10-Boundary-Value-Problems-for-ODE.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Boundary-Value-Problems-for-ODE"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 10.01 Boundary Value Problems"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The ODE $y' = f(t,y)$ does not determine a unique solution.\n",
    "* A kth order ODE requires $k$ side conditions $y_i$ where $i = 1, \\cdots, k$ to determine a unique solution.\n",
    "* If the $y_i$ are specified at values of $t \\neq 0$, then the conditions are referred to as **boundary conditions** (BC).\n",
    "\n",
    "General first-order two-point BVP with $f: \\mathbb{R}^{n+1} \\rightarrow \\mathbb{R}^n$ given by: \n",
    "$$\n",
    "y' = f(t, y), \\qquad a < t < b\n",
    "$$\n",
    "\n",
    "and BC $g: \\mathbb{R}^{2n} \\rightarrow \\mathbb{R}^n$ given by:\n",
    "$$\n",
    "g(y(a), y(b)) = 0\n",
    "$$\n",
    "\n",
    "#### Boundary Conditions\n",
    "**Separated** (def): $g$ contains solution values only at $a$ or $b$, but not both.\n",
    "\n",
    "**Linear** (def): form sum of matrix-vector products\n",
    "$$\n",
    "B_a y(a) + B_b y(b) = c\n",
    "$$\n",
    "where\n",
    "* $B_a$ and $B_b$ are $n \\times n$ matrices\n",
    "* $c$ is a $n \\times 1$ vector\n",
    "\n",
    "#### Existence and Uniqueness\n",
    "Solvability of BVP with **nonlinear BC** is difficult to establish.\n",
    "\n",
    "BVP with **linear BC** has unique solution IFF matrix $Q$ is nonsingular:\n",
    "$$\n",
    "Q \\equiv B_a Y(a) + B_b Y(b)\n",
    "$$\n",
    "where\n",
    "* $Y(t)$ is the matrix whose ith column is $y_i(t)$ and $Y$ is called the fundamental solution matrix for the ODE, and its columns are called solution modes.\n",
    "\n",
    "#### Conditioning and Stability\n",
    "For BVP solution is determined everywhere simultaneously.\n",
    "* For BVP to be well-conditioned growing and decaying modes must be controlled by BC."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Integrate the two-point BVP for the second-order ODE $u'' = 6t$ and $0 < t < 1$ with BC $u(0) = 0$ and $u(1) = 1$.\n",
    "\n",
    "Note: From calculus we know that the exact solution is $u(t) = t^3$.\n",
    "* Since $\\int 6t\\, dt = 3t^2 + C$ and $\\int 3t^2\\, dt = t^3 + C$.  \n",
    "* Based on the initial conditions, the value of $C = 0$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1. 3.]\n"
     ]
    }
   ],
   "source": [
    "import math\n",
    "import numpy as np\n",
    "import scipy.integrate as integrate\n",
    "\n",
    "def fty(t, y):\n",
    "    \"\"\"\n",
    "    Evaluate the n differential equations y^{(i)}(t) at t for i=1..n.\n",
    "    \n",
    "    y is an input matrix with the following components\n",
    "        y_0 = u\n",
    "        y_1 = u'\n",
    "\n",
    "    Returns a matrix of derivatives for each component of y.\n",
    "        y_0' = u'\n",
    "        y_1' = u''\n",
    "    \"\"\"\n",
    "    return np.vstack((y[1], 6.*t))\n",
    "\n",
    "\n",
    "# Setup boundary conditions.\n",
    "bc = lambda ya, yb: np.array([ya[0], yb[0]-1.])\n",
    "\n",
    "# Integrate from [0, 1] with step size 0.5.\n",
    "t0, tn, nsteps = 0., 1., 2\n",
    "t_eval = np.linspace(t0, tn, nsteps+1)\n",
    "y0 = np.zeros((2, t_eval.size))  # Initial guess.\n",
    "result = integrate.solve_bvp(fty, bc, t_eval, y0)\n",
    "np.testing.assert_equal(result.success, True)\n",
    "\n",
    "# Verify boundary conditions are satisfied at t=0 and t=1.\n",
    "np.testing.assert_equal(result.sol(0)[0], 0.)\n",
    "np.testing.assert_equal(result.sol(1)[0], 1.)\n",
    "\n",
