0/2.0+np.sqrt(21.0)/14.0, \\\n",
        "                    1.0])\n",
        "N4 = np.array([[0.278, -0.202, 0.169, -0.071], \\\n",
        "               [0.398,  0.069, 0.064, -0.031], \\\n",
        "               [0.387,  0.234, 0.278, -0.071], \\\n",
        "               [0.389,  0.222, 0.389,  0.000]])\n",
        "M4 = inv(tf * N4)\n",
        "# Solve linear system of equations by matrix manipulation\n",
        "P4 = inv(tau*M4 + np.eye(4))\n",
        "Q4 = np.dot(tau*M4,np.ones(4)*x0) + k*np.ones(4)\n",
        "v4 = np.dot(P4, Q4)\n",
        "x4 = np.insert(v4,0,x0)\n",
        "\n",
        "# compare results\n",
        "import matplotlib.pyplot as plt\n",
        "plt.figure(1)\n",
        "plt.subplot(2,1,1)\n",
        "plt.plot(time,x,'r-',linewidth=3)\n",
        "plt.plot(t2,x2,'b-.',linewidth=2,markersize=20)\n",
        "plt.plot(t3,x3,'k:.',linewidth=2,markersize=20)\n",
        "plt.plot(t4,x4,'g.-',linewidth=2,markersize=20)\n",
        "plt.xlabel('Time')\n",
        "plt.ylabel('Value')\n",
        "plt.legend(['Exact','3 Points','4 Points','5 Points'])\n",
        "plt.text(4,0.5,'tau dx/dt = -x + k u')\n",
        "\n",
        "plt.subplot(2,1,2)\n",
        "# exact solutions at collocation points\n",
        "y2 = k*(1-np.exp(-t2/tau))\n",
        "y3 = k*(1-np.exp(-t3/tau))\n",
        "y4 = k*(1-np.exp(-t4/tau))\n",
        "plt.plot(t2,x2-y2,'b--.',linewidth=2,markersize=20)\n",
        "plt.plot(t3,x3-y3,'k:.',linewidth=2,markersize=20)\n",
        "plt.plot(t4,x4-y4,'g.-',linewidth=2,markersize=20)\n",
        "plt.xlabel('Time')\n",
        "plt.ylabel('Error')\n",
        "plt.legend(['3 Points','4 Points','5 Points'])\n",
        "plt.show()\n"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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vHuLjb6BWq3F397CZwQxiO2ChZCiuhKPdHopmWyjqvXuQ9LZd4BfLy937KwLh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m4s5o9/VOYkm5en4q23tGzc6I7ZrMLd3UZUlBkPDxnRL0RdSQJwm8i9z3D414w67Wu0GOi44Z4K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\nIVzQ/wNpNDAEDHh0KwAAAABJRU5ErkJggg==\n",
            "text/plain": [
              "<Figure size 576x396 with 2 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "metadata": {
        "id": "iaCfw754_SOA",
        "colab_type": "code",
        "cellView": "form",
        "outputId": "e72c4d1a-6b21-4696-eec6-fcb9b3781f51",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 372
        }
      },
      "cell_type": "code",
      "source": [
        "#@title\n",
        "%%html\n",
        "<div><img src='http://apmonitor.com/do/uploads/Main/download.png' alt='' title='' /> <a class='urllink' href='http://apmonitor.com/do/uploads/Main/orthogonal_collocation.zip' rel='nofollow'>Orthogonal Collocation in MATLAB</a></div>\n",
        "<div class='vspace'></div>\n",
        "<iframe width=\"560\" height=\"315\" src=\"https://www.youtube.com/embed/sxFTAF_xtLg?rel=0\" frameborder=\"0\" allowfullscreen></iframe>\n"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/html": [
              "<div><img src='http://apmonitor.com/do/uploads/Main/download.png' alt='' title='' /> <a class='urllink' href='http://apmonitor.com/do/uploads/Main/orthogonal_collocation.zip' rel='nofollow'>Orthogonal Collocation in MATLAB</a></div>\n",
              "<div class='vspace'></div>\n",
              "<iframe width=\"560\" height=\"315\" src=\"https://www.youtube.com/embed/sxFTAF_xtLg?rel=0\" frameborder=\"0\" allowfullscreen></iframe>"
            ],
            "text/plain": [
              "<IPython.core.display.HTML object>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "metadata": {
        "id": "EUOuDf_pACGQ",
        "colab_type": "text"
      },
      "cell_type": "markdown",
      "source": [
        "##Exercise 1\n",
        "\n",
        "Objective: Solve a differential equation with orthogonal collocation on finite elements. Create a MATLAB or Python script to simulate and display the results. Estimated Time: 2-3 hours\n",
        "\n",
        "Solve the following differential equation from time 0 to 1 with orthogonal collocation on finite elements with 4 nodes for discretization in time.\n",
        "\n",
        "$5 dx/dt = -x2 + u$\n",
        " \n",
        "Specify the initial condition for x as 0 and the value of the input, u, as 4. Compare the solution result with 2-6 time points (nodes). Report the solution at the final time for each and comment on how the solution changes with an increase in the number of nodes."
