06_Boundary_Value_Problems.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Lecture 6: Boundary Value Problems\n",
    "===\n",
    "\n",
    "Shooting Method\n",
    "---\n",
    "Example: We consider the BVP\n",
    "$$\n",
    "    u''(x) = \\frac{3}{2} u^2(x), \\qquad u(0) = 4, \\quad u(1) = 1\n",
    "$$\n",
    "\n",
    "The Shooting Method converts this into an IVP,\n",
    "\n",
    "$$\n",
    "u''(x) = \\frac{3}{2} u^2(x), \\qquad u(0) = 4,\\quad u'(0) = u_{guess}\n",
    "$$\n",
    "\n",
    "and the non-linear root-finding problem\n",
    "\n",
    "$$\n",
    "u(1; u_{guess}) - \\underbrace{1}_{=u_{desired}(1)} = 0.\n",
    "$$\n",
    "\n",
    "Here, we use the Julia packages DifferentialEquations.jl and Roots.jl\n",
    "to implement the shooting method for this problem."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "using PyPlot\n",
    "PyPlot.svg(true)\n",
    "\n",
    "using DifferentialEquations\n",
    "using Roots\n",
    "using LinearAlgebra"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "solve_ivp (generic function with 1 method)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function solve_ivp(u_guess; saveat=(0.0, 1.0))\n",
    "    # Differential equation in first-order form\n",
    "    f(u, p, t) = [u[2], 3.0/2.0 * u[1]^2]\n",
    "    \n",
    "    # initial conditions\n",
    "    u0 = [4.0, u_guess]\n",
    "    \n",
    "    # time interval on which to solve\n",
    "    tspan = (0.0, 1.0)\n",
    "    \n",
    "    prob = ODEProblem(f, u0, tspan)\n",
    "    solution = solve(prob, saveat=saveat)\n",
    "    \n",
    "    # return u(1; u_guess)\n",
    "    return solution\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "retcode: Success\n",
       "Interpolation: 1st order linear\n",
       "t: 2-element Array{Float64,1}:\n",
       " 0.0\n",
       " 1.0\n",
       "u: 2-element Array{Array{Float64,1},1}:\n",
       " [4.0, -8.0]         \n",
       " [0.999984, -1.00005]"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve_ivp(-8)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "root_finding_problem (generic function with 1 method)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# define the root-finding problem\n",
    "# solving this problem solves the BVP-- this is the shooting method\n",
    "function root_finding_problem(u_guess)\n",
    "    sln = solve_ivp(u_guess)\n",
    "    u_1 = sln.u[end][1]\n",
    "    u_1_desired = 1.0\n",
    "    \n",
    "    return u_1 - u_1_desired\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1.2878587085651816e-14"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "root_finding_problem(-35.85775621541935)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-7.9999927342830786"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Solve the root-finding problem using Roots.jl\n",
    "# and the Bisection method (will revisit the details, for now this is a black box)\n",
    "\n",
    "root = find_zero(root_finding_problem, (-10.0, 0.0), Bisection())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-35.85775621541935"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Turns out there is a second solution!\n",
    "\n",
    "root2 = find_zero(root_finding_problem, (-40.0, -30.0), Bisection())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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    "# check the solution and plot it\n",
    "solution_shooting = solve_ivp(root; saveat=LinRange(0, 1, 100))\n",
    "solution_shooting2 = solve_ivp(root2; saveat=LinRange(0, 1, 100))\n",
    "\n",
    "\n",
    "figure()\n",
    "plot(solution_shooting.t, solution_shooting[1,:], label=\"solution 1\")\n",
    "plot(solution_shooting2.t, solution_shooting2[1,:], label=\"solution 2\")\n",
    "\n",
    "xlabel(L\"x\")\n",
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    "legend()"
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   "source": [
    "Finite-Difference Method\n",
    "---\n",
    "\n",
    "The finite-difference method solves the BVP by sampling the solution interval\n",
    "and converting the differential operator (and boundary conditions) into\n",
    "a matrix equation.\n",
    "\n",
    "Here, we consider the beam-bending example\n",
    "\n",
    "$$\n",
    "\\alpha \\frac{d^4 f}{dx^4} = q(x), \\qquad q(x) = -x^2 \\\\\n",
    "f(a) = f'(a) = f(b) = f'(b) = 0\n",
    "$$\n",
    "\n",
    "We discretize the solution interval into $N+1$ samples and define\n",
    "$$\n",
    "x_n = a + n h, \\quad f_n = f(a + n h), \\quad f^{(4)}_n = f^{(4)}(a + nh),\n",
    "\\quad q_n = q(a + nh), \\qquad h = \\frac{b - a}{N+1}.\n",
    "$$\n",
    "\n",
    "Then the finite-difference matrix implementing the fourth derivative\n",
    "using a central-difference stencil of order 2 is the $(N+1)\\times (N+1)$ matrix\n",
    "$$\n",
    "A = \\frac{1}{h^4} \\begin{pmatrix}\n",
    "6 & -4 & 1 & 0 & \\dots & & & & 0\\\\\n",
    "-4 & 6 & -4 & 1 & 0 & \\dots & & & 0 \\\\\n",
    "1 & -4 & 6 & -4 & 1 & 0 & \\dots & & 0 \\\\\n",
    "0 & 1 & -4 & 6 & -4 & 1 & 0 & \\dots & 0 \\\\\n",
    "\\vdots &  & \\ddots & \\ddots & \\ddots & \\ddots & \\ddots &  & \\vdots \\\\\n",
    "0 & \\dots & & & & 0 & -4 & 6 & -4 \\\\\n",
    "0 & \\dots & & & & 0 & 0 & -4 & 6 \\\\\n",
