Predavanje 13 - Numerično reševanje diferencialnih enačb - robni problem.ipynb

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   "source": [
    "<font size=\"2\"> [github.com/jankoslavic/pypinm](http://github.com/jankoslavic/pypinm)</font>\n",
    "\n",
    "[Prof. dr. Janko Slavič](http://www.ladisk.si/~slavic)\n",
    "\n",
    "<center>\n",
    "<img width=300 src=\"./fig/ladisk.png\">\n",
    "<br>\n",
    "<font size=\"7\" color=\"f00e0e\" face=\"garamond\"><b>Numerično reševanje diferencialnih enačb - robni problem</b></font>\n",
    "<img src=\"\">\n",
    "<font size=\"2\">Februar 2022</font>\n",
    "</center>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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    "toc": "true"
   },
   "source": [
    "<h1>Kazalo<span class=\"tocSkip\"></span></h1>\n",
    "<div class=\"toc\"><ul class=\"toc-item\"><li><span><a href=\"#Reševanje-dvotočkovnih-robnih-problemov\" data-toc-modified-id=\"Reševanje-dvotočkovnih-robnih-problemov-1\"><span class=\"toc-item-num\">1&nbsp;&nbsp;</span>Reševanje dvotočkovnih robnih problemov</a></span></li><li><span><a href=\"#Strelska-metoda\" data-toc-modified-id=\"Strelska-metoda-2\"><span class=\"toc-item-num\">2&nbsp;&nbsp;</span>Strelska metoda</a></span><ul class=\"toc-item\"><li><span><a href=\"#Numerični-zgled:-poševni-met\" data-toc-modified-id=\"Numerični-zgled:-poševni-met-2.1\"><span class=\"toc-item-num\">2.1&nbsp;&nbsp;</span>Numerični zgled: poševni met</a></span><ul class=\"toc-item\"><li><span><a href=\"#Vertikalni-met\" data-toc-modified-id=\"Vertikalni-met-2.1.1\"><span class=\"toc-item-num\">2.1.1&nbsp;&nbsp;</span>Vertikalni met</a></span></li><li><span><a href=\"#Poševni-met\" data-toc-modified-id=\"Poševni-met-2.1.2\"><span class=\"toc-item-num\">2.1.2&nbsp;&nbsp;</span>Poševni met</a></span></li></ul></li><li><span><a href=\"#Uporaba-scipy.integrate.solve_bvp\" data-toc-modified-id=\"Uporaba-scipy.integrate.solve_bvp-2.2\"><span class=\"toc-item-num\">2.2&nbsp;&nbsp;</span>Uporaba <code>scipy.integrate.solve_bvp</code></a></span></li><li><span><a href=\"#Numerični-zgled:-nosilec-z-obremenitvijo\" data-toc-modified-id=\"Numerični-zgled:-nosilec-z-obremenitvijo-2.3\"><span class=\"toc-item-num\">2.3&nbsp;&nbsp;</span>Numerični zgled: nosilec z obremenitvijo</a></span></li></ul></li><li><span><a href=\"#Metoda-končnih-razlik\" data-toc-modified-id=\"Metoda-končnih-razlik-3\"><span class=\"toc-item-num\">3&nbsp;&nbsp;</span>Metoda končnih razlik</a></span><ul class=\"toc-item\"><li><ul class=\"toc-item\"><li><span><a href=\"#Ocena-napake\" data-toc-modified-id=\"Ocena-napake-3.0.1\"><span class=\"toc-item-num\">3.0.1&nbsp;&nbsp;</span>Ocena napake</a></span></li></ul></li><li><span><a href=\"#Numerični-zgled:-vertikalni-met\" data-toc-modified-id=\"Numerični-zgled:-vertikalni-met-3.1\"><span class=\"toc-item-num\">3.1&nbsp;&nbsp;</span>Numerični zgled: vertikalni met</a></span></li><li><span><a