      ]
    },
    {
      "metadata": {
        "id": "85iw0bAzGHBp",
        "colab_type": "text"
      },
      "cell_type": "markdown",
      "source": [
        "##Solution 1\n",
        "\n",
        "<div><img src='http://apmonitor.com/do/uploads/Main/download.png' alt='' title='' /> <a class='urllink' href='http://apmonitor.com/do/uploads/Main/collocation_exercise.zip' rel='nofollow'>Collocation in MATLAB and Python (APM Solution)</a></div>\n",
        "\n"
      ]
    },
    {
      "metadata": {
        "id": "GDCaTFmjA5L5",
        "colab_type": "code",
        "cellView": "form",
        "colab": {}
      },
      "cell_type": "code",
      "source": [
        "#@markdown Solution with GEKKO\n",
        "\n",
        "try:\n",
        "    from pip import main as pipmain\n",
        "except:\n",
        "    from pip._internal import main as pipmain\n",
        "pipmain(['install','gekko'])\n",
        "\n",
        "from __future__ import division\n",
        "import numpy as np\n",
        "from scipy.optimize import fsolve\n",
        "from scipy.integrate import odeint\n",
        "import matplotlib.pyplot as plt\n",
        "\n",
        "# final time\n",
        "tf = 1.0\n",
        "\n",
        "# solve with ODEINT (for comparison)\n",
        "def model(x,t):\n",
        "    u = 4.0\n",
        "    return (-x**2 + u)/5.0\n",
        "t = np.linspace(0,tf,20)\n",
        "y0 = 0\n",
        "y = odeint(model,y0,t)\n",
        "plt.figure(1)\n",
        "plt.plot(t,y,'r-',label='ODEINT')\n",
        "\n",
        "# ----------------------------------------------------\n",
        "# Approach #1 - Write the model equations in Python\n",
        "# ----------------------------------------------------\n",
        "# define collocation matrices\n",
        "def colloc(n):\n",
        "    if (n==2):\n",
        "        NC = np.array([[1.0]])\n",
        "    if (n==3):\n",
        "        NC = np.array([[0.75,-0.25], \\\n",
        "                       [1.00, 0.00]])\n",
        "    if (n==4):\n",
        "        NC = np.array([[0.436,-0.281, 0.121], \\\n",
        "                       [0.614, 0.064, 0.0461], \\\n",
        "                       [0.603, 0.230, 0.167]])\n",
        "    if (n==5):\n",
        "        NC = np.array([[0.278, -0.202, 0.169, -0.071], \\\n",
        "                       [0.398,  0.069, 0.064, -0.031], \\\n",
        "                       [0.387,  0.234, 0.278, -0.071], \\\n",
        "                       [0.389,  0.222, 0.389,  0.000]])\n",
        "    if (n==6):\n",
        "        NC = np.array([[0.191, -0.147, 0.139, -0.113, 0.047],\n",
        "                       [0.276,  0.059, 0.051, -0.050, 0.022],\n",
        "                       [0.267,  0.193, 0.252, -0.114, 0.045],\n",
        "                       [0.269,  0.178, 0.384,  0.032, 0.019],\n",
        "                       [0.269,  0.181, 0.374,  0.110, 0.067]])\n",
        "    return NC\n",
        "\n",
        "# define collocation points from Lobatto quadrature\n",
        "def tc(n):\n",
        "    if (n==2):\n",
        "        time = np.array([0.0,1.0])\n",
        "    if (n==3):\n",
        "        time = np.array([0.0,0.5,1.0])\n",
        "    if (n==4):\n",
        "        time = np.array([0.0, \\\n",
        "                         0.5-np.sqrt(5)/10.0, \\\n",
        "                         0.5+np.sqrt(5)/10.0, \\\n",
        "                         1.0])\n",
        "    if (n==5):\n",
