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    "# Initial Value Problems\n",
    "\n",
    "An initial value problem is an ordinary differential equation of the form $y'(t) = f(y, t)$ with $y(0) = c$, where $y$ can be a single or muliti-valued.\n",
    "\n",
    "The idea is that you specifty the starting point of a system and the rules that govern the system, and let the simulation go from there."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Your Zebra Sanctuary\n",
    "\n",
    "Let's say you are really enthusiastic about zebras.\n",
    "\n",
    "<p><a href=\"https://commons.wikimedia.org/wiki/File:Grevy%27s_Zebra_Stallion.jpg#/media/File:Grevy's_Zebra_Stallion.jpg\"><img src=\"https://upload.wikimedia.org/wikipedia/commons/7/74/Grevy%27s_Zebra_Stallion.jpg\" alt=\"Grevy's Zebra Stallion.jpg\" style=\"height:300px;\"></a>\n",
    "<br>\n",
    "<center>By <a rel=\"nofollow\" class=\"external text\" href=\"https://www.flickr.com/photos/rainbirder/\">Rainbirder</a> - <a rel=\"nofollow\" class=\"external text\" href=\"https://www.flickr.com/photos/rainbirder/5299622573/\">Grevy's Zebra Stallion</a>, <a href=\"https://creativecommons.org/licenses/by-sa/2.0\" title=\"Creative Commons Attribution-Share Alike 2.0\">CC BY-SA 2.0</a>, <a href=\"https://commons.wikimedia.org/w/index.php?curid=13267554\">Link</a></center></p>\n",
    "\n",
    "You decide to start a zebra sanctuary in your backyard.  Soon you have a bunch of zebras having a great time eating your grass and shrubbery."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### A Growing Problem\n",
    "\n",
    "Your zebras keep having babies.  This is manageable for now, and the baby zebras are cute.  However, you're a bit worried about the number of zerbas you're going to need to feed, and you want to figure out what will happen if you let things go unchecked.  Naturally, you decide to write a program to simulate your zebra problem.\n",
    "\n",
    "Let's say you have $z_0$ zebras this month. The number of zebra babies that will appear next month is proportional to the current population, so \n",
    "\\begin{equation}\n",
    "z_1 = z_0 + \\Delta(z_0)\n",
    "\\end{equation}\n",
    "where $\\Delta(z_0) = \\alpha z_0$.  You estimate $\\alpha$ is about $0.02$. \n",
    "\n",
    "We can write this as a differential equation\n",
    "\\begin{equation}\n",
    "z'(t) = \\alpha z(t)\n",
    "\\end{equation}\n",
    "\n",
    "Now it is time to fire up your Python interpreter.  We'll use `solve_ivp` in `scipy.integrate` - this is a high-level wrapper with lots of options for solving initial value problems.  The important arguments to provide are:\n",
    "* `f(t, y)` - a Python function that returns the right-hand side of the ODE - this can be a multivalued function\n",
    "* `t_span` - a tuple `(t0, t1)` which gives the start and end times of the simulation\n",
    "* `y0` - the state of the system at `t0`\n",
    "\n",
    "`solve_ivp` will do a lot of work for you - deciding on a backend algorithm, chooing time steps, etc.  You might find it useful to specify `t_eval`, which gives an array of points to evaluate the function on."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy.integrate import solve_ivp\n",
    "\n",
    "z0 = np.array([50])\n",
    "alpha = 0.02\n",
    "\n",
    "f = lambda t, z : alpha * z\n",
    "t_span = (0, 100)\n",
    "t_eval = np.linspace(0,100, 200)\n",
    "\n",
    "sol = solve_ivp(f, t_span, z0, t_eval=t_eval)\n",
    "    \n",
    "plt.plot(sol.t, sol.y[0], c='k')\n",
    "plt.title(\"The Zebra Problem\")\n",
    "plt.ylabel(\"zebras\")\n",
    "plt.xlabel(\"months\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Things do not look good.  There is no way you can afford to feed even 100 zerbas while paying tuition at the University of Chicago.  Hard choices will need to be made.\n",