href=\"#Numerični-zgled:-nosilec-z-obremenitvijo\" data-toc-modified-id=\"Numerični-zgled:-nosilec-z-obremenitvijo-3.2\"><span class=\"toc-item-num\">3.2&nbsp;&nbsp;</span>Numerični zgled: nosilec z obremenitvijo</a></span></li></ul></li><li><span><a href=\"#Dodatno:-simbolna-rešitev-nosilca\" data-toc-modified-id=\"Dodatno:-simbolna-rešitev-nosilca-4\"><span class=\"toc-item-num\">4&nbsp;&nbsp;</span>Dodatno: simbolna rešitev nosilca</a></span></li></ul></div>"
   ]
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  {
   "cell_type": "markdown",
   "metadata": {
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   },
   "source": [
    "# Reševanje dvotočkovnih robnih problemov"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Pod dvotočkovnim robnim problemom razumemo navadno diferencialno enačbo drugega reda oblike:\n",
    "\n",
    "$$\\ddot y=f(t, y, \\dot y),$$\n",
    "\n",
    "ob predpisanih *robnih pogojih*:\n",
    "\n",
    "$$y(a)=\\alpha\\qquad\\textrm{in}\\qquad y(b)=\\beta.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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   "source": [
    "Metode, ki smo jih spoznali pri reševanju *začetnih problemov*, tukaj neposredno niso uporabne, ker nimamo podanega odvoda v začetni točki pri $t=a$.\n",
    "\n",
    "V nadaljevanju si bomo pogledali dva različna pristopa k reševanju robnih problemov:\n",
    "\n",
    "1. t. i. **strelska metoda**,\n",
    "2. **metoda končnih razlik**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
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   },
   "source": [
    "# Strelska metoda"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Rešujemo robni problem:\n",
    "\n",
    "$$\\ddot y=f(t, y, \\dot y),\\qquad y(a)=\\alpha,\\quad y(b)=\\beta,$$\n",
    "\n",
    "ki ga prevedemo na začetni problem tako, da si izberemo:\n",
    "\n",
    "$$\\dot y (a)=u.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Problem rešimo z numeričnimi metodami reševanja začetnega problema in rešitev označimo z $\\theta(u, t)$.\n",
    "\n",
    "Robni problem rešimo, ko izberemo $u$ tako, da velja:\n",
    "\n",
    "$$r(u)=\\theta(u,b)-\\beta=0.$$\n",
    "\n",
    "Dobili smo nelinearno enačbo, ki jo moramo rešiti; za izračun vrednosti mejnih preostankov $r(u)$ moramo numerično rešiti začetni problem."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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   "source": [
    "Za rešitev enačbe $r(u)=0$ lahko uporabimo sekantno metodo. Izberemo $u_0$ in $u_1$ in na $i$-tem koraku izračunamo:\n",
    "\n",
    "$$u_{i+1}=u_i-r(u_i)\\,\\frac{u_{i}-u_{i-1}}{r(u_{i})-r(u_{i-1})},\\qquad i=2,3,\\dots$$\n",
    "\n",
    "Zaključimo, ko je:\n",
    "$$\\left|r(u_{i+1})\\right|<\\epsilon.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Rešitev strelske metode je obremenjena z napako metode reševanja nelinearne enačbe $\\epsilon$ in z napako numerične metode za reševanje začetnega problema."
