Lab-10 finite difference.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Lab 10\n",
    "~Rohit R Jorige"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [],
   "source": [
    "#libraries\n",
    "import math\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np # only used to plot the graph\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "#gauss seidel method without using numpy\n",
    "def norm(x):\n",
    "    return max(x)\n",
    "def subtract_list_elem(a,b):\n",
    "    ans=[]\n",
    "    for i in range(len(a)):\n",
    "        ans.append(a[i]-b[i])\n",
    "    return ans\n",
    "def summation_seidel(A, i, curr_x, new_x):\n",
    "    final_sum = 0\n",
    "    for j in range(0, i):\n",
    "        final_sum += new_x[j] * A[i][j]\n",
    "    for j in range(i+1, len(A[i])):\n",
    "        final_sum += curr_x[j] * A[i][j]\n",
    "    return final_sum\n",
    "\n",
    "def Seidel_alt(A, b, x, tol):\n",
    "    prev_x = x\n",
    "    curr_x = x\n",
    "    while True:\n",
    "        new_x = []\n",
    "        for i in range(len(A)):\n",
    "            x_i = (b[i] - summation_seidel(A, i, curr_x, new_x)) / A[i][i]\n",
    "            new_x.append(x_i)\n",
    "        if norm(subtract_list_elem(new_x, curr_x)) / norm(new_x) < tol:\n",
    "            return new_x\n",
    "        curr_x = new_x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "#general finite difference method\n",
    "#inputs : endpoints (a,b) , boundary conditions (y(a),y(b)) , integrer N>=2, differential equation(p(x),q(x),r(x))\n",
    "#ouput : approximations w_i to y(x_i) for each i = 0, 1, ... , N + 1\n",
    "#the differential equation is of the form y' = p(x)y' + q(x)y + r(x)\n",
    "def construct_matrix(N):\n",
    "    matrix_A=[]\n",
    "    matrix_B=[]\n",
    "    matrix_X=[]\n",
    "    for i in range(N):\n",
    "        matrix_A.append([])\n",
    "        matrix_B.append(0)\n",
    "        matrix_X.append(0)\n",
    "    for  i in range(N):\n",
    "        for j in range(N):\n",
    "            matrix_A[i].append(0)\n",
    "    return matrix_A,matrix_B,matrix_X\n",
    "def finite_difference(a,b,alpha,beta,N,p,q,r):\n",
    "    matrix_A,matrix_B,matrix_X=construct_matrix(N)\n",
    "    h=(b-a)/(N+1)\n",
    "    x=a+h\n",
    "    #calculating  all x_1 values\n",
    "    a_1=2 + (h**2)*q(x)\n",
    "    b_1= -1 + (h/2)*p(x)\n",
    "    d_1=-(h**2)*r(x) + (1 + (h/2)*p(x))*alpha\n",
    "    matrix_A[0][0]=a_1\n",
    "    matrix_A[0][1]=b_1\n",
    "    matrix_B[0]=d_1\n",
    "    #calculating all x_i values from 2 to N-1\n",
    "    for i in range(1,N-1):\n",
    "        x=a+(i+1)*h\n",
    "        a_i= -1 - (h/2)*p(x)\n",
    "        b_i=  2 + (h**2)*q(x)\n",
    "        c_i= -1 + (h/2)*p(x)\n",
    "        d_i=-(h**2)*r(x)\n",
    "        matrix_A[i][i-1]=a_i\n",
    "        matrix_A[i][i]=b_i\n",
    "        matrix_A[i][i+1]=c_i\n",
    "        matrix_B[i]=d_i\n",
    "    #calculating  all x_N values\n",
    "    x=b-h\n",
    "    a_N= -1 - (h/2)*p(x)\n",
    "    b_N=  2 + (h**2)*q(x)\n",
    "    d_N=  -(h**2)*r(x) + (1 - (h/2)*p(x))*beta\n",
    "    matrix_A[N-1][-2]=a_N\n",
    "    matrix_A[N-1][-1]=b_N\n",
    "    matrix_B[-1]=d_N\n",
    "    #print(\"Matrix A is\",matrix_A)\n",
    "    #print(\"Matrix B is\",matrix_B)\n",
    "    return Seidel_alt(matrix_A,matrix_B,matrix_X,0.00000000001)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Q1.** Use the Linear Finite-Difference Algorithm with N = 9 to approximate the solution to the\n",
    "boundary value problem,\n",
    "\n",
    "$y^{\"} = -\\frac{2}{x}y^{'} +\\frac{2}{x^2}y+\\frac{sin(\\ln x)}{x^2}$\n",
