Finite Difference Method on u''(x) + sin(x) = 0.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "I have numerically solved for $ u(x) $ as a function of $ x $ for the equation $ \\frac{d^2}{dx^2}u(x) + sin(x) = 0 $ .Boundary conditions: $ x{\\epsilon}[0,1] ;$ $ u(0)=u(1)=0 $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from pylab import *\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# first generate the matrix containing values of x in [0,1] #\n",
    "\n",
    "n_divisions = 100 # number of divisions of [0,1]\n",
    "h = 1/n_divisions # spacial separation between consecutive divisions\n",
    "x_matrix = np.zeros(n_divisions-1)\n",
    "for i in range(n_divisions-1):\n",
    "    x_matrix[i] = 1*(i+1)/n_divisions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "# generate the matrix containing the values of f(x) #\n",
    "\n",
    "fx_matrix = np.zeros((n_divisions-1,1), 'double')\n",
    "for i in range(n_divisions-1):\n",
    "    fx_matrix[i]=sin(x_matrix[i])\n",
    "\n",
    "#fx_matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "for i in range(1,n_divisions-1):\n",
    "    fx_matrix[i] += (fx_matrix[i-1])*i/(i+1)\n",
    "\n",
    "#fx_matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "for i in range(1,n_divisions-1):\n",
    "    fx_matrix[n_divisions-2-i] += (fx_matrix[n_divisions-1-i])*(n_divisions-i)/(n_divisions+1-i)\n",
    "    \n",
    "#fx_matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "# generate the matrix containing the values of u(x) #\n",
    "\n",
    "ux_matrix = np.zeros((n_divisions-1,1), 'double')\n",
    "\n",
    "for i in range(n_divisions-1):\n",
    "    ux_matrix[i] = h*h*fx_matrix[i]*(i+1)/(i+2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7fe5944c8350>]"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "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": [
    "# plot the numerical values of u(x) versus x #\n",
    "\n",
    "plt.plot(x_matrix,ux_matrix)"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "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": "3.7.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}