   ]
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   },
   "source": [
    "## Numerični zgled: poševni met"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Na sliki je prikazan izstrelek mase $m$, ki ga izstrelimo s hitrostjo $\\textbf{v}_0$.\n",
    "<img src=\"./fig/posevni_met.png\" width=300>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
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    }
   },
   "source": [
    "Velikost sile upora zraka je $|\\textbf{F}|=c\\,|\\textbf{v}|^2$, potem sta gibalni enačbi:\n",
    "\n",
    "$$x''(t)=-F_x/m\\qquad\\ddot y(t)=-F_y/m-g.$$\n",
    "\n",
    "Komponente sile so (glejte izpeljavo pri poglavju iz reševanja začetnega problema sistema diferencialnih enačb):\n",
    "$$F_x=-c\\,x'\\,\\sqrt{x'^2+y'^2},\\qquad F_y=-c\\,y'\\,\\sqrt{x'^2+y'^2}.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
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   "source": [
    "### Vertikalni met\n",
    "\n",
    "Najprej predpostavimo, da je $\\varphi=90$° in torej v $x$ smeri nimamo gibanja. Zanima nas rešitev, ko izstrelek izstrelimo iz višine $y=0\\,$m in mora pri času $t=b=1\\,$s biti na višini $y(b)=10\\,$m. Definirali smo robni problem:\n",
    "\n",
    "$$y''(t)=F_y\\,/m-g,\\qquad y(0)=0,\\quad y(1)=10.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Najprej moramo enačbo drugega reda:\n",
    "\n",
    "$$y''=f(t, y, y') = F_y/m-g$$\n",
    "\n",
    "preoblikovati na sistem dveh enačb prvega reda. Uporabimo $y_i=y^{i}$ ter upoštevamo $F_y=-c\\,y'\\,\\sqrt{y'^2}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Odvajamo $y_i'=y_{i+1}$ in pripravimo sistem enačb prvega reda:\n",
    "\n",
    "$$\n",
    "\\begin{array}{rcl}\n",
    "y'_0&=&y_1\\\\\n",
    "y'_1&=&-c\\,y'\\,\\sqrt{y'^2}/m-g\n",
    "\\end{array}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Pripravimo seznam funkcij desnih strani:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "def f_vert(t, y, g=9.81, m=1., c=0.5):\n",
    "    return np.array([y[1], -g-c*y[1]*np.sqrt(y[1]**2)/m])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Uvozimo modula `numpy` in `scipy.integrate.solve_ivp`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from scipy.integrate import solve_ivp"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "ter pripravimo funkcijo za izračun mejnega preostanka pri času $b$ v odvisnosti od začetne hitrosti $v_0$ (privzeti zračni upor je $c=0.1$):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def r(v0=100., t=None, ciljna_lega=10., g=9.81, m=1., c=0.1):\n",
    "    def f_vert_tmp(t, y): # pripravimo funkcijo z željenimi parametri\n",
    "        return f_vert(t, y, g=g, m=m, c=c)\n",
    "    sol = solve_ivp(f_vert_tmp, t_span=(t[0], t[-1]), y0=np.array([0., v0]))\n",
    "    r = sol.y[0,-1] - ciljna_lega\n",
    "    return r"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Preverimo mejni preostanek pri začetnem pogoju $v_0=y'=50$ m/s:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5.634550939997352"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t = np.linspace(0, 1, 100)\n",
    "r(v0=50., t=t)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Opazimo, da je masa pri 1 sekundi 5,635 m nad ciljno višino.\n",
    "\n",
    "Naš cilj je, da pri 1 sekundi masa doseže lego 10 m z natančnostjo `1e-6`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "ciljna_lega = 10\n",
    "epsilon = 1e-6"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Izvedimo sedaj sekantno metodo:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Novi približek je 5.644178759347092, napaka je -9.672536331804437.\n",
      "Novi približek je 33.67258709458858, napaka je 2.2187017725280267.\n",
