p17.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# p17: Helmholtz Equation "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ \n",
    "u_{xx} + u_{yy} + k^{2}u = f, \\qquad \\mbox{on} \\qquad [-1,1] \\times [-1,1]\n",
    "$$\n",
    "A minor modification of p16 to solve such problem for the particular choices as follows:\n",
    "\n",
    "$$\n",
    "k = 9, \\qquad f(x,y) = \\exp (-10[(y-1)^2 +(x-1/2)^2])\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "%config InlineBackend.figure_format='svg'\n",
    "from chebPy import cheb\n",
    "from numpy import meshgrid,sin,dot,eye,kron,zeros,reshape,exp,linspace\n",
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "from matplotlib.pyplot import figure,subplot,plot,title,axis,xlabel,ylabel,contour\n",
    "from matplotlib import cm\n",
    "from scipy.linalg import solve\n",
    "from scipy.interpolate import interp2d"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
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   "source": [
    "N = 24; D,x = cheb(N); y = x;\n",
    "xx,yy = meshgrid(x[1:N],y[1:N])\n",
    "xx = reshape(xx,(N-1)**2)\n",
    "yy = reshape(yy,(N-1)**2)\n",
    "f = exp(-10*((yy-1)**2 + (xx - 0.5)**2 ))\n",
    "D2 = dot(D,D); D2 = D2[1:N,1:N]; I = eye(N-1)\n",
    "k = 9\n",
    "L = kron(I,D2) + kron(D2,I) + k**2*eye((N-1)**2)\n",
    "# Solve Lu=f\n",
    "u = solve(L,f)\n",
    "# Convert 1-d vectors to 2-d\n",
    "uu = zeros((N+1,N+1)); uu[1:N,1:N] = reshape(u,(N-1,N-1))\n",
    "[xx,yy] = meshgrid(x,y)\n",
    "value = uu[N//2,N//2]\n",
    "\n",
    "f = interp2d(x,y,uu,kind='cubic')\n",
    "xxx = linspace(-1.0,1.0,50)\n",
    "uuu = f(xxx,xxx)\n",
    "\n",
    "fig = figure(figsize=(8,8))\n",
    "ax = fig.add_subplot(111, projection='3d')\n",
    "[X ,Y] = meshgrid(xxx,xxx)\n",
    "ax.plot_surface(X,Y,uuu,rstride=1,cstride=1,cmap=cm.jet,edgecolor='black')\n",
    "title(\"$u(0,0)$=\"+str(value))\n",
    "xlabel(\"x\"); ylabel(\"y\");\n",
    "\n",
    "figure(figsize = (8,8))\n",
    "contour(X,Y,uuu);"
   ]
  }
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