welcome.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Welcome to MAT230, Nonlinear differential equations.\n",
    "This is the first Jupyter notebook. The idea is to see if everone can get it to work and do a first example.\n",
    "\n",
    "We need to load a couple of libraries first. lick inside the cell below and hold *shift* and hit *return*."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import dsolve, Eq, symbols, Function, lambdify,solve, classify_ode\n",
    "from sympy.plotting import plot\n",
    "from scipy.integrate import solve_ivp\n",
    "import numpy as np\n",
    "from matplotlib import pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## A simple first order example\n",
    "So now we have the libraries loaded. Let's demonstrate the use of *sympy* on this first order differential equation\n",
    "$$\\frac{d x(t)}{d t} = 1-x(t)$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{d}{d t} x{\\left(t \\right)} = 1 - x{\\left(t \\right)}$"
      ],
      "text/plain": [
       "Eq(Derivative(x(t), t), 1 - x(t))"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t = symbols('t')\n",
    "x = symbols('x', cls=Function)\n",
    "\n",
    "deqn1 = Eq(x(t).diff(t), 1-x(t))\n",
    "deqn1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Lets classify the equation"
   ]
  },
  {
   "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_undetermined_coefficients',\n",
       " 'nth_linear_constant_coeff_variation_of_parameters',\n",
       " 'separable_Integral',\n",
       " '1st_exact_Integral',\n",
       " '1st_linear_Integral',\n",
       " 'Bernoulli_Integral',\n",
       " 'almost_linear_Integral',\n",
       " 'nth_linear_constant_coeff_variation_of_parameters_Integral')"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "classify_ode(deqn1, x(t))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and solve it"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x{\\left(t \\right)} = C_{1} e^{- t} + 1$"
      ],
      "text/plain": [
       "Eq(x(t), C1*exp(-t) + 1)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sol1 = dsolve(deqn1, x(t))\n",
    "\n",
    "sol1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is the general solution, the constant $C_1$ will be detemined by the initial conditions \n",
    "$$ x(t_0)=X_0.$$\n",
    "\n",
    "Let's solve the Initial Value Problem (IVP).  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x{\\left(t \\right)} = 1 + e^{- t}$"
      ],
      "text/plain": [
       "Eq(x(t), 1 + exp(-t))"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ics={x(0):2}\n",
    "sol2 = dsolve(deqn1, x(t),ics={x(0):2})\n",
    "\n",
    "sol2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We could also use the general solution. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The solution with initial conditions  {x(0): 2}  we set C=  [1]\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x{\\left(t \\right)} = 1 + e^{- t}$"
      ],
      "text/plain": [
       "Eq(x(t), 1 + exp(-t))"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C1=symbols('C1')\n",
    "X0=symbols('X0')\n",
    "C=symbols('C')\n",
    "C=solve(sol1.subs(t,0).subs(x(0),ics[x(0)]),C1)\n",
    "print('The solution with initial conditions ', ics, ' we set C= ', C)\n",
    "sol1.subs(C1,C[0])\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Plotting the solution of the initial condition\n",
    "We use the *lambdify* routine "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [],
   "source": [
    "f=lambdify(t,sol2.rhs,'numpy')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and we can evaluate the solution at different *t* values. Try to change the number below."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.3328710836980795"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(1.1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can create an array of *t*, using *np.linspace(start,end,n)*. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.        , 0.04040404, 0.08080808, 0.12121212, 0.16161616,\n",
       "       0.2020202 , 0.24242424, 0.28282828, 0.32323232, 0.36363636,\n",
       "       0.4040404 , 0.44444444, 0.48484848, 0.52525253, 0.56565657,\n",
       "       0.60606061, 0.64646465, 0.68686869, 0.72727273, 0.76767677,\n",
       "       0.80808081, 0.84848485, 0.88888889, 0.92929293, 0.96969697,\n",
       "       1.01010101, 1.05050505, 1.09090909, 1.13131313, 1.17171717,\n",
       "       1.21212121, 1.25252525, 1.29292929, 1.33333333, 1.37373737,\n",
       "       1.41414141, 1.45454545, 1.49494949, 1.53535354, 1.57575758,\n",
       "       1.61616162, 1.65656566, 1.6969697 , 1.73737374, 1.77777778,\n",
       "       1.81818182, 1.85858586, 1.8989899 , 1.93939394, 1.97979798,\n",
       "       2.02020202, 2.06060606, 2.1010101 , 2.14141414, 2.18181818,\n",
       "       2.22222222, 2.26262626, 2.3030303 , 2.34343434, 2.38383838,\n",
       "       2.42424242, 2.46464646, 2.50505051, 2.54545455, 2.58585859,\n",
       "       2.62626263, 2.66666667, 2.70707071, 2.74747475, 2.78787879,\n",
       "       2.82828283, 2.86868687, 2.90909091, 2.94949495, 2.98989899,\n",
       "       3.03030303, 3.07070707, 3.11111111, 3.15151515, 3.19191919,\n",
       "       3.23232323, 3.27272727, 3.31313131, 3.35353535, 3.39393939,\n",
       "       3.43434343, 3.47474747, 3.51515152, 3.55555556, 3.5959596 ,\n",
       "       3.63636364, 3.67676768, 3.71717172, 3.75757576, 3.7979798 ,\n",
       "       3.83838384, 3.87878788, 3.91919192, 3.95959596, 4.        ])"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tval=np.linspace(0,4,100)\n",
    "tval"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and finally, plot the solution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f926c17c5e0>]"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(tval,f(tval))"
   ]
  }
 ],
 "metadata": {
  "interpreter": {
   "hash": "cc5aea2381060e6c07a5bcea4572f763dfe34e6a40ece333d15e520321d95b42"
  },
  "kernelspec": {
   "display_name": "Python 3.8.5 64-bit ('venv': venv)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.5"
  },
  "orig_nbformat": 4
 },
 "nbformat": 4,
 "nbformat_minor": 2
}