      "Novi približek je 28.442965190719505, napaka je 0.8225839451367349.\n",
      "Novi približek je 25.361704474139287, napaka je -0.10153714557801585.\n",
      "Novi približek je 25.700255797214542, napaka je 0.004276366407129473.\n",
      "Novi približek je 25.686573522825128, napaka je 2.1630891925639162e-05.\n",
      "Novi približek je 25.686503962732015, napaka je -4.623824878535743e-09.\n",
      "Rešitev 25.686503962732015\n"
     ]
    }
   ],
   "source": [
    "x0 = 100\n",
    "x1 = 50\n",
    "for i in range(10):\n",
    "    f0 = r(v0=x0, t=t, ciljna_lega=ciljna_lega)\n",
    "    f1 = r(v0=x1, t=t, ciljna_lega=ciljna_lega)\n",
    "    x2 = x1 - f1 * (x1 - x0)/(f1 - f0)\n",
    "    err = r(v0=x2, t=t, ciljna_lega=ciljna_lega)\n",
    "    print(f'Novi približek je {x2}, napaka je {err}.')\n",
    "    x0 = x1\n",
    "    x1 = x2\n",
    "    if abs(err)<epsilon:\n",
    "        rešitev = x2\n",
    "        print(f'Rešitev {rešitev}')\n",
    "        break"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Poglejmo si sedaj izračunano rešitev:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "sol = solve_ivp(lambda t, y: f_vert(t, y, c=0.1),\n",
    "                t_span=(t[0], t[-1]), y0=np.array([0., rešitev]),\n",
    "                t_eval=t\n",
    "               )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Zgoraj smo uporabili `lambda` izraz ([dokumentacija](https://docs.python.org/3/tutorial/controlflow.html#lambda-expressions)). Izraz `lambda t, y: f_vert(t, y, c=0.1)` je ekvivalenten:\n",
    "\n",
    "```python\n",
    "def ime_funkcije(t, y):\n",
    "    return f_vert(t, y, c=0.1)\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Uvozimo `matplotlib`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Prikažemo rezultat:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.title('Vertikalni met')\n",
    "plt.hlines(ciljna_lega,0., 1, 'C3', linestyles='dashed', label='Ciljna lega', alpha=0.5)\n",
    "plt.plot(sol.t, sol.y[0], label='Lega $y$')\n",
    "plt.xlabel('$t$ [s]')\n",
    "plt.legend(loc=4)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Poševni met\n",
    "\n",
    "Poglejmo si sedaj splošen poševni met (diferencialne enačbe so definirana že zgoraj). Najprej moramo sistem diferencialnih enačb drugega reda preoblikovati v sistem enačb prvega reda. \n",
    "\n",
    "Uporabimo:\n",
    "$$y_0=x,~ y_1=x',~ y_2=y,~ y_3=y'$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "in dobimo sistem diferencialnih enačb prvega reda:\n",
    "\n",
    "$$\n",
    "\\begin{array}{rcl}\n",
    "y_0'&=&y_1\\\\\n",
    "y_1'&=&F_x/m\\\\\n",
    "y_2'&=&y_3\\\\\n",
    "y_3'&=&F_y/m-g.\\\\\n",
    "\\end{array}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Pripravimo seznam funkcij desnih strani:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "def f_poševno(t, y, g=9.81, m=1., c=0.1):\n",
    "    x, vx, yy, vy = y\n",
    "    return np.array([vx, -c*vx*np.sqrt(vx**2+vy**2)/m, vy, -g-c*vy*np.sqrt(vx**2+vy**2)/m])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Pripravimo še funkcijo za izračun mejnega preostanka pri času $b=1$ v odvisnosti od vektorja začetne hitrosti $\\mathbf{v}_0$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "def r_poševno(v0=[5., 100.], t=None, ciljna_lega=np.array([10, 5.]), g=9.81, m=1., c=0.1):\n",
    "    sol = solve_ivp(lambda t, y: f_poševno(t, y, g=g, m=m, c=c),\n",
    "                      t_span=(t[0], t[-1]),\n",
    "                      y0=[0, v0[0], 0, v0[1]])\n",
    "    r = sol.y[0:3:2, -1] - ciljna_lega\n",