    "# Verify solution at some other point t=0.5 based on calculus.\n",
    "np.testing.assert_equal(result.sol(0.5), \n",
    "                        np.array([math.pow(0.5, 3), 3.*0.5*0.5]))\n",
    "\n",
    "# Print the solution u and u' at t=1.\n",
    "print(result.sol(1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Integrate the two-point BVP for the third-order ODE $y''' = 2y'' + 6ty$ and $0 < t < 5$ with BC $y(0) = 2$ and $y(5) = 0$ and $y'(5) = 0$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import math\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "import scipy.integrate as integrate\n",
    "\n",
    "%matplotlib inline\n",
    "\n",
    "def fty(t, y):\n",
    "    \"\"\"\n",
    "    Evaluate the n differential equations y^{(i)}(t) at t for i=1..n.\n",
    "    \n",
    "    y is an input matrix with the following components\n",
    "        y_0 = u\n",
    "        y_1 = u'\n",
    "        y_2 = u''\n",
    "\n",
    "    Returns a matrix of derivatives for each component of y.\n",
    "        y_0' = u'\n",
    "        y_1' = u''\n",
    "        y_2' = u'''\n",
    "    \"\"\"\n",
    "    return np.vstack((y[1], y[2], 2*y[2] + 6.*t*y[0]))\n",
    "\n",
    "\n",
    "# Setup boundary conditions.\n",
    "bc = lambda ya, yb: np.array([ya[0]-2., yb[0], yb[1]])\n",
    "\n",
    "# Integrate from [0, 1] with step size 0.5.\n",
    "t0, tn, nsteps = 0., 5., 25\n",
    "t_eval = np.linspace(t0, tn, nsteps+1)\n",
    "y0 = np.zeros((3, t_eval.size))  # Initial guess.\n",
    "result = integrate.solve_bvp(fty, bc, t_eval, y0)\n",
    "np.testing.assert_equal(result.success, True)\n",
    "\n",
    "# Verify boundary conditions are satisfied at t=0 and t=5.\n",
    "np.testing.assert_equal(result.sol(0)[0], 2.)\n",
    "np.testing.assert_almost_equal(result.sol(5.)[:2], np.zeros(2))\n",
    "\n",
    "# Plot approximate solution.\n",
    "plt.plot(t_eval, result.sol(t_eval)[0,:], marker='o', label='y(t)')\n",
    "plt.plot(t_eval, result.sol(t_eval)[1,:], marker='o', label=\"y'(t)\")\n",
    "plt.plot(t_eval, result.sol(t_eval)[2,:], marker='o', label=\"y''(t)\")\n",
    "plt.title(\"Third Order ODE, $y''' = 2y'' + 6ty$\")\n",
    "plt.ylabel('y')\n",
    "plt.xlabel('t')\n",
    "plt.grid()\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 10.02 Numerical Methods for BVP"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Shooting Method\n",
    "Given BC $u(a) = \\alpha$ and $u(b) = \\beta$.\n",
    "1. **Guess** values for $u'(a)$ and solve the resulting IVP.\n",
    "2. Compare solution from previous step at $b$ with the BC $u(b) = \\beta$.\n",
    "3. Repeat step 1 until the solution obtained matches the BC at $b$.\n",
    "  * Use root-finding method to obtain the next guess for $u'(a)$ that is successively closer to $u(b) = \\beta$. \n",
    "\n",
    "Shooting method inherits stability of associated IVP.\n",
    "* Can be unstable even when BVP is stable.\n",
    "* Workaround: Divide integration interval into subintervals. Perform shooting on each subinterval.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 10.03 Finite Difference Method"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Finite Difference Method\n",
    "Convert BVP into system of algebraic equations by replacing all derivatives with finite difference approximations.\n",
    "\n",
    "Given $u'' = f(t, u, u')$ and $a < t < b$ with BC $u(a) = \\alpha$ and $u(b) = \\beta$.\n",
    "1. Compute mesh points $t_i = a + ih$ where step size $h = (b-a)/(n+1)$ and $i = 0, \\cdots, n+1$.\n",
    "2. Compute the finite difference approximation of derivative at each mesh point.\n",