        "        time = np.array([0.0,0.5-np.sqrt(21)/14.0, \\\n",
        "                         0.5,0.5+np.sqrt(21)/14.0, 1])\n",
        "    if (n==6):\n",
        "        time = np.array([0.0, \\\n",
        "                         0.5-np.sqrt((7.0+2.0*np.sqrt(7.0))/21.0)/2.0, \\\n",
        "                         0.5-np.sqrt((7.0-2.0*np.sqrt(7.0))/21.0)/2.0, \\\n",
        "                         0.5+np.sqrt((7.0-2.0*np.sqrt(7.0))/21.0)/2.0, \\\n",
        "                         0.5+np.sqrt((7.0+2.0*np.sqrt(7.0))/21.0)/2.0, \\\n",
        "                         1.0])\n",
        "    return time*tf\n",
        "\n",
        "# solve with SciPy fsolve\n",
        "def myFunction(z,*param):\n",
        "    n = param[0]\n",
        "    m = param[1]\n",
        "    # rename z as x and xdot variables\n",
        "    x = np.empty(n-1)\n",
        "    xdot = np.empty(n-1)\n",
        "    x[0:n-1] = z[0:n-1]\n",
        "    xdot[0:n-1] = z[n-1:m]\n",
        "\n",
        "    # initial condition (x0)\n",
        "    x0 = 0.0\n",
        "    # input parameter (u)\n",
        "    u = 4.0\n",
        "    # final time\n",
        "    tn = tf\n",
        "\n",
        "    # function evaluation residuals\n",
        "    F = np.empty(m)\n",
        "    # nonlinear differential equations at each node\n",
        "    # 5 dx/dt = -x^2 + u\n",
        "    F[0:n-1] = 5.0 * xdot[0:n-1] + x[0:n-1]**2 - u\n",
        "    # collocation equations\n",
        "    # tn * NC * xdot = x - x0\n",
        "    NC = colloc(n)\n",
        "    F[n-1:m] = tn * np.dot(NC,xdot) - x + x0 * np.ones(n-1)\n",
        "    return F\n",
        "\n",
        "sol_py = np.empty(5) # store 5 results\n",
        "for i in range(2,7):\n",
        "    n = i\n",
        "    m = (i-1)*2\n",
        "    zGuess = np.ones(m)\n",
        "    z = fsolve(myFunction,zGuess,args=(n,m))\n",
        "    # add to plot\n",
        "    yc = np.insert(z[0:n-1],0,0)\n",
        "    plt.plot(tc(n),yc,'o',markersize=10,label='Nodes = '+str(i))\n",
        "    # store just the last x[n] value\n",
        "    sol_py[i-2] = z[n-2]\n",
        "plt.legend(loc='best')\n",
        "\n",
        "# ----------------------------------------------------\n",
        "# Approach #2 - Write model in APMonitor and let\n",
        "#   modeling language create the collocation equations\n",
        "# ----------------------------------------------------\n",
        "# load GEKKO\n",
        "from gekko import GEKKO\n",
        "\n",
        "sol_apm = np.empty(5) # store 5 results\n",
        "i = 0\n",
        "for nodes in range(2,7):\n",
        "    m = GEKKO()\n",
        "\n",
        "    u = m.Param(value=4)\n",
        "    x = m.Var(value=0)\n",
        "    m.Equation(5*x.dt() == -x**2 + u)\n",
        "\n",
        "    m.time = [0,tf]\n",
        "\n",
        "    m.options.imode = 4\n",
        "    m.options.time_shift = 0\n",
        "    m.options.nodes = nodes\n",
        "\n",
        "    m.solve() # solve problem\n",
        "    sol_apm[i] = x.value[-1] # store solution (last point)\n",
        "    i += 1\n",
        "\n",
        "# print the solutions\n",
        "print(sol_py)\n",
        "print(sol_apm)\n",
        "\n",
        "# show plot\n",
        "plt.ylabel('x(t)')\n",
        "plt.xlabel('time')\n",
        "plt.show()"
      ],
      "execution_count": 0,
      "outputs": []
    },
    {
      "metadata": {
        "id": "GZSEbOQnCMDt",
        "colab_type": "code",