    "\n",
    "### A Natural Solution\n",
    "\n",
    "When posting a \"Zebras for sale\" ad on craigslist, you see an add for a family of lions that need a home.  You feel bad for the lions, and realize you can use them to solve your zebra problem.  However, before commiting, you want to make sure the lions won't eat all the zebras (you just want to keep the population controlled, not exterminated).  You read a bit about how lions and zebras might interact and decide to use a [Predator-Prey model](https://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equations).  \n",
    "1. Zebras will continue to have babies at rate $\\alpha$.\n",
    "2. A lion will catch and eat every $100$th zebra it sees.  The number of zebras a lion sees depends on the zebra population, we'll denote the rate at which a lion encounters a single zebra in a month as $\\beta$.\n",
    "3. For every 5 zebras the lions eat, a lion baby is born.  \n",
    "4. Lions die of old age at a rate of $\\gamma$\n",
    "\n",
    "We'll use $x$ to denote the number of lions, and $z$ for the number of zebras.  Now, the amount the zebra population changes every month is $\\Delta_z(z,x) = \\alpha z - \\frac{\\beta}{100} x z$.  The amount the lion population changes every month is $\\Delta_x(x,z) = \\frac{\\beta}{500} x z - \\gamma x$.  \n",
    "\n",
    "In order to solve this problem, we can use a vector $y = [x, z]^T$, and the following ODE system:\n",
    "\\begin{equation}\n",
    "y_0'(t) = \\frac{\\beta}{500} y_0 y_1 - \\gamma y_0\\\\\n",
    "y_1'(t) = \\alpha y_1 - \\frac{\\beta}{100} y_0 y_1\n",
    "\\end{equation}\n",
    "\n",
    "You estimate $\\gamma$ to be $0.02$, and $\\beta$ to be $1.0$.\n",
    "\n",
    "You plan to start with 5 lions and 50 zebras."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "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": [
    "z0 = 50\n",
    "x0 = 5\n",
    "alpha = 0.02\n",
    "beta = 1.0\n",
    "gamma = 0.02\n",
    "\n",
    "y0 = np.array([x0, z0])\n",
    "f = lambda t, y : np.array([\n",
    "    -gamma*y[0] + beta/500 * y[0]*y[1],\n",
    "    alpha*y[1] - beta/100 * y[0]*y[1]\n",
    "] )\n",
    "\n",
    "t_eval = np.linspace(0, 100, 200)\n",
    "\n",
    "sol = solve_ivp(f, t_span, y0, t_eval=t_eval)\n",
    "\n",
    "    \n",
    "plt.plot(sol.t, sol.y[1], label=\"zebras\", c='k')\n",
    "plt.plot(sol.t, sol.y[0], label=\"lions\", c='r')\n",
    "plt.title(\"Lions v.s. Zebras\")\n",
    "plt.ylabel(\"population\")\n",
    "plt.xlabel(\"months\")\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Oh no!  Soon the zebras will be eaten, and you'll be left with a pack of hungry lions.  You wonder what would happen if you do some landscaping to make it easier for the zebras to hide.  Then you might have $\\beta = 0.3$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "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": [
    "z0 = 50\n",
    "x0 = 5\n",
    "alpha = 0.02\n",
    "beta = 0.3\n",
    "gamma = 0.02\n",
    "\n",
    "y0 = np.array([x0, z0])\n",
    "f = lambda t, y : np.array([\n",
    "    -gamma*y[0] + beta/500 * y[0]*y[1],\n",
    "    alpha*y[1] - beta/100 * y[0]*y[1]\n",
    "] )\n",
    "\n",
    "t_eval = np.linspace(0, 100, 200)\n",
    "\n",
    "sol = solve_ivp(f, t_span, y0, t_eval=t_eval)\n",
    "\n",
    "    \n",
    "plt.plot(sol.t, sol.y[1], label=\"zebras\", c='k')\n",
    "plt.plot(sol.t, sol.y[0], label=\"lions\", c='r')\n",
    "plt.title(\"Lions v.s. Zebras\")\n",
    "plt.ylabel(\"population\")\n",
    "plt.xlabel(\"months\")\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "That looks a bit better - what if we run the simulation for longer?"
