FredholmAnalyticalSolution.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Analytical Solution of Fredholm Equation\n",
    "This notebook is the symbolid code I used in my thesis at Univesity of New Brunswick, mainly third chapter, to show that Fourier transform and POD(PCA) are not equivalent in a homogeneous direction.\n",
    "\n",
    "The `sympy` version is 1.3.\n",
    "\n",
    "Total run time is about 30-45 minutes."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import symbols,sin,cos,integrate,expand,Matrix,simplify,pi,collect,I,init_printing\n",
    "from IPython.display import display\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def kernel(U):\n",
    "    # forms the kernel from a base function\n",
    "    V=U.subs(x,y)\n",
    "    UV=expand(U*V.conjugate(),trig=True)\n",
    "    K=simplify(integrate(UV,(t,0,2*pi),conds='none')/pi)\n",
    "    return expand(K,trig=True)\n",
    "def Vjs(K,Ui):\n",
    "    # Factors the degenerate terms of Ui from the kernel\n",
    "    Vj=Ui.copy()\n",
    "    for i in range(Ui.shape[0]):\n",
    "        Vj[i,0]=collect(K,Ui[i,0]).coeff(Ui[i,0])\n",
    "    return Vj\n",
    "def fredholmint(K,Ui):\n",
    "    # Fredholm integral solution\n",
    "    Vj=Vjs(K,Ui)\n",
    "    if simplify((Ui.T*Vj)[0]-K):\n",
    "        raise ValueError('Kernel is not degenerate')\n",
    "    aij=simplify(integrate(Ui*Vj.conjugate().subs(y,x).T,(x,0,2*pi),conds='none')/pi)\n",
    "    return aij,aij.eigenvects(simplify=True)\n",
    "def eig_calc(cjs,ui):\n",
    "    # Recovering eigenvalues\n",
    "    ef = []\n",
    "    ev = []\n",
    "    for cj in cjs:\n",
    "        if cj[0]==0: continue\n",
    "        for i in range(cj[1]):\n",
    "            ev.append(cj[0])\n",
    "            ef.append(cj[2][i].T*ui)\n",
    "#         eigs.append([cj[0],cj[2][0].T*ui])\n",
    "    return Matrix(ev),Matrix(ef)\n",
    "def eig_display(ev,ef):\n",
    "    print('Eigenvalues = ',end=' ')\n",
    "    display(evs)\n",
    "    print('Eigenfunctions = ',end=' ')\n",
    "    display(efs)\n",
    "def fullrun(u,sublist=None):\n",
    "    print('Function:')\n",
    "    if sublist is not None:\n",
    "        display(u.subs(sublist))\n",
    "    else:\n",
    "        display(u)\n",
    "    k = kernel(u)\n",
    "    aij, cjs = fredholmint(k,ui)\n",
    "    evs, efs = eig_calc(cjs,ui)\n",
    "    if sublist is not None:\n",
    "        efs = efs.subs(sublist)\n",
    "    print('Eigenvalues: ',end=' ')\n",
    "    display(evs)\n",
    "    print('Eigenfunctions: ',end=' ')\n",
    "    display(efs)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "c,d,e,x,y,t = symbols('c d e x y t',real=True)\n",
    "a,b,f,g,m,n,o,p = symbols('a b f g m n o p',integer=True,real=True)\n",
    "ui = Matrix([sin(a*x), sin(b*x), sin(f*x), sin(g*x), cos(a*x), cos(b*x), cos(f*x), cos(g*x)])\n",
    "# Averaging in time\n",
    "u1 = sin(a*x+m*t)\n",
    "u2 = sin(a*x+m*t)+d*cos(a*x-m*t)\n",
    "u3 = sin(a*x+m*t)+d*cos(a*x-m*t)+sin(b*x+m*t)+e*cos(b*x-m*t)\n",