    "    return r"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Preverimo mejni preostanek pri začetnem pogoju $\\mathbf{v}_0=[100., 100.]\\,$m/s:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 9.66195325, 11.84726075])"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r_poševno(v0=[100., 100.],t=t)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Za iskanje korena sistema nelinearnih funkcij smo že spoznali funkcijo ``scipy.optimize.root`` ([dokumentacija](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.root.html)):\n",
    "\n",
    "```python\n",
    "root(fun, x0, args=(), method='hybr', jac=None,\n",
    "     tol=None, callback=None, options=None)\n",
    "```\n",
    "Najprej jo uvozimo:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy import optimize"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Potem uporabimo z začetnim ugibanjem $\\mathbf{v}_0$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "rešitev = optimize.root(r_poševno, np.array([100., 100.]), args=(t))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Rešitev je:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "    fjac: array([[-0.93392959,  0.35745703],\n",
       "       [-0.35745703, -0.93392959]])\n",
       "     fun: array([ 1.15463195e-13, -3.25961480e-13])\n",
       " message: 'The solution converged.'\n",
       "    nfev: 17\n",
       "     qtf: array([ 1.06701940e-09, -1.31133975e-09])\n",
       "       r: array([-0.40775373,  0.24087449, -0.37544067])\n",
       "  status: 1\n",
       " success: True\n",
       "       x: array([19.37894314, 16.55482478])"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rešitev"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Atribut `rešitev.x` vsebuje vektor izračunanih rešitev (`[19.37894314,  16.55482478]`). Preverimo mejni preostanek pri izračunani rešitvi:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 1.15463195e-13, -3.25961480e-13])"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r_poševno(v0=rešitev.x, t=t)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Poglejmo si sedaj izračunano rešitev:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "sol = solve_ivp(lambda t, y: f_poševno(t, y, g=9.81, m=1., c=.1),\n",
    "                t_span=(t[0], t[-1]),\n",
    "                y0=[0, rešitev.x[0], 0, rešitev.x[1]],\n",
    "                t_eval=t)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Prikažemo rezultat:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.title('Poševni met')\n",
    "plt.hlines(10,0., 1, 'C3', linestyles='dashed', label='Ciljna lega za $x$', alpha=0.5)\n",
    "plt.hlines(5,0., 1, 'C4', linestyles='dashed', label='Ciljna lega za $y$', alpha=0.5)\n",
    "plt.plot(sol.t, sol.y[0], label='Lega $x$', color='C3')\n",
    "plt.plot(sol.t, sol.y[2], label='Lega $y$', color='C4')\n",
    "plt.xlabel('$t$ [s]')\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Uporaba `scipy.integrate.solve_bvp`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Namesto `scipy.integrate.solve_ivp` in `scipy.optimize` lahko uporabimo vgrajeno funkcijo za reševanje robnih problemov `scipy.integrate.solve_bvp` (BVP - *angl.* Boundary Value Problem):\n",
    "```python\n",
    "scipy.integrate.solve_bvp(fun, bc, x, y, \n",
    "                          p=None, S=None, fun_jac=None, bc_jac=None, \n",
    "                          tol=0.001, max_nodes=1000, verbose=0)\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Pojasnilo vseh argumentov je v [dokumentaciji](https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.solve_bvp.html#scipy-integrate-solve-bvp), tukaj bomo izpostavili nekatere:\n",
    "\n",
    "* `fun` je desna stran (`func(t, y...)`),\n",
    "* `bc` je mejni preostanek: `bc(y(a), y(b)) = 0`\n",
    "* `x` numerično polje (dimenzija `(m)`) neodvisne spremenljivke `x[0]=a` in `x[-1]=b`,\n",