   ]
  },
  {
   "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": [
    "z0 = 50\n",
    "x0 = 5\n",
    "alpha = 0.02\n",
    "beta = 0.3\n",
    "gamma = 0.02\n",
    "\n",
    "\n",
    "y0 = np.array([x0, z0])\n",
    "f = lambda t, y : np.array([\n",
    "    -gamma*y[0] + beta/500 * y[0]*y[1],\n",
    "    alpha*y[1] - beta/100 * y[0]*y[1]\n",
    "] )\n",
    "\n",
    "t_eval = np.linspace(0, 1000, 2000)\n",
    "t_span = (0, 1000)\n",
    "\n",
    "sol = solve_ivp(f, t_span, y0, t_eval=t_eval)\n",
    "\n",
    "    \n",
    "plt.plot(sol.t, sol.y[1], label=\"zebras\", c='k')\n",
    "plt.plot(sol.t, sol.y[0], label=\"lions\", c='r')\n",
    "plt.title(\"Lions v.s. Zebras\")\n",
    "plt.ylabel(\"population\")\n",
    "plt.xlabel(\"months\")\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This seems like a good solution - you install some hiding places for zebras, and give the lions a home.  You start charging interested visitors for tours, and open a gift shop.  Soon you are able to cover the expenses for the zerba hay.  You quit your day job and become a safari guide in your own backyard."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercises\n",
    "\n",
    "1. Come up with agent-based models for the zebra growth model and lion-zebra predator-prey model.  How do these compare to the ODE models in terms of speed?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "## Your code here\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Symbolic Solutions of ODEs\n",
    "\n",
    "You can use SymPy for symbolic solutions of ODEs.  For instance, in our zebra growth problem, we had an ODE of the form\n",
    "\\begin{equation}\n",
    "y'(t) = a y(t)\n",
    "\\end{equation}\n",
    "We can use SymPy to classify the ODEs and find a symbolic solution (if one exists)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy as sym\n",
    "from sympy.solvers import ode"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - a y{\\left(t \\right)} + \\frac{d}{d t} y{\\left(t \\right)}$"
      ],
      "text/plain": [
       "-a*y(t) + Derivative(y(t), t)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a, t = sym.symbols('a t') # symbol\n",
    "y = sym.Function('y') # symbolic function\n",
    "eqn = y(t).diff(t) - a * y(t) # eqn = 0\n",
    "eqn"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "('separable',\n",
       " '1st_exact',\n",
       " '1st_linear',\n",
       " 'Bernoulli',\n",
       " 'almost_linear',\n",
       " '1st_power_series',\n",
       " 'lie_group',\n",
       " 'nth_linear_constant_coeff_homogeneous',\n",
       " 'separable_Integral',\n",
       " '1st_exact_Integral',\n",
       " '1st_linear_Integral',\n",
       " 'Bernoulli_Integral',\n",
       " 'almost_linear_Integral')"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ode.classify_ode(eqn)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You might recognize some of the above from an ODE class. We see that this ODE is fairly simple, and a variety of methods might be used to solve the equation symbolicly."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle y{\\left(t \\right)} = C_{1} e^{a t}$"
      ],
      "text/plain": [
       "Eq(y(t), C1*exp(a*t))"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ode.dsolve(eqn, hint='separable') # we solve the differential equation using the hinted method"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If you have taken a class on ODEs, hopefully this looks correct.  If you don't remember your ODEs, or haven't seen them before, try verifying that the solution satisfies `y'(t) = a * y(t)`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The Logistic Equation\n",
    "\n",
    "Another ODE is the Logistic equation\n",
    "\\begin{equation}\n",
    "y'(t) = y(t) (1 - y(t))\n",
    "\\end{equation}\n",
    "\n",
    "This models growth of a population with resource limits. \n",
    "\n",
    "First, let's use SymPy to find a symbolic solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\left(1 - y{\\left(t \\right)}\\right) y{\\left(t \\right)} + \\frac{d}{d t} y{\\left(t \\right)}$"
      ],
      "text/plain": [
       "-(1 - y(t))*y(t) + Derivative(y(t), t)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t = sym.symbols('t') # symbol\n",
    "y = sym.Function('y') # symbolic function\n",
    "eqn = y(t).diff(t) - y(t) * (1 - y(t))\n",
    "eqn"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "('separable',\n",
       " '1st_exact',\n",
       " 'Bernoulli',\n",
       " '1st_power_series',\n",
       " 'lie_group',\n",
       " 'separable_Integral',\n",
       " '1st_exact_Integral',\n",
       " 'Bernoulli_Integral')"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ode.classify_ode(eqn)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle y{\\left(t \\right)} = \\frac{1}{C_{1} e^{- t} + 1}$"
      ],
      "text/plain": [
       "Eq(y(t), 1/(C1*exp(-t) + 1))"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = ode.dsolve(eqn, hint='separable')\n",
    "f"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's just set the constant $C_1$ to be 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "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"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x7f3cb6e796d0>"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "exp = 1 / (sym.exp(-t) + 1)\n",
    "sym.plot(exp)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's look at solving the ODE with $y(0) = 0.1$ .  We can pass this into SymPy's `ode.dsolve` using the `ics` keyword:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle y{\\left(t \\right)} = \\frac{1}{1 + 9.0 e^{- t}}$"
      ],
      "text/plain": [
       "Eq(y(t), 1/(1 + 9.0*exp(-t)))"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = ode.dsolve(eqn, hint='separable', ics={y(0): 0.1})\n",