    "$$\n",
    "u'(t_i) \\approx \\frac{y_{i+1} - y_{i-1}}{2h} \\\\\n",
    "u''(t_i) \\approx \\frac{y_{i+1} - 2y_i + y_{i-1}}{h^2}\n",
    "$$\n",
    "3. Form the system of equations and solve for $y_i$ and $i = 1, \\cdots, n$.  The second two equations are used at the boundary conditions.\n",
    "$$\n",
    "\\frac{y_{i+1} - 2y_i + y_{i-1}}{h^2} = f \\left( t_i, y_i, \\frac{y_{i+1} - y_{i-1}}{2h} \\right) \\\\\n",
    "y_2 - 2y_1 + a = h^2 f \\left( x_1, y_1, \\frac{y_2 - y_a}{2h} \\right) \\\\\n",
    "b - 2y_n + y_{n+1} = h^2 f \\left( x_n, y_n, \\frac{b - y_{n-1}}{2h} \\right) \\\\\n",
    "$$\n",
    "\n",
    "The form of the system formed from the finite difference equations is **tridiagonal**.\n",
    "* A tridiagonal system of bandwidth $\\beta$ requires only $O(\\beta n)$ storage and $O(\\beta^2 n)$ work to factorize using LU factorization."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Integrate the two-point BVP for the second order ODE $y'' = -4y + 4t$ and $0 < t < \\pi/2$ with BC $y(0) = 0$ and $y'(\\pi/2) = 0$ using the finite difference method and $n = 10$.\n",
    "\n",
    "The linear system used to solve this ODE can be simplified using the steps shown below:\n",
    "$$\n",
    "\\begin{aligned}\n",
    "& y_{i+1} - 2y_i + y_{i-1} - h^2 f \\left( t_i, y_i, \\frac{y_{i+1} - y_{i-1}}{2h} \\right) &&= 0 &&& (1) \\\\\n",
    "& y_{i+1} - 2y_i + y_{i-1} - h^2 (-4y_i + 4t_i) &&= 0 &&& (2) \\\\\n",
    "& y_{i+1} - 2y_i + y_{i-1} + 4 h^2 y_i - 4 h^2 t_i &&= 0 &&& (3) \\\\\n",
    "& y_{i+1} + y_i (-2 + 4 h^2) + y_{i-1} &&= 4 h^2 t_i &&& (4) \\\\\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "The tridiagonal matrix becomes:\n",
    "$$\n",
    "\\begin{bmatrix}\n",
    "1 & 0 & & & \\\\\n",
    "1 & -2 + 4 h^2 & 1 & & \\\\\n",
    "  & \\ddots & \\ddots & \\ddots & \\\\\n",
    "  & & 1 & -2 + 4 h^2 & 1 \\\\\n",
    "  & & & 2 & -2 + 4 h^2 \\\\\n",
    "\\end{bmatrix}\n",
    "\\begin{bmatrix}\n",
    "y_0 \\\\\n",
    "y_1 \\\\\n",
    "\\vdots \\\\\n",
    "y_{n-1} \\\\\n",
    "y_n \\\\\n",
    "\\end{bmatrix} =\n",
    "\\begin{bmatrix}\n",
    "0 \\\\\n",
    "4 h^2 t_1 \\\\\n",
    "\\vdots \\\\\n",
    "4 h^2 t_{n-1} \\\\\n",
    "4 h^2 t_n \\\\\n",
    "\\end{bmatrix}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "import scipy.linalg as la\n",
    "\n",
    "%matplotlib inline\n",
    "\n",
    "def tridiag_bvp(n, a, b):\n",
    "    \"\"\"\n",
    "    Form tridiagonal matrix A and vector b used to solve two-point BVP.\n",
    "    \n",
    "    n is the number of interior mesh points, excluding the BC.\n",
    "    a, b is the interval of integration.\n",
    "    \n",
    "    Returns \n",
    "        A is the matrix of coefficients.\n",
    "        B is the right hand side.\n",
    "        h is the mesh spacing.\n",
    "        t is the location of the mesh points.\n",
    "    \"\"\"\n",
    "    h = (b-a)/n\n",
    "    t = np.linspace(a, b, n+1, endpoint=True)\n",
    "    \n",
    "    # d are lower diagonal, diagonal, and upper diagonal vectors.\n",
    "    d = np.ones(n), np.ones(n+1), np.ones(n)\n",
    "    # First equation should be y_0 = 0\n",
    "    d[2][0] = 0.\n",
    "    # Last equation should be 2 y_{n-1} + ... = 4 h^2 y_n\n",
    "    d[0][n-1] = 2.\n",
    "    # Interior diagonals should be y_{i-1} + y_i (-2 + 4 h^2) + y_{i+1}\n",
    "    d[1][1:] = -2. + 4.*h*h\n",
    "    # Form the tridiagonal matrix A.\n",
    "    A = np.diag(d[0], k=-1) + np.diag(d[1], k=0) + np.diag(d[2], k=1)\n",
    "\n",
    "    # Form the right hand side.\n",
    "    B = 4.*h*h*t\n",
    "\n",
    "    return A, B, h, t\n",