    "\n",
    "$1 \\leq x \\leq 2$ , $y(1) = 1$ , $y(2) = 2$\n",
    "\n",
    "and compare the results to those obtained using the linear shooting method for the same problem.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.0926005206437606\n",
      "1.187043128668555\n",
      "1.2833368700615362\n",
      "1.3814020460698797\n",
      "1.4811202619640589\n",
      "1.5823598955678606\n",
      "1.6849890182890201\n",
      "1.788881746131507\n",
      "1.893921099127733\n"
     ]
    }
   ],
   "source": [
    "def p(x):\n",
    "    ans=-2/x\n",
    "    return ans\n",
    "def q(x):\n",
    "    ans=2/(x**2)\n",
    "    return ans\n",
    "def r(x):\n",
    "    ans= (math.sin(math.log(x)))/x**2\n",
    "    return ans\n",
    "solution=finite_difference(1,2,1,2,9,p,q,r)\n",
    "for i in solution:\n",
    "    print(i)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Q2.** Consider the boundary value problem\n",
    "\n",
    "$y^{\"}=-(x+1)y^{'}+2y+(1-x^2)e^{-x}$\n",
    "\n",
    "$0 \\leq x \\leq 1$ , $y(0) = 1$ , $y(1) = 0$\n",
    "\n",
    "Use N = 9 and N = 19 respectively and apply the Linear Finite-Difference Algorithm with N = 9\n",
    "to approximate the solution to the above boundary value problem\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The solution for N=9 is \n",
      "0.7637068009927273\n",
      "0.5750195530328508\n",
      "0.4259574711601672\n",
      "0.3094199581908733\n",
      "0.21917622033641565\n",
      "0.14983393833148204\n",
      "0.09679150666215999\n",
      "0.05617822574757339\n",
      "0.024786540568339992\n",
      "The solution for N=19 is \n",
      "0.8754842088255258\n",
      "0.7639151932535033\n",
      "0.6642151260394605\n",
      "0.5753604387711247\n",
      "0.4963824866922035\n",
      "0.42636778191720764\n",
      "0.3644578288912448\n",
      "0.30984859700629647\n",
      "0.2617896659160541\n",
      "0.21958307930777204\n",
      "0.18258194271113576\n",
      "0.15018880037577106\n",
      "0.12185382536047765\n",
      "0.09707285578216729\n",
      "0.07538530870738477\n",
      "0.05637200147277722\n",
      "0.03965290833267316\n",
      "0.02488487829202604\n",
      "0.011759337830834283\n"
     ]
    },
    {
     "data": {
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tLbm5ucCJX+/KcHudEStUy3VGyrNvA0zuBqYTBn4C515udSIRkVotPT2dI0eO0KRJE0JDQzF0c1OPMU2T3Nxc9u3bR/369YmOjj5lTEV/fuveNJ7UpC10vQ+WToHU4fDnH8CuX2IREav8PrnidKuES9XVr1//tJNYKko/KT3tssfh11mwbx0sf6OknIiIiCUMwyA6OpomTZpQVFRkdZwaJyAgwCPXf6qMeFpoA7jiSZj3KObXz7IxN4ycnCxCImJpm5CM3aFfchERX7Pb7TVn0kQNpGtGvMFZTN749oTkpZfZnElD9iam0Cl5kEXBREREfMcr96aRiln51f8Izk0/ZXtj8yAdFz/MyvkzLEglIiJSPamMeJizuJiYJaMo73CT7fhF3NFLRuEsLvZpLhERkepKZcTDNiydTyQHS4vHyWwGRHGQDUvn+zaYiIhINaUy4mF5h/d4dJyIiEhNpzLiYSERsR4dJyIiUtOpjHhY24RkMmmI6zRzlFwmZNCQtgnJvg0mIiJSTamMeJjd4WBvYgpAuYXEALZ3ekzrjYiIiBynMuIFnZIHsarby+w3GpbZ7jQNDAOyty7XXSRFRESO06JnXuQsLmbD0vnkHd5DSEQsoeTSYsG9OE2D7y+byRVX9rI6ooiIiNfoRnnVgN3h4ILuvcts25T2Pq33zyfm+8dIj+9GdAP/KVciIiLeoNM0PnbugIlkG2G0YQeLZozQ6RoREan1VEZ8zBHWhNyrngOg75F3+HThdxYnEhERsZbKiAWiug9gV8PuBBnFxP7wGDsOHLU6koiIiGVURqxgGMTeOYU8I4TOxkYWzBiN83QLk4iIiNRwKiMWsUWcQ37PEQDcmv06M7/60eJEIiIi1lAZsVDEZQ+wP6ITdY18mi56kk0Z2VZHEhER8TmVESvZbDS6/VWKCOAyWxpz35lAkdNldSoRERGfUhmxmNG4DQXd/wHA3Udf5bX5P1ucSERExLdURqqBulc+SlZYGxoYOcQu+Re/7j5idSQRERGfURmpDuwBhPefggsbfe2LmfW/18gvclqdSkRExCdURqqL2IsovPgBAIbkTuLleSssDiQiIuIbKiPVSPDVT5Fb9xxijENE/zKWn7YetDqSiIiI16mMVCeBoYT+30QABti/4u1Z75FTUGxxKBEREe9SGaluzr2Mwo53AjA0byLjPk2zNo+IiIiXqYxUQ4G9nqMwpAktbelEpf2HbzbuszqSiIiI16iMVEch9Qns+28A/mz/jNfe/4QjuYUWhxIREfEOlZHq6vw+ONv2xWG4GFY4kVGfrLI6kYiIiFeojFRj9t7jKQ4Mp4NtO03WTGfe6nSrI4mIiHicykh1Vi8Sx7WjAfi7Yw6vfvgl+47mWxxKRETEs1RGqrv4O3C1uJxgo4jhzsk8+cGvmKZpdSoRERGPURmp7gwDW9//4HKEcIltPY1+m8mc5butTiUiIuIxKiP+IKI5tqtGADDc8S5TPl3EniN5FocSERHxDJURf5HwF8yYzoQZeQxzTeOfs9NwuXS6RkRE/J/KiL+w2TFumIhpC+Bq+3Iits/j7Z92WJ1KRESkylRG/ElkO4weQwEYFfAmk79Yxtb9ORaHEhERqRqVEX/T4x+YjdvSyMjmn7zFP95fRbHTZXUqERGRSlMZ8TeOIIy+r2BicJP9B+rt/o5Xv99qdSoREZFKUxnxR3FdMRL+DMDogNdYsXA2e354C7b9AC6nxeFERETcY5h+sIJWdnY24eHhZGVlERYWZnWc6qEgB/M/F2LkHiy7PSwGeo2Ddn2tySUiInJcRX9+68iIv9ry9alFBCA7HWYPhHVzfZ9JRESkElRG/JHLCamPn+bJ4we6UofplI2IiPgFlRF/tGMxZO89wwATsveUjBMREanmVEb8UU6mZ8eJiIhYSGXEH9WN9Ow4ERERC6mM+KNm3UpmzWCU+7RpQiYNORp5sW9ziYiIVILKiD+y2Uum7wInFxITMAz4rrg9z8zb6PNoIiIi7lIZ8Vft+sItb0FYdJnNRnB9AG6w/8ja5Yv4ap2uGxERkerNYXUAqYJ2faFt75JZMzmZJdeInJMIswcQtHEeLwe8wqAPmnFRs2toUCfQ6rQiIiLl0pERf2ezQ4se0OHmkq92B/SdiFk3mpa2dB4smMaIT9ZYnVJEROS0VEZqojoNMW6aionBrY5vYc1HzF11pnVJRERErKMyUlO16InRYygAYwJeY/JHX5OZnW9xKBERkVOpjNRklw/HFXsxYUYuz7omMHzOSvzgvogiIlLLqIzUZPYAbDe/hjOwHp1tv9Fx66vM+nmX1alERETKUBmp6SKaY+8zAYCH7B/zxWdz2HUo19pMIiIif6AyUht0uBlX/B3YDJMxxiv8a9YPuFw6XSMiItWDykgtYbv2eYrqtyTGOMQte5/n9UVbrY4kIiICqIzUHkF1CbjldZyGg2T7L+xaMJHN+45anUpERERlpFaJicd29b8AGG57m/+8+wnFTpe1mUREpNZTGalljEuGUND8SoKNIh48NIZXF661OpKIiNRyKiO1jc1G0M2vkh/UkDa23YT/MIo1e7KsTiUiIrWYykhtVLcJQTdPBeBO+wLm/G8KBcVOi0OJiEhtpTJSSxnnJZHX5a8APHLsZaZ9tsjiRCIiUltVqoxMmjSJ5s2bExwcTEJCAsuWLTvj+CNHjjBkyBCio6MJCgqidevWzJs3r1KBxXNCeo0iK+ICIowcuqx4nF+27rc6koiI1EJul5FZs2YxdOhQUlJSWLFiBR07diQ5OZl9+/aVO76wsJCrr76a7du3M2fOHDZu3Mi0adOIjY2tcnipIkcg4Xe+TYERwiW29ax8L4XcwmKrU4mISC1jmG7eOS0hIYGLL76YiRMnAuByuYiLi+Ohhx5i2LBhp4yfMmUKL7zwAhs2bCAgIKBSIbOzswkPDycrK4uwsLBKvYecXt7PbxPy+YMUmzZebz2Z+++41epIIiJSA1T057dbR0YKCwtZvnw5SUlJJ97AZiMpKYklS5aU+5q5c+eSmJjIkCFDiIyMpH379owePRqnUxdMVhchXe5kX/O+OAwX1216ksVrt1gdSUREahG3ysiBAwdwOp1ERkaW2R4ZGUlGRka5r9m6dStz5szB6XQyb948RowYwYsvvsizzz572s8pKCggOzu7zEO8yDBocuskDgXG0NQ4wLEPHiIrt9DqVCIiUkt4fTaNy+WiSZMmTJ06lc6dO9O/f3+efPJJpkyZctrXjBkzhvDw8NJHXFyct2NKcBiht71BMXaudv3IvHfGW51IRERqCbfKSKNGjbDb7WRmZpbZnpmZSVRUVLmviY6OpnXr1tjt9tJt559/PhkZGRQWlv+v7+HDh5OVlVX62LVrlzsxpZKCW1xCZuehANywZwI//LTY4kQiIlIbuFVGAgMD6dy5MwsXLizd5nK5WLhwIYmJieW+pnv37mzevBmX68Q9UDZt2kR0dDSBgYHlviYoKIiwsLAyD/GN2N7D2R7WhVCjgMapf+XAEZ0iExER73L7NM3QoUOZNm0aM2bMYP369TzwwAMcO3aMwYMHAzBw4ECGDx9eOv6BBx7g0KFDPPLII2zatInPP/+c0aNHM2TIEM/thXiOzU704BlkGfVoyzbS3vgbbk64EhERcYvD3Rf079+f/fv3M3LkSDIyMoiPjyc1NbX0otadO3dis53oOHFxccyfP5+///3vXHjhhcTGxvLII4/w+OOPe24vxKOCIpqScc0EwuffQ1LWByyadyUR9SPIO7yHkIhY2iYkY3e4/UdHRESkXG6vM2IFrTNijVVT76fj3lk4TQO7ceKPSSYN2ZuYQqfkQRamExGR6s4r64xI7eKM7YppUqaIADQ2D9Jx8cOsnD/DomQiIlKTqIxIuZzFxcT+PLrc52xGydfoJaNwFmv5eBERqRqVESnXhqXzieQghlH+8zYDojjIhqXzfRtMRERqHJURKVfe4T0eHSciInI6KiNSrpCIit1VuaLjRERETkdlRMrVNiGZTBriOs1cK9OEDBrSNiHZt8FERKTGURmRctkdDvYmpgCcUkhMEwwDttTrqvVGRESkylRG5LQ6JQ9iVbeX2W80LLP9qBEKQEL2fL5Lfd+KaCIiUoNo0TM5K2dxMRuWzj+xAmvXa1g/dSDt98/jiFmHLf0+pXOnzlbHFBGRaqaiP79VRqRSXIV57Pz3FTTPX89vxGHcs4BWcdFWxxIRkWpEK7CKV9kCQ4i6fw6HbA04j13sfWMg+7PzrI4lIiJ+SGVEKi24QVPst71LIQH0dC3j68l/I6/QaXUsERHxMyojUiXh5yWSlTQegP55M3nrtQk4TzcfWEREpBwqI1JljS+9i4wL7gFgQOY4ps+Za3EiERHxJyoj4hFR//c8+5t0I9Qo4Lq1Q5n13QqrI4mIiJ9QGRHPsDtoPPhdjoTE0dQ4QLOFf+XrtbutTiUiIn5AZUQ8JySC8MHvk28L5RLbejJn/401e7KsTiUiItWcyoh4lNHkfBx/eg0XBrcZC/h0+nPsPaIpvyIicnoqI+JxjvN7U9jzCQAedb7Gi6+9ydH8IotTiYhIdaUyIl4RfMU/yW19AwGGk+FHRzNixhcUOV1WxxIRkWpIZUS8wzAIvXkKeQ3b08jI5r49T/HMR7/gB3cfEBERH1MZEe8JDCVkwEwKgxpwgW0HF68awZRvt1idSkREqhmVEfGu+nEE3v4/nIaDPvafOPrVOD77da/VqUREpBpRGRHva9YNe++SJeMfdbzPp++/zi/bD1kcSkREqguVEfGNLoNxdbkHm2Ey3jaRMTM+YvuBY1anEhGRakBlRHzGdu04nOd0p56Rx3jnOB564xsOHyu0OpaIiFhMZUR8xx6Avf9bOMPiaGHL5J/ZY3ngrWUUFDutTiYiIhZSGRHfqtMI++3v4XKE0NO+mqQ9E3lszq+a8isiUoupjIjvRXXAduMUAO51fIHj15n8e8Emi0OJiIhVHFYHkFrqgn6Q+Rh8/zyjA6Zz2zdRXORawxWxJtSNhGbdwGa3OqWIiPiAyohY5/LhkLmWoI2f837gKOw//eFUTVgM9BoH7fpal09ERHxCp2nEOjYbXHADJmA3TrpmJDsdZg+EdXMtiSYiIr6jMiLWcTnhq39hlPvk8XKSOqxknIiI1FgqI2KdHYsh+0xLw5uQvadknIiI1FgqI2KdnEzPjhMREb+kMiLWqRvp2XEiIuKXVEbEOs26lcyaOd1VIyYcMMOYsSfat7lERMSnVEbEOjZ7yfRdoLxCYhhQhzw+nPcFX63TqRoRkZpKZUSs1a4v3PIWhJ109CMsBrNJO0KMIl4PGMeL733O6t1Z1mQUERGvMkw/uClIdnY24eHhZGVlERYWZnUc8QaXs2TWTE7miRVYi3JxvXk9tvQ09pgNuT9gDFMf7Ets/RCr04qISAVU9Oe3joxI9WCzQ4se0OHmkq82OwTVw3bnB7gatCTWOMiEwlE8PP0rsvOLrE4rIiIepDIi1VudRtgGfoKzbjTn2fYwIiuFv7/1I0VOl9XJRETEQ1RGpPqrH4d94McUB9Un3raFQbueIuWDlfjBGUYREakAlRHxD03a4hjwAcWOUHraV5O4+gkmf7PJ6lQiIuIBKiPiP5p2wXHrOzgNB33sP1Hv6yeYm7bH6lQiIlJFKiPiX1pdhf2mqZgYDHB8xc4PR/Dz9kNWpxIRkSpQGRH/0/4mzOvGA/Cg7QO+nvE02w4csziUiIhUlsqI+CVb13sp6jkcgMfNN/jftBc4dKzQ4lQiIlIZKiPitwKueJzcTvcC8Hj+y0yeNpn8IqfFqURExF0qI+K/DIPQPi+QfV4/AgwnQw8/y3/ffheXS1N+RUT8icqI+DebjbBbX+NwzGWEGIXcs+MxZnz8udWpRETEDSoj4v/sAUTc9R4HI+IJN3K5btUQPv12sdWpRESkglRGpGYIrEPD+z7iQGhLIo0jdPhmMD/9ut7qVCIiUgEqI1JzhDag4V8+42BAFM2NDMI/uJXfdmhRNBGR6k5lRGoUIyyGevd9yhFbfc43tpPz5s3sO3jY6lgiInIGKiNS4wQ2aY19wAfkEEoncx1bp/QnNz/f6lgiInIaKiNSI9Vr0YWc/3ubAgK4pGgpKycOxOl04SwuZu2Pn/PLZ1NZ++PnOIuLrY4qIlLrGaYf3Ic9Ozub8PBwsrKyCAsLszqO+JHN38+ixcI/YzdMVoQkEpu3iUgOlj6fSUP2JqbQKXmQhSlFRGqmiv781pERqdFa9ezPms7PAnBR3hKamAfLPN/YPEjHxQ+zcv4MK+KJiAgqI1ILtL/uLxwlFADDKPuc7fj30UtG6ZSNiIhFVEakxtuwdD71yD3t8zYDojjIhqXzfZhKRER+pzIiNV7e4YqtNVLRcSIi4lkqI1LjhUTEenSciIh4lsqI1HhtE5LJpCGnu5mvaUIGDWmbkOzbYCIiAqiMSC1gdzjYm5gCUG4hMQzYVicew2b3cTIREQGVEaklOiUPYlW3l9lvNCyz/SghACQeW8iC/z5MUbHTingiIrWaFj2TWsVZXMyGpfPJO7yHkIhY2iYks/6DZ2m//iUA5tW7mcuGTKFOcIDFSUVE/J9XFz2bNGkSzZs3Jzg4mISEBJYtW1ah182cORPDMOjXr19lPlakyuwOBxd0702X6+/ngu69sTsctO//L37rPAKA647O4bsJgzh4NM/ipCIitYfbZWTWrFkMHTqUlJQUVqxYQceOHUlOTmbfvn1nfN327dt59NFH6dGjR6XDinjLeX0eZWf3MbgwuC7/c5b95w52HThqdSwRkVrB7TLy73//m/vuu4/BgwfTrl07pkyZQmhoKK+//vppX+N0OrnjjjsYNWoU5557bpUCi3jLOVf/lf1J/6EYG9cWL2TdpP6s333w7C8UEZEqcauMFBYWsnz5cpKSkk68gc1GUlISS5YsOe3rnn76aZo0acI999xToc8pKCggOzu7zEPEFyIvHUTO9dMoxk6y+SN7p/Vn6W97rY4lIlKjuVVGDhw4gNPpJDIyssz2yMhIMjIyyn3NokWLmD59OtOmTavw54wZM4bw8PDSR1xcnDsxRaqkfpebKbj5bQoJ4CrjZwrevpUvV22zOpaISI3l1am9R48eZcCAAUybNo1GjRpV+HXDhw8nKyur9LFr1y4vphQ5VZ32veH2WRQaQfS0raLuB7czc9F6q2OJiNRIDncGN2rUCLvdTmZmZpntmZmZREVFnTJ+y5YtbN++nT59+pRuc7lcJR/scLBx40Zatmx5yuuCgoIICgpyJ5qIxwW2vgrnwI/If/tmurGOwC8HMunoq/y1VyeMk2//KyIilebWkZHAwEA6d+7MwoULS7e5XC4WLlxIYmLiKePbtm3L6tWrSUtLK3307duXK664grS0NJ1+kWrP3qI7QXd/Sr69Hl1sm+ix5G6eff9HnKdbW15ERNzm1pERgKFDhzJo0CC6dOlC165dmTBhAseOHWPw4MEADBw4kNjYWMaMGUNwcDDt27cv8/r69esDnLJdpLoymnYh+N555L/elwuLthGw5i88fuzfPHvnVQQHaAl5EZGqcvuakf79+zN+/HhGjhxJfHw8aWlppKamll7UunPnTtLT0z0eVMRS0RcSfF8q+cGNOd+2iwe2PcwjU+eRlVdkdTIREb+n5eBF3HFwCwWvX0/Qsb3scDVhZPhonr+vD5FhwVYnExGpdry6HLxIrdWwJUH3zaew3jk0s+1jTPZjPDxxDlv251idTETEb6mMiLir/jkE3jefoohWxBiHeKXgSR777yxW7jxsdTIREb+kMiJSGWExBNyTSnHjdjQxjjDNNZJnps3km437wOWEbT/A6jklX11Oq9OKiFRrumZEpCpyD+F860bsGWlkm6G86uzLkLrfEJr/h7V4wmKg1zho19e6nCIiFtA1IyK+ENoA+11zcTVNIMzI5VH7TELyyi4KSHY6zB4I6+Zak1FEpJpTGRGpquBwbHe8j2kPxDDg1MVZjx98TB2mUzYiIuVQGRHxhIxfMZyFZxhgQvYe2LHYZ5FERPyFyoiIJ+Rknn2MO+NERGoRlRERT6gb6dlxIiK1iMqIiCc061Yya4by7+ZrmrDfaMCuevE+jSUi4g9URkQ8wWYvmb4LnFxITEouanU6XTwy5WPWp2f7PJ6ISHWmMiLiKe36wi1vQVh0mc1GnSY4gxsQZTvC9KJhjJsynZ+2HrQopIhI9aNFz0Q8zeUsmTWTk1lyjUizbnDsAMXv3oojfQWFpp0Rrvu5ov8j9Gofffb3ExHxUxX9+a0yIuIrRXk4P/wz9vWfADCp+AbCe4/izsQWFgcTEfEOrcAqUt0EhGD/05u4Ln0UgCGOT4iYdz+vpP6KH/ybQETEa1RGRHzJZsOWNAKz32SchoPe9mX0WDyI0bO/w+lSIRGR2kllRMQCRvzt2AfNpSAgnHjbVu5adzfPvT6b/CItFy8itY/KiIhVmncn6C/fkFO3ObHGQYbuepiXJr1CVl6R1clERHxKZUTESg1bUnfIt2RFJVLXyOexw//i3ZeHk5mVZ3UyERGfURkRsVpIBOH3fcrhtrdhN0weyJvGopfvYkvmEauTiYj4hMqISHVgDyCi/2SOXDoSFwY3OVPJmNyXX7fstDqZiIjXqYyIVBeGQf2kf5BzwxvkE0R3VhH61rX8tHy51clERLxKZUSkmgnrdCPm4C84ZG9IK2M3583tx7dffWZ1LBERr1EZEamGQpp1pu6Q79kVdB4NjWwSf7iLhbMnWR1LRMQrVEZEqqnABk2J/fu3bKjfkyCjiKvWPcH3U/+By+nCWVzM2h8/55fPprL2x89xFhdbHVdEpNJ0bxqR6s7lIu2NR4jf9RYA6wLa06hoL004VDokk4bsTUyhU/Igq1KKiJxC96YRqSlsNuLveYXlF46i2LTRrmgNjc1DZYY0Ng/ScfHDrJw/w6KQIiKVpzIi4ifi+z5ItlEX0wTDKPuc7fj30UtG6ZSNiPgdlRERP7Fh6XwakH1KEfmdzYAoDrJh6XzfBhMRqSKVERE/kXd4j0fHiYhUFyojIn4iJCLWo+NERKoLlRERP9E2IZlMGuI6w/w304TMbatxOl2+CyYiUkUqIyJ+wu5wsDcxBeCUQuIyKb2w9crNY1jyQj92p2dakFJExH0qIyJ+pFPyIFZ1e5n9RsMy2/cZDVl5yX9Ydf5Qik0bl+Z/h3NKT776egF+sJSQiNRyWvRMxA85i4vZsHQ+eYf3EBIRS9uEZOwOBwCZa7/H9sHdNHbtp8AM4MMmf6XXoCeJqBtkcWoRqW0q+vNbZUSkBnIeO8TO1wfR4uD3AHxlJBJy03/p3v5ci5OJSG2iFVhFajF7nQa0eHAuexNGUoydJHMJTWcn8+rMD8gvclodT0SkDJURkZrKMIi59h8UD/qCw4FRNLPt46719/Pai8NYs/uI1elEREqpjIjUcMEtEoj4+0/sb3o1QUYxD+ZPZc/UPzF9QRrOM80TFhHxEZURkdogJILG97zPsSufoxgHybZlXPPDzTw1cQa7DuVanU5EajmVEZHawjCo0/NB7PctICe0KXG2/Yw6OJT3/vM4Hy7fpSnAImIZlRGRWsaIvYi6Dy8mt9X1BBpOHjPeot7Hg3jsne84kltodTwRqYVURkRqo+BwQu94B+e1L+A0ArjavpxHNt/Do/9+jUW/HSgZ43LCth9g9ZySry7NwhER79A6IyK1XfoqCt4bSFD2dopMOy8U30Kr1u3504FJGEf3nhgXFgO9xkG7vtZlFRG/okXPRKTi8rMpnvsIjnUfAiX3ucEAo8yg49/d8pYKiYhUiBY9E5GKCw7D8afXofe/MSm54Z5xyqDj/25JHaZTNiLiUSojIlLCMKBR63JKyB+ZkL0Hdiz2USgRqQ1URkTkhJxMz44TEakAlREROaFuZIWG5QQ09HIQEalNVEZE5IRm3UpmzZzhZE2RaeMfc9awePMB3+USkRpNZURETrDZS6bvAqe7hDXAcDG5aATr33yQcZ8s112ARaTKVEZEpKx2fUum74ZFl90eFovRbwrFF96GzTC5x/EFty6/lZETJrM+PduarCJSI2idEREpn8tZMmsmJ7PkWpJm3UqOnAD89hX5Hz1IcG46AO85kyi4IoWBl3fAZjvzfBwRqT206JmIeFd+NnlfPEXIqhkA7DUb8FbDoQwceC8x9UMsDici1YEWPRMR7woOI+TGlzEHfcrR0KbEGIcYdugpfp7Qn3nL1lqdTkT8iMqIiFSJ0aIn9f62jKyO9+PC4Aa+4+LPr+W1qS+TlVdkdTwR8QMqIyJSdYF1CL/xBVyDUzkU0pzGRhb37h3Bzy/05ec1G6xOJyLVnMqIiHiMo9klNBi6lIwLh+DERpJrMS3fv4qP35pAQVGx1fFEpJpSGRERzwoIJur/RlM4+CvSg1vSwMih39YUVo67ls2bN1qdTkSqIZUREfGKkGadiX70JzZf8AhFOLikeBlN3r6c72f9G5fTBYCzuJi1P37OL59NZe2Pn+Ms1tETkdpIU3tFxOsObUvjyHv3c25hyZGR1UEXkd/qOs5ZO5lIDpaOy6QhexNT6JQ8yKqoIuJBWmdERKoV01nEitmjuWDDKwQbRfz+N4/xhzXSXMe3rer2sgqJSA2gdUZEpFox7AF0vi2FjFu/pMB0YBhliwjA74u3Ri8ZpVM2IrWIyoiI+NSxQ+kEGacvGjYDojjIhqXzfZhKRKykMiIiPpV3eI9Hx4mI/1MZERGfComIrdC4/MN78YNL2kTEA1RGRMSn2iYkk0nD0otVT/Z7/7h0y0v8PO461q391XfhRMQSlSojkyZNonnz5gQHB5OQkMCyZctOO3batGn06NGDiIgIIiIiSEpKOuN4EanZ7A4HexNTAE4pJC4TTGB18MU4TYOu+YtpOfsK5v/nL2zbk+H7sCLiE26XkVmzZjF06FBSUlJYsWIFHTt2JDk5mX379pU7/ttvv+W2227jm2++YcmSJcTFxXHNNdewZ4/OB4vUVp2SB7Gq28vsNxqW2b7PaMiqbi/TYdhXHBjwNZvqdCHIKCb58HvUmZrAB9PHsi8r16LUIuItbq8zkpCQwMUXX8zEiRMBcLlcxMXF8dBDDzFs2LCzvt7pdBIREcHEiRMZOHBghT5T64yI1EzO4mI2LJ1P3uE9hETE0jYhGbvDcWKAabJ76QcEfDWSyOKSf8CsMc9lTYdh9L7+/6gXHGBRchGpiIr+/Hac9plyFBYWsnz5coYPH166zWazkZSUxJIlSyr0Hrm5uRQVFdGgQYPTjikoKKCgoKD0++zsbHdiioifsDscXNC99+kHGAZNL7kZuvRhxxcv0WjFf2jPVtqvuZ/5a97k8KUjuPHyBIIcdt+FFhGPc+s0zYEDB3A6nURGRpbZHhkZSUZGxc7nPv7448TExJCUlHTaMWPGjCE8PLz0ERcX505MEalpHEE06zOM0H+sYleLW3BhkMxi+v3Ql3fH/Jm5P2/CdborYkWk2vPpbJqxY8cyc+ZMPvroI4KDg087bvjw4WRlZZU+du3a5cOUIlJdGXWbEDdoGq77vmNfgy4EG0UMdr7PxZ8lM378M3y7IVPTgUX8kFtlpFGjRtjtdjIzM8tsz8zMJCoq6oyvHT9+PGPHjuXLL7/kwgsvPOPYoKAgwsLCyjxERH7niO1Ik4e+ouD/3iQ7OIZo4xCP5b5I2LvX8dSkGaTtOnJisMsJ236A1XNKvrqcluUWkfK5VUYCAwPp3LkzCxcuLN3mcrlYuHAhiYmJp33d888/zzPPPENqaipdunSpfFoRkd8ZBkEX3kjYP1aS1+NJCm0hXGTbzHMHHmHL1DsYPiOVjJ9mw4T2MON6+OCekq8T2sO6uVanF5E/cHs2zaxZsxg0aBCvvvoqXbt2ZcKECcyePZsNGzYQGRnJwIEDiY2NZcyYMQCMGzeOkSNH8u6779K9e/fS96lbty5169at0GdqNo2InFV2Ose+GEmd9bMBKDAdBFIMBpS9H9/x7255C9r19XVKkVrFa3ft7d+/P+PHj2fkyJHEx8eTlpZGampq6UWtO3fuJD09vXT85MmTKSws5OabbyY6Orr0MX78+ErslojIaYRFU6f/NLjva3KbdCbIKC65M/ApA4//+yt1mE7ZiFQTbh8ZsYKOjIiIW7Z9DzP6nH3coM+gRQ/v5xGppbx2ZEREpNrLKX9F6JO5jmqJeZHqQGVERGqeupFnHwO8Nm8x32g6sIjlVEZEpOZp1g3CYijvihEovWqE+/NfJ/Tdvjwz8dWy04FFxKdURkSk5rHZode449+cXEgMDAwKW1xFsRFIgm0DIw8+Tv60Xrw0bTpb9+f4Oq1IracyIiI1U7u+JdN3w6LLbg+LgVveInDQhzj+voqcjndTbARwiW09f98zlH2vXM20t99mX3a+NblFaiHNphGRms3lhB2LISez5FqSZt1Kjpz8UdYeDn85lrpr3yOAIgCWmO3Z0eEhel9/k+4OLFJJFf35rTIiIvK7rN1kfv4cDTbNJoBiAJbSgX1dhnJNrxt0d2ARN6mMiIhUknl4B7s/fY6orR+UlpJlto7kdf8nPa64Hput/AtjRaQslRERkSoqPridHR8/TbNdH+OgZLXW5Y5OcMVwLup2DYZRUkqcxcVsWDqfvMN7CImIpW1CMnaHw8roItWCyoiIiIfk79vK1o9GcV76pwQcLyVpQV0IveYpjh3YScySUURysHR8Jg3Zm5hCp+RBVkUWqRZURkREPCxrzya2ffQ07fd/jsNwAfD736DGH87cuI5vW9XtZRUSqdW0HLyIiIeFx7Ym/sF3ODB4McvCkjHNkhJinHQJye+XlEQvGYWzuNj3QUX8jMqIiIibopqfT52EQaeUkD+yGRDFQTYsne+7YCJ+SmVERKQS8g7vqdC4nPRNXk4i4v9URkREKiEkIrZC4y5c/Rw/vjyYjWvTvBtIxI+pjIiIVELbhGQyaVh6serJTBOKTDshRhHdD33IebMvZ9noq/km9QPyCnQdicgfqYyIiFSC3eFgb2IKwCmFxGWW3Bl49SX/ZlPy26ytcwk2w6Rr4TKu+Oludoy5iE/efIGtGQdPfWORWkhTe0VEqmDl/BmnrDOSQUPST1pn5PDOtexJfYmWe+cSQgEA+80wvg+/gYieD9Cz0/k47Pr3odQsWmdERMRH3FmB1XnsENu/nETEmhk0cO4HoMAMYL69J1kd7+WaK64kMizYl/FFvEZlRESkOnMWceDn9yn8YSIxx9aWbv7R1Z602NvpdOWfSGzVuHTJ+QrdfVikmlEZERHxB6ZJ4Y6f2P/lS0TvXYCNkpVdt7ii+TSkLxHd7uLm8I3U+foJyN574nVhMdBrHLTra1FwkbNTGRER8TdHdnLwm4mErv4fIa4cAI6ZgYRSCAaUXWPt+He3vKVCItWWloMXEfE39c+h4Y3PE/L4BgquHkt2SBx1jMKSJedPGXz835Gpw0pO4Yj4MZUREZHqJqgeQd0fIOxPk84y0ITsPSXXkoj4MZUREZHq6tj+Cg37/IM3mf/rboqdLi8HEvEOlRERkeqqbmSFhvXOmUOnD7rz7nN3M2PuAjKz870cTMSzdAGriEh15XLChPaQnU7pNSInDwmoQ54rgDrOI6XbfnG1YV1UX1pfMYCEtuecmB4s4mOaTSMiUhOsmwuzBx7/5o9/Xf9hNk3rXhRt+IKDP7xOk8zvS6cH55jBfBd4Ka6Od9Lzit6E1wn0aXQRlRERkZpi3VxIffykdUZiodfYU6f1Zqez78c3saX9j0YFu0o3bzFj2BB1A+dedS/nt25V7se4s5KsSEWojIiI1CTursBqmuRuWUT611OJ3Tuf4OP3wyky7SwPupjijnfSJekWgoOCgPLvsZNJQ/aedI8dEXeojIiICABmfhY7vn8HVrxN8/z1pdv3U59NUX2w14+h6/pxANj+cHnJ73cjXtXtZRUSqRSVEREROcXh7avY8dVUmu2eSwTZpdtNE8q7ztVlwj6jIY2f2qRTNuI2rcAqIiKniGjekfh7JxH25GZWXzqJjY42QPlFBEqOlERxkA1L5/swpdQ2KiMiIrWQPSCIDkl3cjT+3gqNzz2w08uJpDZTGRERqcVCImIrNK7NilEsmXAnv3z9EfkFBV5OJbWNrhkREanFnMXFHHi2NY3Ng2UuXv2daYILA7tx4kfFATOcteGXYbS/kY7driO8brAPE4s/0QWsIiJSISvnz6Dj4oeB8mfTrEx4iZCwhuStnEOrg18TTk7pmH1mfdLqXoZ5QT86dutFVP1QX0aXak5lREREKqy8dUYyaEj6SeuMmMWF7PgllaMrZtN8/zfUM08Ukwwzgp9De1DU9gY6Jl5Dyybl/H3t7nop4tdURkRExC1ur8BaXEjGqvlk/TyLpplfU8c8VvrUXrMBiwMvJa91Xy5MSOLCuPoY6z8tZyXZGOg17tSVZKVGUBkRERHfKS7gyJr5HFo2i+j0rwkxc0uf2m02YoutBT3Nn4HSu+oc94d77KiQ1DgqIyIiYo2ifHI3fMnBpTNpvOdrgs28s7zAKDlC8rfVOmVTw2jRMxERsUZAMKEd+hJ377sEP7GNop7Dz/ICE7L3kLPpe5/Ek+pHa/uKiIj3BIQQ0Lj8uwSfbPe7D7Is4nrqdujNJV0uJqZ+iJfDSXWhMiIiIt5VN7JCw9radtM2awosmsKW76P5OPQSaJ1M+8RkWkU38HJIsZLKiIiIeFezbiXXhGSnA+VdpmhA3SYc6vhn8talEnl4OS1t6bTM/wh+/YjsVSF8G3AROedcRbOEvlxwXits5a3Qdpzbs4LEcrqAVUREvG/dXJg98Pg3f/yxU85smvwsstYu4OCKuTRK/44w15HS0S7TYK3tPDIje9KgU186dL6UAMeJi17LWy8lk4bsPWm9FPENzaYREZHqZd3cctYZiYVeY08/rdfl4tj2ZexZ+jEh278iruC3Mk9n0oAt9bsTdP61mIU5XPTLY0D5K8mu6vayComPqYyIiEj1U8UVWAsO7WLbko9xbUylRfbPhHDipn2//zQzyjmD4zJhn9GQxk9t0ikbH1IZERGRGs1ZmMfmZalk//oZLfZ9RSOOnPU1a69+lwu69/Z+OAG0zoiIiNRw9sAQ2lx6Ixf/dTrbL3qiQq8pWvQyK7+eQ1bWEe+GE7foWJWIiPi9kIZNKzQuPu8n+P4nCr+zsyagLfsbX0JI6ys476LLaRhez8sp5XR0mkZERPyes7iYA8+2prF5kPJm/bpMyDbq8lv4pZyT/QuR5oEyz+eaQax1tONgk0sIbnMlbTt2Jyqizuk/UHcfrhBdMyIiIrXKyvkz6Lj4YeAss2lMk4M7N7AnbT7Gtu+Jy/qF+mZWmfc6YtbhV0cHDjZOILTtVbTr0IWmDUIxDOM0s4J09+HyqIyIiEitU946Ixk0JP1M64y4XGTvWMXetPnYtn9P0+wVhJ50c79Msz5p9gsJDWvApUc+BnT34YpQGRERkVqpyiuwOos5tv1nMld9iW37D8RkpxFI0VlfZmJg6O7DZaiMiIiIeEJRPvnbFpO9+E2abP/krMN/aP0E4ZfeTbvYhjjstXvSakV/fms2jYiIyJkEBBPc+kqCCw5BBcpIj02jyd34IstpRUZ4R2znXELTDpdxQctzCHRUvJzUpnvs1My9EhER8bQK3n04n2BCjXwSWAvZa2HNu7hWG/xGU3bV6UBxbFcaX9CTdud3JCSo/B/Dv1/7csEf77GzoObeY0enaURERCrC5YQJ7c989+GwGHh4Fc4Dm8hc+z25mxcRdmAFTYr2njJ6vxnO1uALyI3sQnibS2ndqQd1Q0MrPivID+iaEREREU9z5+7Df2AezSRj7Xcc3vgDIem/0DR/EwEUlxmTbwaw2XEezYu3Uof8GnGPHZURERERb6jM3YdPYhblkblhCfvWfo9tzzKaHv2V+hytcAR/uceOyoiIiIi3eHoFVtMkc9sadnz2Al0Pnf0i2f2NEmiceDvEXARNzgd7QOU+18sryaqMiIiI+Jm1P37OBQtud+9F9iCI