p33.ipynb

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   "source": [
    "# p33: Solve linear BVP, Neumann bc\n",
    "\n",
    "$$\n",
    "u_{xx} = \\exp(4x)\n",
    "$$, \n",
    "$$\n",
    "u'(-1)=0, \\qquad u(1)=0\n",
    "$$"
   ]
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   "execution_count": 3,
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      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
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     },
     "metadata": {},
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   ],
   "source": [
    "%config InlineBackend.figure_format='svg'\n",
    "from chebPy import *\n",
    "from numpy import dot,exp,zeros,max,linspace,polyval,polyfit,inf\n",
    "from numpy.linalg import norm\n",
    "from scipy.linalg import solve\n",
    "from matplotlib.pyplot import title,plot,xlabel,ylabel,grid\n",
    "\n",
    "N = 16\n",
    "\n",
    "# Build matrix\n",
    "D,x = cheb(N)\n",
    "D2 = dot(D,D)\n",
    "D2[N,:] = D[N,:] # Last eqn has neumann bc\n",
    "D2 = D2[1:N+1,1:N+1]\n",
    "\n",
    "# RHS\n",
    "f = zeros(N)\n",
    "f[0:-1] = exp(4.0*x[1:N])\n",
    "\n",
    "# Solve\n",
    "u = solve(D2,f)\n",
    "s = zeros(N+1)\n",
    "s[1:N+1] = u\n",
    "\n",
    "# Compute error\n",
    "xx = linspace(-1.0,1.0,200)\n",
    "uu = polyval(polyfit(x,s,N),xx)    # interpolate grid data\n",
    "exact = (exp(4.0*xx) - 4.0*exp(-4.0)*(xx-1.0) - exp(4.0))/16.0\n",
    "maxerr = norm(uu-exact,inf)\n",
    "\n",
    "title('max err = %e' % maxerr)\n",
    "plot(x,s,'o',xx,exact)\n",
    "xlabel('x'); ylabel('u'); grid(True);"
   ]
  }
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