    "f"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "now, because we actually solved the IVP, there are no constants to fill in, and we can plot the right hand side of the equation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "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"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x7f3cb6d2bf40>"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.plot(f.rhs)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, let's solve the IVP numerically using `scipy`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "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": [
    "y0 = np.array([0.1])\n",
    "\n",
    "f = lambda t, y : y * (1 - y)\n",
    "t_span = (0, 10)\n",
    "t_eval = np.linspace(0,10, 200)\n",
    "\n",
    "sol = solve_ivp(f, t_span, y0, t_eval=t_eval)\n",
    "    \n",
    "plt.plot(sol.t, sol.y[0], c='k')\n",
    "plt.title(\"Logistic Function\")\n",
    "plt.ylabel(\"y\")\n",
    "plt.xlabel(\"t\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercises\n",
    "\n",
    "1. Get an exact expression for the zebra growth IVP using SymPy.  What is the worst-case error over the 100-month simulation we conducted?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "## Your code here\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Euler Method\n",
    "\n",
    "The [(forward) Euler Method](https://en.wikipedia.org/wiki/Euler_method) is a numerical method for solving Initial Value Problems.  The basic idea is to choose a step size `h`, and iteratively compute `y(t0), y(t0+h), y(t0+2h), ...` using `y(t0 + (k+1)h) = y(t0 + kh) + h f(t, y(t0 + hk))`\n",
    "\n",
    "Here's an example implementation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "def forward_euler(f, y0, t0=0, h=1e-2, n=100):\n",
    "    \"\"\"\n",
    "    compute n forward euler steps\n",
    "    \"\"\"\n",
    "    y = [y0]\n",
    "    t = [t0]\n",
    "    for k in range(n-1):\n",
    "        fk = f(t[-1], y[-1])\n",
    "        y.append(y[-1] + h*fk)\n",
    "        t.append(t[-1]+h)\n",
    "        \n",
    "    return np.array(y), np.array(t)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's use this function to compute the solution to the Logistic equation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "f = lambda t, y : y * (1 - y)\n",
    "\n",
    "y, t = forward_euler(f, 0.1, h=1e-1, n=100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "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": [
    "plt.plot(t, y, c='k')\n",
    "plt.title(\"Logistic Function\")\n",
    "plt.ylabel(\"y\")\n",
    "plt.xlabel(\"t\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise\n",
    "\n",
    "1. write a version of `forward_euler` that will take in arguments in the same manner as `solve_ivp` (`f`, `tspan`,  `y0`, and `t_eval`).  You can define `h` for every step in `t_eval`.  Return a solution class which has fields `y` and `t`, just like `solve_ivp`.  Compare the output of your function to the output of `solve_ivp` on the logistic equation IVP above.  How accurate is either compared to a ground truth logistic function?\n",
    "\n",
    "2. Try using some of the different methods in the [`solve_ivp` documentation](https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.solve_ivp.html).  How does your accuracy change in both the Logistic equation and growth equation?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Events\n",
    "\n",
    "You can ask `solve_ivp` to find the location of events. An event is defined as a function `event(t, y)`, and `solve_ivp` will find values of `t` where `event(t,y) = 0`.  For instance, in the zebra growth problem, we might want to find the value of `t` where the number of zebras reaches 100.  We can pass in an event or list of events using the `events` keyword."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "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 numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy.integrate import solve_ivp\n",
    "\n",
    "z0 = np.array([50])\n",
    "alpha = 0.02\n",
    "\n",
    "f = lambda t, z : alpha * z\n",
    "t_span = (0, 100)\n",
    "t_eval = np.linspace(0,100, 200)\n",
    "\n",
    "def zebra100(t, y): \n",
    "    return y - 100\n",
    "\n",
    "sol = solve_ivp(f, t_span, z0, t_eval=t_eval, events=zebra100)\n",
    "    \n",
    "plt.plot(sol.t, sol.y[0], c='k')\n",
    "plt.title(\"The Zebra Problem\")\n",
    "plt.ylabel(\"zebras\")\n",
    "plt.xlabel(\"months\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "we can access the time at which events occur using the `t_events` field of the solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[array([34.65385735])]"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sol.t_events"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If you want to end the simulation when an event occurs, you can set a flag `terminal=True`.  You can also specify a direction flag to only trigger an event when the root of the function goes from negative to positive (`direction=1`), or positive to negative (`direction=-1`)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "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 numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy.integrate import solve_ivp\n",
    "\n",
    "z0 = np.array([50])\n",
    "alpha = 0.02\n",
    "\n",
    "f = lambda t, z : alpha * z\n",
    "t_span = (0, 100)\n",
    "t_eval = np.linspace(0,100, 200)\n",
    "\n",
    "def zebra100(t, y): \n",
    "    return y - 100\n",
    "\n",
    "zebra100.terminal=True # we can set this a flag on the function object\n",
    "zebra100.direction=1\n",
    "\n",
    "sol = solve_ivp(f, t_span, z0, t_eval=t_eval, events=zebra100)\n",
    "    \n",
    "plt.plot(sol.t, sol.y[0], c='k')\n",
    "plt.title(\"The Zebra Problem\")\n",
    "plt.ylabel(\"zebras\")\n",
    "plt.xlabel(\"months\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[array([34.65385735])]"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sol.t_events"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python (pycourse)",
   "language": "python",
   "name": "pycourse"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.17"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}