6gAxnU48Grc5e6nwwUqyKiMiIiJ+pmL32KlHWOJgbBmrYG8aFGSdOjAgFKI7li0oDVqC7fjU4tJrX06uAJ5dSVbrjIiIiPgZu8PB3sQUGi9+GJdZ/myabYnPnZhNY5pwaCvsXXnikb4KCnNg55KSx+8C60FMfElJSfsf5c8IMgEDUodB294+W0lWR0ZERESqmUrdY+d3Licc3HyinOxZARm/QnG+eyEGfQYtelQi/Qk6TSMiIuLHPLoCq7MY9m8oKSer58C2b8/+mpumQ4ebK/d5x1X053elFs2fNGkSzZs3Jzg4mISEBJYtW3bG8e+//z5t27YlODiYDh06MG/evMp8rIiISK1hdzi4oHtvulx/Pxd07121dUXsDohqDxcNgJ6PVuw1FVxx1hPcLiOzZs1i6NChpKSksGLFCjp27EhycjL79u0rd/zixYu57bbbuOeee1i5ciX9+vWjX79+rFmzpsrhRURExE3NupXMmqGcK2ShZHtYbMk4H3H7NE1CQgIXX3wxEydOBMDlchEXF8dDDz3EsGHDThnfv39/jh07xmeffVa67ZJLLiE+Pp4pU6ZU6DN1mkZERMSDKrmSrLu8cpqmsLCQ5cuXk5SUdOINbDaSkpJYsmRJua9ZsmRJmfEAycnJpx0PUFBQQHZ2dpmHiIiIeEi7viWFIyy67PawGI8VEXe4dQLqwIEDOJ1OIiPLnkeKjIxkw4YN5b4mIyOj3PEZGRmn/ZwxY8YwatQod6KJiIiIO9r1LZm+68UVWCuqUhewetvw4cPJysoqfezatcvqSCIiIjWPzV4yfbfDzSVfLSgi4OaRkUaNGmG328nMzCyzPTMzk6ioqHJfExUV5dZ4gKCgIIKCgtyJJiIiIn7KrSMjgYGBdO7cmYULF5Zuc7lcLFy4kMTExHJfk5iYWGY8wIIFC047XkRERGoXtyctDx06lEGDBtGlSxe6du3KhAkTOHbsGIMHDwZg4MCBxMbGMmbMGAAeeeQRLrvsMl588UV69+7NzJkz+eWXX5g6dapn90RERET8kttlpH///uzfv5+RI0eSkZFBfHw8qamppRep7ty5E5vtxAGXbt268e677/LUU0/xxBNPcN555/Hxxx/Tvn17z+2FiIiI+C0tBy8iIiJe4dXl4EVEREQ8RWVERERELKUyIiIiIpZSGRERERFLVeF+xL7z+zW2ukeNiIiI//j95/bZ5sr4RRk5evQoAHFxcRYnEREREXcdPXqU8PDw0z7vF1N7XS4Xe/fupV69ehiGYXUcn8jOziYuLo5du3bVyunM2v/au/+1ed9B+6/9r1n7b5omR48eJSYmpswaZCfziyMjNpuNpk2bWh3DEmFhYTXiD2Rlaf9r7/7X5n0H7b/2v+bs/5mOiPxOF7CKiIiIpVRGRERExFIqI9VUUFAQKSkpBAUFWR3FEtr/2rv/tXnfQfuv/a+d++8XF7CKiIhIzaUjIyIiImIplRERERGxlMqIiIiIWEplRERERCylMmKhSZMm0bx5c4KDg0lISGDZsmWnHbt27VpuuukmmjdvjmEYTJgwwXdBvcSd/Z82bRo9evQgIiKCiIgIkpKSzjjeH7iz/x9++CFdunShfv361KlTh/j4eN5++20fpvUsd/b9j2bOnIlhGPTr18+7Ab3Mnf1/8803MQyjzCM4ONiHaT3P3d//I0eOMGTIEKKjowkKCqJ169bMmzfPR2k9z539v/zyy0/5/TcMg969e/swsQ+YYomZM2eagYGB5uuvv26uXbvWvO+++8z69eubmZmZ5Y5ftmyZ+eijj5rvvfeeGRUVZb700ku+Dexh7u7/7bffbk6aNMlcuXKluX79evOuu+4yw8PDzd27d/s4uWe4u//ffPON+eGHH5rr1q0zN2/ebE6YMMG02+1mamqqj5NXnbv7/rtt27aZsbGxZo8ePcwbbrjBN2G9wN39f+ONN8ywsDAzPT299JGRkeHj1J7j7v4XFBSYXbp0Ma+77jpz0aJF5rZt28xvv/3WTEtL83Fyz3B3/w8ePFjm937NmjWm3W4333jjDd8G9zKVEYt07drVHDJkSOn3TqfTjImJMceMGXPW1zZr1szvy0hV9t80TbO4uNisV6+eOWPGDG9F9Kqq7r9pmmanTp3Mp556yhvxvKoy+15cXGx269bNfO2118xBgwb5dRlxd//feOMNMzw83EfpvM/d/Z88ebJ57rnnmoWFhb6K6FVV/X//pZdeMuvVq2fm5OR4K6IldJrGAoWFhSxfvpykpKTSbTabjaSkJJYsWWJhMt/wxP7n5uZSVFREgwYNvBXTa6q6/6ZpsnDhQjZu3EjPnj29GdXjKrvvTz/9NE2aNOGee+7xRUyvqez+5+Tk0KxZM+Li4rjhhhtYu3atL+J6XGX2f+7cuSQmJjJkyBAiIyNp3749o0ePxul0+iq2x3ji777p06dz6623UqdOHW/FtITKiAUOHDiA0+kkMjKyzPbIyEgyMjIsSuU7ntj/xx9/nJiYmDL/U/uLyu5/VlYWdevWJTAwkN69e/PKK69w9dVXezuuR1Vm3xctWsT06dOZNm2aLyJ6VWX2v02bNrz++ut88sknvPPOO7hcLrp168bu3bt9EdmjKrP/W7duZc6cOTidTubNm8eIESN48cUXefbZZ30R2aOq+nffsmXLWLNmDffee6+3IlrGL+7aK/JHY8eOZebMmXz77bd+fyGfO+rVq0daWho5OTksXLiQoUOHcu6553L55ZdbHc1rjh49yoABA5g2bRqNGjWyOo4lEhMTSUxMLP2+W7dunH/++bz66qs888wzFibzDZfLRZMmTZg6dSp2u53OnTuzZ88eXnjhBVJSUqyO51PTp0+nQ4cOdO3a1eooHqcyYoFGjRpht9vJzMwssz0zM5OoqCiLUvlOVfZ//PjxjB07lq+++ooLL7zQmzG9prL7b7PZaNWqFQDx8fGsX7+eMWPG+FUZcXfft2zZwvbt2+nTp0/pNpfLBYDD4WDjxo20bNnSu6E9yBP/7wcEBNCpUyc2b97sjYheVZn9j46OJiAgALvdXrrt/PPPJyMjg8LCQgIDA72a2ZOq8vt/7NgxZs6cydNPP+3NiJbRaRoLBAYG0rlzZxYuXFi6zeVysXDhwjL/AqqpKrv/zz//PM888wypqal06dLFF1G9wlO//y6Xi4KCAm9E9Bp3971t27asXr2atLS00kffvn254oorSEtLIy4uzpfxq8wTv/dOp5PVq1cTHR3trZheU5n97969O5s3by4toQCbNm0iOjrar4oIVO33//3336egoIA777zT2zGtYfUVtLXVzJkzzaCgIPPNN980161bZ95///1m/fr1S6fsDRgwwBw2bFjp+IKCAnPlypXmypUrzejoaPPRRx81V65caf72229W7UKVuLv/Y8eONQMDA805c+aUmeZ29OhRq3ahStzd/9GjR5tffvmluWXLFnPdunXm+PHjTYfDYU6bNs2qXag0d/f9ZP4+m8bd/R81apQ5f/58c8uWLeby5cvNW2+91QwODjbXrl1r1S5Uibv7v3PnTrNevXrmgw8+aG7cuNH87LPPzCZNmpjPPvusVbtQJZX983/ppZea/fv393Vcn1EZsdArr7xinnPOOWZgYKDZtWtX86effip97rLLLjMHDRpU+v22bdtM4JTHZZdd5vvgHuLO/jdr1qzc/U9JSfF9cA9xZ/+ffPJJs1WrVmZwcLAZERFhJiYmmjNnzrQgtWe4s+8n8/cyYpru7f/f/va30rGRkZHmddddZ65YscKC1J7j7u//4sWLzYSEBDMoKMg899xzzeeee84sLi72cWrPcXf/N2zYYALml19+6eOkvmOYpmladFBGRERERNeMiIiIiLVURkRERMRSKiMiIiJiKZURERERsZTKiIiIiFhKZUREREQspTIiIiIillIZEREREUupjIiIiIilVEZERETEUiojIiIiYimVEREREbHU/wN+tu+t6MerbgAAAABJRU5ErkJggg==",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def p(x):\n",
