p14.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# p14: Solve nonlinear BVP \n",
    "\n",
    "$$\n",
    "u_{xx} = exp(u), \\qquad u(-1)=u(1)=0\n",
    "$$\n",
    "\n",
    "by iteration"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%config InlineBackend.figure_format='svg'\n",
    "from numpy import dot,exp,zeros,linspace,polyval,polyfit,inf\n",
    "from numpy.linalg import norm\n",
    "from chebPy import cheb\n",
    "from scipy.linalg import solve\n",
    "from matplotlib.pyplot import title,plot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
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      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
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     "metadata": {},
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   ],
   "source": [
    "N = 16 # N must be even\n",
    "D,x = cheb(N)\n",
    "D2 = dot(D,D)\n",
    "D2 = D2[1:N,1:N]\n",
    "\n",
    "u = zeros(N-1)\n",
    "err = zeros(N-1)\n",
    "change, it = 1.0, 0\n",
    "\n",
    "while change > 1.0e-15:\n",
    "    unew = solve(D2,exp(u))\n",
    "    change = norm(unew-u, inf)\n",
    "    u = unew\n",
    "    it += 1\n",
    "\n",
    "# Add bounday values to u vector\n",
    "s = zeros(N+1); s[1:N] = u; u = s;\n",
    "\n",
    "xx = linspace(-1.0,1.0,201)\n",
    "uu = polyval(polyfit(x,u,N),xx)    # interpolate grid data\n",
    "\n",
    "title('no. steps = %d,  u(0) = %18.14f' %(it,u[N//2]) )\n",
    "plot(x,u,'o',xx,uu,'b');"
   ]
  }
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