    "u4 = sin(a*x+m*t)+d*cos(a*x-m*t)+sin(b*x+n*t)+e*cos(b*x-n*t)\n",
    "# Positive and negative frequencies of Fourier transform of u3 in time\n",
    "Sp = +c*cos(a*x)+d*cos(b*x)+cos(a*x)+cos(b*x)-I*(c*sin(a*x)+d*sin(b*x)-sin(a*x)-sin(b*x))\n",
    "Sn=-c*cos(a*x)-d*cos(b*x)-cos(a*x)+cos(b*x)+I*(c*sin(a*x)+d*sin(b*x)-sin(a*x)+sin(b*x))\n",
    "# Numerical test function\n",
    "u5 = sin(a*x+m*t)+2*cos(a*x-m*t)+sin(b*x+n*t)+4*cos(b*x-n*t)+2.5*sin(f*x+o*t)+3.5*cos(f*x-o*t)+2*sin(g*x+p*t)+2*cos(g*x-p*t)\n",
    "sublist = [(a,3), (b,5), (f,7), (g,11), (m,1), (n,2), (o,3), (p,6)]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Analytical Test Functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Function:\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\sin{\\left (a x + m t \\right )}$$"
      ],
      "text/plain": [
       "sin(a⋅x + m⋅t)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Eigenvalues:  "
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABYAAAAyBAMAAACqpzYoAAAAJ1BMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAilU6eAAAADHRSTlMA74lUIhAyq2Z2mc3nXb/RAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAASklEQVQoFWOQOXOIAQR0zhxkEHZxBbNDXAwZxMAsEJGIyTYMgImzFtXA2QwMPXRkz0HYtVxmVwNWd4Kcj8399BQf5mGFnGaQ0hIAtho3xcYc7GMAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1\\\\1\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1⎤\n",
       "⎢ ⎥\n",
       "⎣1⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Eigenfunctions:  "
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\sin{\\left (a x \\right )}\\\\\\cos{\\left (a x \\right )}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡sin(a⋅x)⎤\n",
       "⎢        ⎥\n",
       "⎣cos(a⋅x)⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fullrun(u1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Function:\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$d \\cos{\\left (a x - m t \\right )} + \\sin{\\left (a x + m t \\right )}$$"
      ],
      "text/plain": [
       "d⋅cos(a⋅x - m⋅t) + sin(a⋅x + m⋅t)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Eigenvalues:  "
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}d^{2} - 2 d + 1\\\\d^{2} + 2 d + 1\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡ 2          ⎤\n",
       "⎢d  - 2⋅d + 1⎥\n",
       "⎢            ⎥\n",
       "⎢ 2          ⎥\n",
       "⎣d  + 2⋅d + 1⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Eigenfunctions:  "
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}- \\sin{\\left (a x \\right )} + \\cos{\\left (a x \\right )}\\\\\\sin{\\left (a x \\right )} + \\cos{\\left (a x \\right )}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡-sin(a⋅x) + cos(a⋅x)⎤\n",
       "⎢                    ⎥\n",
       "⎣sin(a⋅x) + cos(a⋅x) ⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fullrun(u2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Function:\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$d \\cos{\\left (a x - m t \\right )} + e \\cos{\\left (b x - m t \\right )} + \\sin{\\left (a x + m t \\right )} + \\sin{\\left (b x + m t \\right )}$$"
      ],
      "text/plain": [
       "d⋅cos(a⋅x - m⋅t) + e⋅cos(b⋅x - m⋅t) + sin(a⋅x + m⋅t) + sin(b⋅x + m⋅t)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Eigenvalues:  "
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}d^{2} - 2 d + e^{2} - 2 e + 2\\\\d^{2} + 2 d + e^{2} + 2 e + 2\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡ 2          2          ⎤\n",
       "⎢d  - 2⋅d + e  - 2⋅e + 2⎥\n",
       "⎢                       ⎥\n",
       "⎢ 2          2          ⎥\n",
       "⎣d  + 2⋅d + e  + 2⋅e + 2⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Eigenfunctions:  "
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{\\left(- d + 1\\right) \\sin{\\left (a x \\right )}}{e - 1} + \\frac{\\left(d - 1\\right) \\cos{\\left (a x \\right )}}{e - 1} - \\sin{\\left (b x \\right )} + \\cos{\\left (b x \\right )}\\\\\\frac{\\left(d + 1\\right) \\sin{\\left (a x \\right )}}{e + 1} + \\frac{\\left(d + 1\\right) \\cos{\\left (a x \\right )}}{e + 1} + \\sin{\\left (b x \\right )} + \\cos{\\left (b x \\right )}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡(-d + 1)⋅sin(a⋅x)   (d - 1)⋅cos(a⋅x)                      ⎤\n",
       "⎢───────────────── + ──────────────── - sin(b⋅x) + cos(b⋅x)⎥\n",
       "⎢      e - 1              e - 1                            ⎥\n",
       "⎢                                                          ⎥\n",
       "⎢(d + 1)⋅sin(a⋅x)   (d + 1)⋅cos(a⋅x)                       ⎥\n",
       "⎢──────────────── + ──────────────── + sin(b⋅x) + cos(b⋅x) ⎥\n",
       "⎣     e + 1              e + 1                             ⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fullrun(u3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Function:\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$d \\cos{\\left (a x - m t \\right )} + e \\cos{\\left (b x - n t \\right )} + \\sin{\\left (a x + m t \\right )} + \\sin{\\left (b x + n t \\right )}$$"
      ],
      "text/plain": [
       "d⋅cos(a⋅x - m⋅t) + e⋅cos(b⋅x - n⋅t) + sin(a⋅x + m⋅t) + sin(b⋅x + n⋅t)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Eigenvalues:  "
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}d^{2} - 2 d + 1\\\\d^{2} + 2 d + 1\\\\e^{2} - 2 e + 1\\\\e^{2} + 2 e + 1\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡ 2          ⎤\n",
       "⎢d  - 2⋅d + 1⎥\n",
       "⎢            ⎥\n",
       "⎢ 2          ⎥\n",
       "⎢d  + 2⋅d + 1⎥\n",
       "⎢            ⎥\n",
       "⎢ 2          ⎥\n",
       "⎢e  - 2⋅e + 1⎥\n",
       "⎢            ⎥\n",
       "⎢ 2          ⎥\n",
       "⎣e  + 2⋅e + 1⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Eigenfunctions:  "
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}- \\sin{\\left (a x \\right )} + \\cos{\\left (a x \\right )}\\\\\\sin{\\left (a x \\right )} + \\cos{\\left (a x \\right )}\\\\- \\sin{\\left (b x \\right )} + \\cos{\\left (b x \\right )}\\\\\\sin{\\left (b x \\right )} + \\cos{\\left (b x \\right )}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡-sin(a⋅x) + cos(a⋅x)⎤\n",