    "\\end{pmatrix}.\n",
    "$$\n",
    "\n",
    "The solution of the BVP is then given by\n",
    "$$\n",
    "\\mathbf{f} = A^{-1} \\left( \\frac{1}{\\alpha} \\mathbf{q} \\right),\n",
    "$$\n",
    "where $\\mathbf{q}$ is the vector of beam loadings."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "fourth_derivative (generic function with 1 method)"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function fourth_derivative(N, h)\n",
    "    # construct the matrix A implementing the fourth derivative\n",
    "    A = diagm( 0 => 6*ones(N),\n",
    "               1 => -4*ones(N-1),\n",
    "              -1 => -4*ones(N-1), \n",
    "               2 => ones(N-2),\n",
    "              -2 => ones(N-2))\n",
    "    \n",
    "    return A/h^4\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "100×100 Array{Float64,2}:\n",
       "  62436.2  -41624.2   10406.0       0.0  …       0.0       0.0       0.0\n",
       " -41624.2   62436.2  -41624.2   10406.0          0.0       0.0       0.0\n",
       "  10406.0  -41624.2   62436.2  -41624.2          0.0       0.0       0.0\n",
       "      0.0   10406.0  -41624.2   62436.2          0.0       0.0       0.0\n",
       "      0.0       0.0   10406.0  -41624.2          0.0       0.0       0.0\n",
       "      0.0       0.0       0.0   10406.0  …       0.0       0.0       0.0\n",
       "      0.0       0.0       0.0       0.0          0.0       0.0       0.0\n",
       "      0.0       0.0       0.0       0.0          0.0       0.0       0.0\n",
       "      0.0       0.0       0.0       0.0          0.0       0.0       0.0\n",
       "      0.0       0.0       0.0       0.0          0.0       0.0       0.0\n",
       "      0.0       0.0       0.0       0.0  …       0.0       0.0       0.0\n",
       "      0.0       0.0       0.0       0.0          0.0       0.0       0.0\n",
       "      0.0       0.0       0.0       0.0          0.0       0.0       0.0\n",
       "      ⋮                                  ⋱                              \n",
       "      0.0       0.0       0.0       0.0          0.0       0.0       0.0\n",
       "      0.0       0.0       0.0       0.0          0.0       0.0       0.0\n",
       "      0.0       0.0       0.0       0.0  …       0.0       0.0       0.0\n",
       "      0.0       0.0       0.0       0.0          0.0       0.0       0.0\n",
       "      0.0       0.0       0.0       0.0          0.0       0.0       0.0\n",
       "      0.0       0.0       0.0       0.0          0.0       0.0       0.0\n",
       "      0.0       0.0       0.0       0.0          0.0       0.0       0.0\n",
       "      0.0       0.0       0.0       0.0  …   10406.0       0.0       0.0\n",
       "      0.0       0.0       0.0       0.0     -41624.2   10406.0       0.0\n",
       "      0.0       0.0       0.0       0.0      62436.2  -41624.2   10406.0\n",
       "      0.0       0.0       0.0       0.0     -41624.2   62436.2  -41624.2\n",
       "      0.0       0.0       0.0       0.0      10406.0  -41624.2   62436.2"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# construct discretization and finite difference matrix\n",
    "N = 100\n",
    "a = 0.0\n",
    "b = 10.0\n",
    "\n",
    "h = (b - a)/(N+1)\n",
    "\n",
    "A = fourth_derivative(N, h)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "100-element Array{Float64,1}:\n",
       " -0.06216963743148053\n",
       " -0.12409875261325935\n",
       " -0.18554775382949326\n",
       " -0.24627890683200138\n",
       " -0.3060572545878873 \n",
       " -0.3646515262826552 \n",
       " -0.42183503206206224\n",
       " -0.4773865400511271 \n",
       " -0.531091132257275  \n",
       " -0.5827410360463008 \n",
       " -0.6321364279743265 \n",
       " -0.6790862068658859 \n",
       " -0.7234087331472497 \n",
       "  ⋮                  \n",
       "  0.6790862068658856 \n",
       "  0.6321364279743258 \n",
       "  0.5827410360463008 \n",
       "  0.5310911322572747 \n",
       "  0.47738654005112635\n",
       "  0.4218350320620621 \n",
       "  0.3646515262826547 \n",
       "  0.30605725458788735\n",
       "  0.24627890683200196\n",
       "  0.18554775382949354\n",
       "  0.12409875261325934\n",
       "  0.06216963743148109"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# forcing vector\n",
    "α = 1.0\n",
    "x = zeros(N)\n",
    "q = zeros(N)\n",
    "\n",
    "for i=1:N\n",
    "    x[i] = a + i*h\n",
    "    q[i] = -sin(2π * x[i]/10)#x[i]/α\n",
    "#     q[i] = -exp(-(x[i] - 5)^2/(2*4))\n",
    "end\n",
    "\n",
    "q"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "Figure(PyObject <Figure size 640x480 with 1 Axes>)"
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   ],
   "source": [
    "# solve the BVP. We will revisit the backslash operator later and understand what it really does internally\n",
    "f = A \\ q\n",
    "\n",
    "figure()\n",
    "plot(x, q, \"--\", label=L\"beam load $q(x)$\")\n",
    "plot(x, f, \".-\", label=L\"solution $f(x)$\")\n",
    "\n",
    "xlabel(L\"x\")\n",
    "legend()"
   ]
  },
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   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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