    "* `y` numerično polje (dimenzija `(n, m)`) začetnih vrednosti."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Rezultat klicanja `solve_bvp` je objekt z atributi (izbrani):\n",
    "\n",
    "* `x` vrednosti neodvisne spremenljivke pri katerih je izračunan rezultat,\n",
    "* `y` rezultat,\n",
    "* `sol` rezultat v obliki kubičnega zlepka,\n",
    "* `success` je `True`, če je bila rešitev konvergirala."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Poglejmo primer:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.integrate import solve_bvp"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Definirajmo neodvisno spremenljivko"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "t = np.linspace(0, 1, 15)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "In mejni preostanek (želimo, da je pri času `b` lega $x=10$ m in $y=5$ m):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "def mejni_preostanek(ya, yb):\n",
    "    return np.array([ya[0], ya[2], yb[0]-10, yb[2]-5])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Definirajmo še začetne vrednosti hitrosti (kot začetno ugibanje):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "y0 = np.zeros((4, t.size))\n",
    "y0[1] = 5   # hitrost v x smeri\n",
    "y0[3] = 100 # hitrost v y smeri"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "In rešimo robni problem in opis rezultata"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "sol = solve_bvp(lambda t, y: f_poševno(t, y, g=9.81, m=1., c=.1), \n",
    "                bc=mejni_preostanek, x=t, y=y0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'The algorithm converged to the desired accuracy.'"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sol.message"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Prikažimo rezultat:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.title('Poševni met')\n",
    "plt.hlines(10,0., 1, 'C3', linestyles='dashed', label='Ciljna lega za $x$', alpha=0.5)\n",
    "plt.hlines(5,0., 1, 'C4', linestyles='dashed', label='Ciljna lega za $y$', alpha=0.5)\n",
    "plt.plot(sol.x, sol.y[0], label='Lega $x$', color='C3')\n",
    "plt.plot(sol.x, sol.y[2], label='Lega $y$', color='C4')\n",
    "plt.xlabel('$t$ [s]')\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Numerični zgled: nosilec z obremenitvijo"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "Poglejmo si nosilec:\n",
    "<img src='./fig/nosilec.png' width=400>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Poves $w(x)$ nosilca popiše diferencialna enačba četrtega reda:\n",
    "\n",
    "$$-E\\,I\\frac{\\textrm{d}^4}{\\textrm{d}x^4}w(x)+q(x)=0.$$\n",
    "\n",
    "Znane konstante so $E,I,l$ in je $q(x)$ porazdeljena obremenitev.\n",
    "\n",
    "Robni pogoji (členkasto vpet nosilec):\n",
    "$$w(0)=w(l)=0\\quad\\textrm{in}\\quad w''(0)=w''(l)=0$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Parametri so: \n",
    "\n",
    "* $I=2.1\\cdot10^{-5}\\,$m$^4$, \n",
    "* $E=2.1\\cdot10^{11}\\,N/$m$^2$, \n",
    "* $l=10\\,$m. \n",
    "\n",
    "Porazdeljena obremenitev $q(x)$ bo definirana pozneje."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Najprej moramo diferencialno enačbo četrtega reda preoblikovati v sistem diferencialnih enačb prvega reda. \n",
    "Uporabimo:\n",
    "$$y_0=w,~ y_1=w',~ y_2=w'',~ y_3=w'''$$\n",
    "\n",
    "in dobimo sistem diferencialnih enačb prvega reda:\n",
    "\n",
    "$$\n",
    "\\begin{array}{rcl}\n",
    "y_0'&=&y_1\\\\\n",
    "y_1'&=&y_2\\\\\n",
    "y_2'&=&y_3\\\\\n",
    "y_3'&=&q(x)/(EI).\\\\\n",
    "\\end{array}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Pripravimo različne porazdeljene obremenitve:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [],
   "source": [
    "def q_konstanta(x, F_0=1e3, l=10):\n",
    "    return -F_0*np.ones_like(x)\n",
    "\n",