       "⎢                    ⎥\n",
       "⎢sin(a⋅x) + cos(a⋅x) ⎥\n",
       "⎢                    ⎥\n",
       "⎢-sin(b⋅x) + cos(b⋅x)⎥\n",
       "⎢                    ⎥\n",
       "⎣sin(b⋅x) + cos(b⋅x) ⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fullrun(u4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Positive and Negative Frequencies of u3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Function:\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAvsAAAAUBAMAAADhIu8DAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAiXYyIt1Uu82rmRDvZkSKCuMdAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAHo0lEQVRYCZ1Za4hUZRh+xpmdmfW467gYyIZ2XCNSuyx2FaQ9EBKZlop0gaAhjH5UuCxCFwwnLBgtctn8EYU4WpR/qtEILwkraDfF2CQhBGmT+ietN9bU0p73O2fO+b5zvrmsL+yZ9/3e2/M+3znfmd0F6sgyfT3r6VYdPTOiOVZreqAaft098VaSrWc1xWf2toDT8SRrN64/geJGaAMYTr8BaIthpddVDNs39sBZt7y23u7VNH62q2J7tJXMscho2EqFPXDEj45l7fkuLGLi85eXareD3pt4vDAxVBrXttUPU5sX10LrcRSGKCXvGfZew8LUomkr620gX6mtO6Waxs92RQT9oaTPhSoat1Jx4350PGtTWCSGT633FUI39N6AAS4IalzbVj+s3ry4HlqHozBEKSdNM+WKnVVXKmNVMU3pqEDfli7TC4jfKtZWjHw+jO68GqqhIlnBrlBLubzUl3jvBDgzNVk75UpEOL8YodiLR+jDQJ8D7datD2OVlkTVOSh22P4msWLS5hnbciDmhvitYm3FyGiA1PJkIrO0XfHxJaOClXjvBDgzM1k7Nr8Rbi8eodeCm3FUC83EB35fPFk38EdPfbDAj8X80bYlPxq5lCb+UDL3RKq1Fd3RALXnNZal74rCF9ZUysO9oW305moCHNCktjl/WFgUe/EIvRZs46irhzj3T/sMTtfsUSx45XbGT6rwsqT7jip2z318tgt00w7oz2yfI0+9s727iF09s5C+7XiJ9jz+bOxhlZ4tp/lUpSq0fdm9Qz7FD+kl8Y9cxoNHFsx1uWRvRUdtgCVdfUXYsvJD84oGPiYFwiky6wfT733SMyQr0juzfcdRCzhxWxDFa2vzx6iqU1yhb4GjxcVcCZnvsci7z8MAv4a0EVBuhM/2YWyGU+hY2VYCfhOY/t1/xsvJWby0mD6cXoWcyxe/3MBvckf+qQ57u3DCfZrHXIlrvhTOyCf9UL1U/EbgKTdzkWv2VnQE9BPG+ipgyRp+17lo4GOSL2qK/CDaBvCExyXpfcbLn0USnCS0UDuaP05VneIKfQscrUB7EfkSXu5dAfQ9dBSTCKitCBwYwedIVVNXOz3gBS769DvHMLlEo4zsobYC0hfOVOHSfpL+q5jq3oxbqn9Sly1RkinulE/6oXqp+NeBdcB/XLO3oiOgnzAk35K1porzBj6G+dIpU+QGkTuIqbyRpLfA7kcCnIpvoXY0f5yqOsUV+uYcTVbfKoYrfATH+dWl8NyAR0hTRoGvqlgJB7l+2uhDqlx+58dyeZAJZB3OFVktAufaLt0qET9DtmW4UMWrYqbPYm9ZZIOD8zW/6qXiSf9rUA5LK4bPKJefLZf5jAgM3hewZO2Ecy7CR0iqX7mX0TKF0F/i/UBTsBG2iwgcavBaqx3NH6fKVjxA3wJHU3x2CTJ7jfT379osvAonP8C5QFXhJ/0UdfgwYZjetHC6hsolZ8b1CnWOyFfkWNEnVeivCR8KCv1+LxVP+rlLUsPeio7g7icMAoMlaxM65Y5x6fXxieKLTCH094f0C2zZGHUraOAkvoXa0fx9rklVMFi8uKBvgaPhggAIt/TTKn4fVSeCcw5ZobBPQOMvuSj6mbCGerizVx5Dh/DDB5wPOu/VtGwaOpfLVQl5oNDv91LxEf08fCytmODTX/Mls3h0yaOr4VPd5JKRKTT6/d4COwFOwluorc1fYRXZ9oCqYLB4cUHfAkekDOrsR/EIZ/nbVb+j5kbgjKO9tDpbvRuTqvqrd7iID7OenP3ITqkgfe00sJW9+HqbOsoT9qVJZzHfePUOFwuBX/VS8RH99lZMCOhXMIBkFq7x9WTgY5IvnTKFRj+x+bCT4CShhdrRq5dnv0GVDG4prtA352jyIPb3gn/xSHmLevEWX1o5z//aOIATg4Wx4nosI8Lt/PHv/pzb8UyKxtIRTE8fQr7AF9da2t3yx4VcpXM8d1b+sDRZKPdlTP2hiX6/l4rnyRMcPqkKkGzFTJ9+36devWslRcvaijd42Gn4gn589GQKnjy1w4e9FWwLOMmxIIrXjuaPUyWDW4or9C1wNO2USwT7dkznN+Pu0c6Zd86iKUfHvp57h0b39ez+gmRiA3+Cbz5Dcx69i4bz0WwPS6Z9iT+O/zJKm4+ns2MmnKH5807RzPfy4ku72Or4Ur0kfuH1jxde/+DFfwfrtWJCQL+CAdUllrV/9ghBavj8drzKFOltl7/edvn+9T/5vR2BbQEnORZE8drR/HGq1ODJ4gp96xwJDE34e7cuDrcjoF9f1vU2T7fwjWHRiPkjt7UV3QH9UaChmVk+PiNAN+K9E+D0YCBZu9H89uJW9LHQ+jBOmoBSrthOVa51pKNiOLoMi0bMH7mtrej+NgqxaGZWyrWEREvx3glwUahoydopV9bt89uLW9HHQuvDkBeAJns1vZ66Vnc4Jd1SuuHXvDfQitlmVjN8Zm8LOA2PrXbD+hMoboQ2gBH7H4iRZkCNjD2RGp9AeQy/FnsDrZhtZjXDZ/Y2t06DEqjJ2g3rT6C4EdoIhnzdCaXDC9X6ivp+U3P/WlOiT8MfLZv/2kNLrSRbB9g0yextAafjSdZuXH8CxY1Qwvgfgn02nxMIUsIAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$c \\cos{\\left (a x \\right )} + d \\cos{\\left (b x \\right )} - i \\left(c \\sin{\\left (a x \\right )} + d \\sin{\\left (b x \\right )} - \\sin{\\left (a x \\right )} - \\sin{\\left (b x \\right )}\\right) + \\cos{\\left (a x \\right )} + \\cos{\\left (b x \\right )}$$"
      ],
      "text/plain": [
       "c⋅cos(a⋅x) + d⋅cos(b⋅x) - ⅈ⋅(c⋅sin(a⋅x) + d⋅sin(b⋅x) - sin(a⋅x) - sin(b⋅x)) + \n",
       "cos(a⋅x) + cos(b⋅x)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Eigenvalues:  "
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}4 c^{2} + 4 d^{2} + 8\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡   2      2    ⎤\n",
       "⎣4⋅c  + 4⋅d  + 8⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Eigenfunctions:  "
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{i \\left(- c + 1\\right) \\sin{\\left (a x \\right )}}{d + 1} + \\frac{\\left(c + 1\\right) \\cos{\\left (a x \\right )}}{d + 1} + \\frac{i \\left(- d + 1\\right) \\sin{\\left (b x \\right )}}{d + 1} + \\cos{\\left (b x \\right )}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡ⅈ⋅(-c + 1)⋅sin(a⋅x)   (c + 1)⋅cos(a⋅x)   ⅈ⋅(-d + 1)⋅sin(b⋅x)           ⎤\n",
       "⎢─────────────────── + ──────────────── + ─────────────────── + cos(b⋅x)⎥\n",
       "⎣       d + 1               d + 1                d + 1                  ⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fullrun(Sp)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Function:\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$- c \\cos{\\left (a x \\right )} - d \\cos{\\left (b x \\right )} + i \\left(c \\sin{\\left (a x \\right )} + d \\sin{\\left (b x \\right )} - \\sin{\\left (a x \\right )} + \\sin{\\left (b x \\right )}\\right) - \\cos{\\left (a x \\right )} + \\cos{\\left (b x \\right )}$$"
      ],
      "text/plain": [
       "-c⋅cos(a⋅x) - d⋅cos(b⋅x) + ⅈ⋅(c⋅sin(a⋅x) + d⋅sin(b⋅x) - sin(a⋅x) + sin(b⋅x)) -\n",
       " cos(a⋅x) + cos(b⋅x)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Eigenvalues:  "
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}4 c^{2} + 4 d^{2} + 8\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡   2      2    ⎤\n",
       "⎣4⋅c  + 4⋅d  + 8⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Eigenfunctions:  "
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{i \\left(- c + 1\\right) \\sin{\\left (a x \\right )}}{d - 1} + \\frac{\\left(c + 1\\right) \\cos{\\left (a x \\right )}}{d - 1} + \\cos{\\left (b x \\right )} - \\frac{i \\left(d + 1\\right) \\sin{\\left (b x \\right )}}{d - 1}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡ⅈ⋅(-c + 1)⋅sin(a⋅x)   (c + 1)⋅cos(a⋅x)              ⅈ⋅(d + 1)⋅sin(b⋅x)⎤\n",
       "⎢─────────────────── + ──────────────── + cos(b⋅x) - ──────────────────⎥\n",