    "\n",
    "\n",
    "# Obtain the system of equations for the BVP.\n",
    "n, a, b = 10, 0., np.pi/2.\n",
    "A, B, h, t = tridiag_bvp(n, a, b)\n",
    "\n",
    "# Factorize A into [L|U].\n",
    "# NOTE(mmorais): Use P^T to reorder the right-hand side.\n",
    "P, L, U = la.lu(A)\n",
    "\n",
    "# Solve Ay = b for x.\n",
    "# 1. Solve Lx=B for x.\n",
    "x = la.solve_triangular(L, np.matmul(P.T, B), lower=True)\n",
    "# 2. Solve Uy=x for y.\n",
    "y = la.solve_triangular(U, x, lower=False)\n",
    "\n",
    "# Compute the exact solution\n",
    "ytrue = t + np.sin(t) * np.cos(t)\n",
    "\n",
    "# Plot the true and approximate solution.\n",
    "plt.plot(t, ytrue, marker='o', lw=2., label=\"true solution\")\n",
    "plt.plot(t, y, marker='o', ls='--', label='approximate solution')\n",
    "plt.suptitle(\"Second Order Linear ODE, $y'' = -4y + 4t$\")\n",
    "plt.title('Finite Difference Method')\n",
    "plt.ylabel('y')\n",
    "plt.xlabel('t')\n",
    "plt.grid()\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Integrate the two-point BVP for the second order ODE $y'' = -3yy'$ and $0 < t < 2$ with BC $y(0) = 0$ and $y(2) = 1$ using the finite difference method and $n = 10$.\n",
    "\n",
    "Since the system of equations is nonlinear we will use a nonlinear iterative solver.\n",
    "$$\n",
    "y_{i+1} - 2y_i + y_{i-1} - h^2 f \\left( t_i, y_i, \\frac{y_{i+1} - y_{i-1}}{2h} \\right) = 0\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "import scipy.integrate as integrate\n",
    "import scipy.optimize as opt\n",
    "\n",
    "%matplotlib inline\n",
    "\n",
    "def residual(ftyyp, t, h, y):\n",
    "    \"\"\"\n",
    "    Compute residual of nonlinear system used to solve two-point BVP.\n",
    "    \"\"\"\n",
    "    n = t.size\n",
    "    r = np.zeros(n)\n",
    "    # Residual at first mesh point is y_0 - 0.\n",
    "    r[0] = y[0]\n",
    "    # Interior mesh points should be y_{i-1} - 2y_i + y_{i+1} - h^2 f.\n",
    "    f = ftyyp(t[1:n-1], y[1:n-1], (y[2:n] - y[0:n-2])/(2.*h))\n",
    "    r[1:n-1] = y[0:n-2] - 2.*y[1:n-1] + y[2:n] - h*h*f\n",
    "    # Residual at last mesh point is y_n -  1.\n",
    "    r[n-1] = y[n-1] - 1.\n",
    "    return r\n",
    "\n",
    "\n",
    "def ftyyp(t, y, yp):\n",
    "    \"\"\"\n",
    "    Evaluate the function at t, y, and y'.\n",
    "    \"\"\"\n",
    "    return -3.*y*yp\n",
    "\n",
    "\n",
    "# Setup the number of mesh points and bounday values.\n",
    "n, a, b = 10, 0., 2.\n",
    "\n",
    "# Compute the mesh points.\n",
    "h = (b-a)/n\n",
    "t = np.linspace(a, b, n+1, endpoint=True)\n",
    "\n",
    "# Use a straight line from a to b with y(b) = 1 as initial guess.\n",
    "yinitial = a + 1./(b-a)*t\n",
    "\n",
    "# Solve the nonlinear system of equations using Broyden's good method.\n",
    "soln = opt.broyden1(lambda y: residual(ftyyp, t, h, y), yinitial)\n",
    "\n",
    "# Verify boundary conditions are satisfied at t=0 and t=2.\n",
    "np.testing.assert_equal(soln[0], 0.)\n",
    "np.testing.assert_equal(soln[n], 1.)\n",
    "\n",
    "# Compare solution obtained for nonlinear system to scipy BVP solver.\n",
    "fty = lambda t, y: np.vstack((y[1], -3.*y[0]*y[1]))\n",
    "bc = lambda ya, yb: np.array([ya[0], yb[0]-1.])\n",
    "y0 = np.zeros((2, t.size))  # Initial guess.\n",
    "spy_soln = integrate.solve_bvp(fty, bc, t, y0)\n",
    "np.testing.assert_equal(spy_soln.success, True)\n",
    "\n",
    "# Plot approximate solution.\n",
    "plt.plot(t, spy_soln.sol(t)[0,:], marker='o', label='scipy y(t)')\n",
    "plt.plot(t, soln, marker='o', label='broyden1 y(t)')\n",