        "outputId": "0166db02-dda6-4f6d-be8c-3f7928fdbb0a",
        "cellView": "form",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 336
        }
      },
      "cell_type": "code",
      "source": [
        "#@title\n",
        "%%html\n",
        "<iframe width=\"560\" height=\"315\" src=\"https://www.youtube.com/embed/DBjmW4Lwpjc?rel=0\" frameborder=\"0\" allowfullscreen></iframe>\n"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/html": [
              "<iframe width=\"560\" height=\"315\" src=\"https://www.youtube.com/embed/DBjmW4Lwpjc?rel=0\" frameborder=\"0\" allowfullscreen></iframe>"
            ],
            "text/plain": [
              "<IPython.core.display.HTML object>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "metadata": {
        "id": "IdD-j0XyaPHR",
        "colab_type": "text"
      },
      "cell_type": "markdown",
      "source": [
        "##Exercise 2\n",
        "\n",
        "<p>Compare orthogonal collocation on finite elements with 3 nodes with a numerical integrator (e.g. ODE15s in MATLAB or ODEINT in Python). Calculate the error at each of the solution points for the equation (same as for Exercise 1):\n",
        "</p>\n",
        "<div class='vspace'></div><pre> 5 dx/dt = -x<sup>2</sup> + u\n",
        "</pre><div class='vspace'></div>\n"
      ]
    },
    {
      "metadata": {
        "id": "y4kVql43wCpd",
        "colab_type": "text"
      },
      "cell_type": "markdown",
      "source": [
        "\n",
        "##Solution 2\n",
        "<div><img src='http://apmonitor.com/do/uploads/Main/collocation_compare.png' alt='' title='' /></div>\n",
        "<div><img src='http://apmonitor.com/do/uploads/Main/download.png' alt='' title='' /> <a class='urllink' href='http://apmonitor.com/do/uploads/Main/collocation_compare.zip' rel='nofollow'>Collocation Solution (APM, ODE15s (MATLAB), ODEINT (Python))</a></div>"
      ]
    },
    {
      "metadata": {
        "id": "vSeIiZcYaLvd",
        "colab_type": "code",
        "outputId": "7718127d-e987-4860-f17a-0d1e03fb1081",
        "cellView": "form",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 336
        }
      },
      "cell_type": "code",
      "source": [
        "#@title\n",
        "%%html\n",
        "\n",
        "<iframe width=\"560\" height=\"315\" src=\"https://www.youtube.com/embed/8LmIqeuHGL0\" frameborder=\"0\" allowfullscreen></iframe>\n"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/html": [
              "\n",
              "<iframe width=\"560\" height=\"315\" src=\"https://www.youtube.com/embed/8LmIqeuHGL0\" frameborder=\"0\" allowfullscreen></iframe>"
            ],
            "text/plain": [
              "<IPython.core.display.HTML object>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "metadata": {
        "id": "xegdIaAUakDe",
        "colab_type": "code",
        "cellView": "form",
        "colab": {}
      },
      "cell_type": "code",
      "source": [
        "#@markdown Solution ODEINT Python\n",
        "\n",
        "from scipy.integrate import odeint\n",
        "import numpy as np\n",
        "\n",
        "\n",
        "def collocation(x,t):\n",
        "    u = 4\n",
        "    xdot = -x**2 / 5.0 + u / 5.0\n",
        "\n",
        "    return xdot\n",
        "  \n",
        "y3 = odeint(collocation,[0],[0, 0.5, 1])\n",
        "\n",
        "print('ODEINT solution')\n",
        "print('x1 = ' + str(y3[1]))\n",
        "print('x2 = ' + str(y3[2]))\n",
        "print(' ')"
      ],
      "execution_count": 0,
      "outputs": []
    }
  ]
}