    "    ans=-x-1\n",
    "    return ans\n",
    "def q(x):\n",
    "    ans=2\n",
    "    return ans\n",
    "def r(x):\n",
    "    ans= (1-x**2)*math.exp(-x)\n",
    "    return ans\n",
    "N=9\n",
    "solution1=finite_difference(0,1,1,0,N,p,q,r)\n",
    "x_values1 = np.linspace(0, (math.pi)/4, N+2)[1:-1] \n",
    "print(\"The solution for N={} is \".format(N))\n",
    "for i in solution1:\n",
    "    print(i)\n",
    "\n",
    "N=19\n",
    "solution2=finite_difference(0,1,1,0,N,p,q,r)\n",
    "x_values2 = np.linspace(0, (math.pi)/4, N+2)[1:-1] \n",
    "print(\"The solution for N={} is \".format(N))\n",
    "for i in solution2:\n",
    "    print(i)\n",
    "\n",
    "plt.plot(x_values1, solution1, 'o-', label='N=9')\n",
    "plt.plot(x_values2, solution2, 'o-', label='N=19')\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Q3.** Use the Linear Finite-Difference Algorithm with N = 4 to approximate the solution of the\n",
    "boundary value problem\n",
    "\n",
    "$y^{\"}+4y=cos(x)$\n",
    "\n",
    "$0 \\leq x \\leq \\frac{\\pi}{4}$ , $y(0) = 0$ , $y(\\frac{\\pi}{4}) = 0$\n",
    "\n",
    "and compare the results to the actual solution where the actual solution is given by\n",
    "\n",
    "$y(x)=-\\frac{1}{3}cos(2x) -\\frac{\\sqrt 2}{6}sin(2x)+\\frac{1}{3}cos(x)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The solution for N=4 is \n",
      "-0.06141844977636928\n",
      "-0.09240490854870977\n",
      "-0.09080498945392074\n",
      "-0.058258272342832616\n",
      "The real solution for N=4 is \n",
      "0.0\n",
      "-0.07123076739371831\n",
      "-0.09440908289175365\n",
      "-0.06816510083357769\n",
      "-2.7755575615628914e-17\n",
      "0.0\n"
     ]
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def p(x):\n",
    "    ans=0\n",
    "    return ans\n",
    "def q(x):\n",
    "    ans=-4\n",
    "    return ans\n",
    "def r(x):\n",
    "    ans= math.cos(x)\n",
    "    return ans\n",
    "def real_solution(a,b,N):\n",
    "    solution=[]\n",
    "    solution.append(-math.cos(2*a)/3 -(math.sqrt(2)*math.sin(2*a))/6 +math.cos(a)/3)\n",
    "    h=(b-a)/N\n",
    "    for k in range(1,N+1):\n",
    "        x=a+h*k\n",
    "        ans=-math.cos(2*x)/3 -(math.sqrt(2)*math.sin(2*x))/6 +math.cos(x)/3\n",
    "        solution.append(ans)\n",
    "    solution.append(-math.cos(2*a)/3 -(math.sqrt(2)*math.sin(2*a))/6 +math.cos(a)/3)\n",
    "    return solution\n",
    "\n",
    "N=4\n",
    "#finite difference function is called here\n",
    "solution1=finite_difference(0,(math.pi)/4,0,0,N,p,q,r)\n",
    "x_values1 = np.linspace(0, (math.pi)/4, N+2)[1:-1] \n",
    "print(\"The solution for N={} is \".format(N))\n",
    "for i in solution1:\n",
    "    print(i)\n",
    "#real solution fuction is called here\n",
    "solution2=real_solution(0,(math.pi)/4,N)\n",
    "x_values2 = np.linspace(0, (math.pi)/4, N+2) \n",
    "print(\"The real solution for N={} is \".format(N))\n",
    "for i in solution2:\n",
    "    print(i)\n",
    "\n",
    "plt.plot(x_values1, solution1, 'o-', label='approx sol')\n",
    "plt.plot(x_values2, solution2, 'o-', label='real solution')\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  }
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