{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# p15 - Solve eigenvalue BVP $u_{xx} = \\lambda u$, $u(-1)=u(1)=0$"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"%matplotlib inline\n",
"%config InlineBackend.figure_format='svg'\n",
"from numpy import dot,argsort,linspace,shape,zeros,polyval,polyfit,pi,real\n",
"from chebPy import cheb\n",
"from scipy.linalg import solve,eig\n",
"from matplotlib.pyplot import figure,subplot,plot,title,axis"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
"image/svg+xml": [
"\n",
"\n",
"\n",
"\n"
],
"text/plain": [
""
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"N = 36\n",
"D,x = cheb(N)\n",
"D2 = dot(D,D)\n",
"D2 = D2[1:N,1:N]\n",
"\n",
"lam,V = eig(D2) \n",
"ii = argsort(-lam)\n",
"lam = real(lam[ii])\n",
"V = V[:,ii]\n",
"\n",
"fig = figure(figsize=(10,15))\n",
"for j in range(5,35,5): \n",
" lv = shape(V)[0]+2\n",
" u = zeros(lv)\n",
" u[1:lv-1] = V[:,int(j)] \n",
" subplot(6,1,j//5)\n",
" plot(x,u,'bo')\n",
" xx = linspace(-1.0,1.0,501)\n",
" uu = polyval(polyfit(x,u,N),xx) # interpolate grid data\n",
" s = 'eig %d = %20.13f * 4/$\\pi^2$' %(j,lam[j-1]*4/pi**2)\n",
" s = s + '\\t\\t %4.1f ppw' % (4*N/(pi*j))\n",
" title(s)\n",
" plot(xx,uu,'b')\n",
" axis('off')"
]
}
],
"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": "3.7.9"
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"nbformat": 4,
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