    "plt.suptitle(\"Second Order Nonlinear ODE, $y'' = -3yy'$\")\n",
    "plt.title('Finite Difference Method')\n",
    "plt.ylabel('y')\n",
    "plt.xlabel('t')\n",
    "plt.grid()\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 10.04 Collocation Method"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Collocation Method\n",
    "Approximate solution to ODE by linear combinations of basis functions.\n",
    "* Requires solution to satisfy ODE at discrete set of collocation points.\n",
    "\n",
    "Given $u'' = f(t, u, u')$ and $a < t < b$ with BC $u(a) = \\alpha$ and $u(b) = \\beta$, seek a solution of the form:\n",
    "$$\n",
    "u(t) \\approx v(t, x) = \\sum_{i=1}^n x_i \\phi_i(t)\n",
    "$$\n",
    "where\n",
    "* $\\phi_i$ are basis functions (eg polynomials, trig functions, B-splines) defined on $[a, b]$\n",
    "  * Spectral method basis functions with global support (eg polynomials or trig functions)\n",
    "  * Finite element method basis functions with highly localized support (eg B-splines)\n",
    "* $x$ is n-vector of parameters\n",
    "\n",
    "\n",
    "1. Define $n$ collocation points $a = t_1 < \\cdots < t_n = b$ at which $v(t, x)$ satisfies the ODE and BC.\n",
    "2. Form the $n \\times n$ system of equations and solve for $x_i$ and $i=2,\\cdots,n-1$. The second two equations are used at the boundary conditions.\n",
    "$$\n",
    "\\begin{aligned}\n",
    "v''(t_i, x) &= f(t_i, v(t_i, x), v'(t_i, x)) \\\\\n",
    "v(t_1, x) &= \\alpha \\\\\n",
    "v(t_n, x) &= \\beta \\\\\n",
    "\\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Integrate the two-point BVP for the second-order ODE $u'' = 6t$ and $0 < t < 1$ with BC $u(0) = 0$ and $u(1) = 1$ using the collocation method with the number of points $n = 3, 4, 5, 6$.\n",
    "\n",
    "Note: From calculus we know that the exact solution is $u(t) = t^3$.\n",
    "* Since $\\int 6t\\, dt = 3t^2 + C$ and $\\int 3t^2\\, dt = t^3 + C$.  \n",
    "* Based on the initial conditions, the value of $C = 0$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "import scipy.optimize as opt\n",
    "\n",
    "%matplotlib inline\n",
    "\n",
    "def vander_bvp(t, order, vdt):\n",
    "    \"\"\"\n",
    "    Form Vandermonde matrix A used as basis for collocation method.\n",
    "    \n",
    "    t is the vector of collocation points.\n",
    "    order is the order of the derivative at the interior points.\n",
    "    vdt is the vector of derivative coefficients at the interior points.\n",
    "    \"\"\"\n",
    "    n = t.size\n",
    "    A = np.vander(t, n, increasing=True)\n",
    "    # Roll coefficients of Vandermonde according to order.\n",
    "    A[1:n-1] = np.roll(A[1:n-1], order)\n",
    "    A[1:n-1,:order] = 0.  # Vanishing terms of the derivative.\n",
    "    A[1:n-1,:] = np.multiply(A[1:n-1,:], vdt)\n",
    "    return A\n",
    "\n",
    "\n",
    "def residual(t, ua, ub, A, x):\n",
    "    \"\"\"\n",
    "    Compute residual of nonlinear system used to solve two-point BVP.\n",
    "    \n",
    "    t is the vector of collocation points.\n",
    "    ua, ub are the values of u(t) at the boundary conditions.\n",
    "    A is the Vandermonde matrix used as basis for collocation method.\n",
    "    x are the parameters.\n",
    "    \"\"\"\n",
    "    n = t.size\n",
    "    b = np.zeros(n)\n",
    "    # Residual at first collocation point is BC.\n",
    "    b[0] = ua\n",
    "    # Residual at the interior points satisfies ODE.\n",
    "    b[1:n-1] = 6.*t[1:n-1]\n",
    "    # Residual at last collocation point is BC.\n",
    "    b[n-1] = ub\n",