       "⎣       d - 1               d - 1                          d - 1       ⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fullrun(Sn)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Numerical Test Function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Function:\n"
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\sin{\\left (t + 3 x \\right )} + \\sin{\\left (2 t + 5 x \\right )} + 2.5 \\sin{\\left (3 t + 7 x \\right )} + 2 \\sin{\\left (6 t + 11 x \\right )} + 2 \\cos{\\left (t - 3 x \\right )} + 4 \\cos{\\left (2 t - 5 x \\right )} + 3.5 \\cos{\\left (3 t - 7 x \\right )} + 2 \\cos{\\left (6 t - 11 x \\right )}$$"
      ],
      "text/plain": [
       "sin(t + 3⋅x) + sin(2⋅t + 5⋅x) + 2.5⋅sin(3⋅t + 7⋅x) + 2⋅sin(6⋅t + 11⋅x) + 2⋅cos\n",
       "(t - 3⋅x) + 4⋅cos(2⋅t - 5⋅x) + 3.5⋅cos(3⋅t - 7⋅x) + 2⋅cos(6⋅t - 11⋅x)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Eigenvalues:  "
     ]
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1.0\\\\1.0\\\\9.0\\\\9.0\\\\16.0\\\\25.0\\\\36.0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1.0 ⎤\n",
       "⎢    ⎥\n",
       "⎢1.0 ⎥\n",
       "⎢    ⎥\n",
       "⎢9.0 ⎥\n",
       "⎢    ⎥\n",
       "⎢9.0 ⎥\n",
       "⎢    ⎥\n",
       "⎢16.0⎥\n",
       "⎢    ⎥\n",
       "⎢25.0⎥\n",
       "⎢    ⎥\n",
       "⎣36.0⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Eigenfunctions:  "
     ]
    },
    {
     "data": {
      "image/png": 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6l9sCR7bEsajAkQUtB0zp0GksULrv0xFHCGTpcMCQ1RqLFhxt03LAlA6dxoIV91KHHMjS4YAhqzUWLXiHg7bSIYe0RI0FCaahSjOABA2As1Z9YhtZrbFowcmA1jhRUjqUGot/kKEsMgzjcKB31CDmlfYVxRQdJQcSsl5jUYKjBUJb8BY+8dcoc5QcPIGYXWIExyu34OCnsQCxooRQlTmyG/lq0g6+hR/Iu4KINr5KAFz/01iABGEh76extJzMtwXcGD+N6Vrn/g/mn6nm9Lu0AQ/YCPufO+5U88H9P+rjINRk63sYAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}- 1.0 \\sin{\\left (3 x \\right )} + 1.0 \\cos{\\left (3 x \\right )}\\\\- 1.0 \\sin{\\left (7 x \\right )} + 1.0 \\cos{\\left (7 x \\right )}\\\\1.0 \\sin{\\left (3 x \\right )} + 1.0 \\cos{\\left (3 x \\right )}\\\\- 1.0 \\sin{\\left (5 x \\right )} + 1.0 \\cos{\\left (5 x \\right )}\\\\1.0 \\sin{\\left (11 x \\right )} + 1.0 \\cos{\\left (11 x \\right )}\\\\1.0 \\sin{\\left (5 x \\right )} + 1.0 \\cos{\\left (5 x \\right )}\\\\1.0 \\sin{\\left (7 x \\right )} + 1.0 \\cos{\\left (7 x \\right )}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡-1.0⋅sin(3⋅x) + 1.0⋅cos(3⋅x) ⎤\n",
       "⎢                             ⎥\n",
       "⎢-1.0⋅sin(7⋅x) + 1.0⋅cos(7⋅x) ⎥\n",
       "⎢                             ⎥\n",
       "⎢ 1.0⋅sin(3⋅x) + 1.0⋅cos(3⋅x) ⎥\n",
       "⎢                             ⎥\n",
       "⎢-1.0⋅sin(5⋅x) + 1.0⋅cos(5⋅x) ⎥\n",
       "⎢                             ⎥\n",
       "⎢1.0⋅sin(11⋅x) + 1.0⋅cos(11⋅x)⎥\n",
       "⎢                             ⎥\n",
       "⎢ 1.0⋅sin(5⋅x) + 1.0⋅cos(5⋅x) ⎥\n",
       "⎢                             ⎥\n",
       "⎣ 1.0⋅sin(7⋅x) + 1.0⋅cos(7⋅x) ⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fullrun(u5,sublist)"
   ]
  }
 ],
 "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": "3.7.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}