    "def q_trikotna(x, F_0=1e3, l=10):\n",
    "    return -F_0*x/l\n",
    "\n",
    "def q_pol_sinusna(x, F_0=1e3, l=10):\n",
    "    return -F_0*np.sin(np.pi*x/l)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Definirajmo dolžino (ostale parametre, $I, E$, bomo uporabili privzete):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "l = 10."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(0, l, 15)\n",
    "plt.fill_between(x, q_konstanta(x), alpha=0.4, label='Konstantna')\n",
    "plt.fill_between(x, q_pol_sinusna(x), alpha=0.4, label='Pol-sinusna')\n",
    "plt.fill_between(x, q_trikotna(x), alpha=0.4, label='Trikotna')\n",
    "plt.title('Porazdeljena obremenitev $q(x)$')\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Pripravimo seznam funkcij desne strani:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [],
   "source": [
    "def f_nosilec_konstanta(x, y, E=2.1e11, I=2.1e-5):\n",
    "    return np.array([y[1], y[2], y[3], q_konstanta(x)/(E*I)], dtype=float)\n",
    "def f_nosilec_pol_sinusna(x, y, E=2.1e11, I=2.1e-5):\n",
    "    return np.array([y[1], y[2], y[3], q_pol_sinusna(x)/(E*I)])\n",
    "def f_nosilec_trikotna(x, y, E=2.1e11, I=2.1e-5):\n",
    "    return np.array([y[1], y[2], y[3], q_trikotna(x)/(E*I)])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Definirajmo sedaj robne pogoje oz. mejni preostanek (poves in moment sta na robovih enaka nič):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "def mejni_nosilec(ya, yb):\n",
    "    return np.array([ya[0], ya[2], yb[0], yb[2]])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Definirajmo še začetne vrednosti hitrosti (kot začetno ugibanje):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [],
   "source": [
    "y0 = np.zeros((4, x.size), dtype=float)\n",
    "y0[1] = 1. # w'\n",
    "y0[3] = 1. # w'''"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "In rešimo robni problem (za vse tri tipe obremenitve):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [],
   "source": [
    "sol_ko = solve_bvp(f_nosilec_konstanta, bc=mejni_nosilec, x=x, y=y0)\n",
    "sol_pol_sin = solve_bvp(f_nosilec_pol_sinusna, bc=mejni_nosilec, x=x, y=y0)\n",
    "sol_trikotna = solve_bvp(f_nosilec_trikotna, bc=mejni_nosilec, x=x, y=y0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Rezultat prikažimo za konstantno obremenitev:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.fill_between(sol_ko.x, sol_ko.y[0], label='Poves $w$', alpha=0.5)\n",
    "plt.fill_between(sol_ko.x, sol_ko.y[1], label='Naklon $w\\'$', alpha=0.5)\n",
    "plt.title('Konstantna obremenitev')\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "In primerjavo povesa za različne tipe obremenitve:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.fill_between(sol_ko.x, sol_ko.y[0], label='Konstantna $w$', alpha=0.5)\n",
    "plt.fill_between(sol_pol_sin.x, sol_pol_sin.y[0], label='Pol-sinusna $w$', alpha=0.5)\n",
    "plt.fill_between(sol_trikotna.x, sol_trikotna.y[0], label='Trikotna $w$', alpha=0.5)\n",
    "plt.title('Poves pri različnem tipu obremenitve')\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Metoda končnih razlik"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/jpeg": 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      "text/html": [
       "\n",
       "        <iframe\n",
       "            width=\"800\"\n",
       "            height=\"300\"\n",
       "            src=\"https://www.youtube.com/embed/YozKGsNYZ6Y\"\n",
       "            frameborder=\"0\"\n",
       "            allowfullscreen\n",
       "        ></iframe>\n",
       "        "
      ],
      "text/plain": [
       "<IPython.lib.display.YouTubeVideo at 0x11612f11130>"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from IPython.display import YouTubeVideo\n",