    "    return np.matmul(A, x) - b\n",
    "\n",
    "\n",
    "def utx(t, x):\n",
    "    \"\"\"\n",
    "    Evaluate the solution of u(t) using the vector of parameters.\n",
    "    \n",
    "    t is the value where the solution is evaluated.\n",
    "    x is the vector of parameters returned from the collocation method.\n",
    "    \"\"\"\n",
    "    exps = np.arange(0, x.size)\n",
    "    return np.dot(x, np.power(t, exps))\n",
    "\n",
    "\n",
    "# Setup the number of collocation points and boundary values.\n",
    "npoints, a, b, ua, ub = [3, 6], 0., 1., 0., 1.\n",
    "\n",
    "# Store the solution associated with each n.\n",
    "soltns = []\n",
    "\n",
    "# Solve the ODE for each value of n.\n",
    "for n in npoints:\n",
    "\n",
    "    # Compute the collocation points.\n",
    "    t = np.linspace(a, b, n, endpoint=True)\n",
    "\n",
    "    # Form the system of equations.\n",
    "    order, vdt = 2, np.array([0., 0., 2., 6., 12., 20.])\n",
    "    A = vander_bvp(t, order, vdt[:n])\n",
    "\n",
    "    # Use a straight line from a to b with y(b) = 1 as initial guess.\n",
    "    x0 = a + 1./(b-a)*t\n",
    "\n",
    "    # Solve the system of equations.\n",
    "    soltn = opt.fsolve(lambda x: residual(t, ua, ub, A, x), x0)\n",
    "    soltns.append((t, soltn))\n",
    "\n",
    "# Plot the solution obtained for each number of collocation points.\n",
    "for n, soltn in zip(npoints, soltns):\n",
    "    (t, x) = soltn\n",
    "    vutx = np.vectorize(lambda t: utx(t, x))\n",
    "    plt.plot(t, vutx(t), marker='o', label='$u(t)$, n={0}'.format(n))\n",
    "\n",
    "# Plot the exact value.\n",
    "t = np.linspace(a, b, 100)\n",
    "vutx = np.vectorize(lambda t: utx(t, x))\n",
    "plt.plot(t, vutx(t), ls='-', lw=1., label='exact solution')\n",
    "\n",
    "plt.suptitle(\"Second Order Linear ODE, $u'' = 6t$\")\n",
    "plt.title('Collocation Method')\n",
    "plt.ylabel('u')\n",
    "plt.xlabel('t')\n",
    "plt.grid()\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 10.05 Galerkin Method"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Galerkin Method\n",
    "Rather than forcing residual to be zero at collocation points, minimize the residual over the entire interval of integration eg via least squares."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 10.06 Eigenvalue Problems"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Standard eigenvalue problem for second-order ODE\n",
    "$$\n",
    "u'' = \\lambda f(t, u, u'), \\qquad a < t < b\n",
    "$$\n",
    "\n",
    "with boundary conditions\n",
    "$$\n",
    "u(a) = \\alpha, \\qquad u(b) = \\beta\n",
    "$$\n",
    "\n",
    "where\n",
    "* $\\lambda$ is the **eigenvalue** of this two-point BVP\n",
    "* $u$ is the **eigenfunction**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Summary: Boundary Value Problems for ODE"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* Two-point BVP for ODE specifies BC at both endpoints of interval.\n",
    "* Shooting method replaces the BVP by a sequence of IVP formed by guessing values of $u'(a)$ until the second boundary condition $u(b)$ is satisfied.\n",
    "* Finite difference method replaces the derivatives in the ODE by finite differences on a mesh of points.\n",
    "* Collocation method approximates the solution to the ODE by linear combinations of basis functions.\n",
    "* Galerkin method approximates the solution to the ODE by minimizing the residual over the entire interval of integration."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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