    "YouTubeVideo('YozKGsNYZ6Y', width=800, height=300)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Rešujemo robni problem:\n",
    "\n",
    "$$y''=f(t, y, y'),\\qquad y(a)=\\alpha,\\quad y(b)=\\beta.$$\n",
    "\n",
    "Velja torej: $$a=t_0,\\quad b=t_{n-1}.$$\n",
    "\n",
    "Pri metodi končnih razlik za reševanje robnega problema uporabimo diferenčno shemo. Predpostavimo, da imamo interval $[a,b]$, na katerem rešujemo diferencialno enačbo (neodvisna spremenljivka) razdeljeno na enake podintervale (točk je $n$):\n",
    "\n",
    "$$t=[t_0, t_1,\\dots, t_{n-1}].$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Odvode nadomestimo s centralno diferenčno shemo:\n",
    "\n",
    "|    Odvod$\\downarrow$ $\\backslash$ Vrednosti $\\rightarrow$   |$y_{i-2}$|$y_{i-1}$ | $y_{i}$   |$y_{i+1}$ |$y_{i+2}$|\n",
    "|:--------:|:-------------------:|:----------:|:----------:|:----------:|:----------:|\n",
    "|$y_i'=\\frac{1}{h}\\cdot$| 0     | -0.5 | 0 | 0.5 | 0|\n",
    "|$y_i''=\\frac{1}{h^2}\\cdot$| 0 | 1 | -2 | 1 | 0|\n",
    "|$y_i'''=\\frac{1}{h^3}\\cdot$| -0.5 | 1 | 0 | -1 | 0.5|\n",
    "|$y_i^{(4)}=\\frac{1}{h^4}\\cdot$| 1 | -4 | 6 | -4 | 1|\n",
    "\n",
    "Prikazana centralna diferenčna shema ima napako drugega reda $\\mathcal{O}(h^{2})$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "V $i$-ti točki navadno differencialno enačbo drugega reda s centralimi diferencami zapišemo:\n",
    "\n",
    "$$\\frac{1}{h^2}\\left(y_{i-1}-2\\,y_{i}+y_{i+1}\\right)+\\mathcal{O}(h^{2})=f\\left(t_i, y_i, \n",
    "\\frac{1}{2h}\\left(-y_{i-1}+y_{i+1}\\right)+\\mathcal{O}(h^{2})\\right).$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Če zanemarimo napako metode:\n",
    "\n",
    "$$\\frac{1}{h^2}\\left(y_{i-1}-2\\,y_{i}+y_{i+1}\\right)=f\\left(t_i, y_i, \n",
    "\\frac{1}{2h}\\left(-y_{i-1}+y_{i+1}\\right)\\right),\\qquad i=1,2,\\dots,n-2.$$\n",
    "\n",
    "Zgornjo enačbo lahko zapišemo za $n-2$ notranjih točk, kar pomeni, da nam do rešljivega sistema enačb za $n$ neznank manjkata še dve enačbi. Ti dve enačbi sta robna pogoja:\n",
    "\n",
    "$$y_0=\\alpha,\\quad y_{n-1}=\\beta.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "V primeru linearnega robnega problema moramo za izračun $n$ neznank $y_i$ rešiti sistem $n$ linearnih enačb (če je pa nelinearen, pa sistem nelinearnih enačb)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Ocena napake\n",
    "\n",
    "Točen rezultat $y(t_{i})$ pri velikosti koraka $h$ je:\n",
    "$$y(t_{i})=y_{i,h}+E_{i,h},$$\n",
    "kjer je $y_{i,h}$ numerični približek in $E_h$ napaka metode. Ker je globalna napaka drugega reda, lahko napako zapišemo:\n",
    "$$E_{i,h}=k\\,h^2,$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Podobno lahko za velikost koraka $2h$ zapišemo:\n",
    "$$y(t_{j})=y_{j,2h}+E_{j,2h},$$\n",
    "kjer je $y_{j,2h}$ numerični približek in $E_{2h}$ napaka metode in je:\n",
    "$$E_{j,2h}=k\\,(2\\,h)^2=4\\,k\\,h^2$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Ob predpostavki, da je konstanta $k$ pri koraku $h$ in koraku $2h$ enaka, lahko določimo oceno napake pri boljšem približku $E_h$. Najprej izenačimo točna rezultat $y(t_{i})$ pri koraku $h$ in rezultat $y(t_{j})$ pri koraku $2h$ (velja $i=2\\,j$, $j=1,2,\\dots$):\n",
    "\n",
    "$$y_{i,h}+k\\,h^2=y_{j,2h}+4\\,k\\,h^2$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "sledi:\n",
    "\n",
    "$$3\\,k\\,h^2=y_{2j,h}-y_{j,2h}$$\n",
    "\n",
    "in nato izračunamo *oceno napake*:\n",
    "\n",
    "$$E_{j,h}=\\frac{y_{2j,h}-y_{j,2h}}{3}.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Numerični zgled: vertikalni met\n",
    "\n",
    "Zgoraj smo vertikalni met rešili s strelsko metodo; uporabimo sedaj metodo končnih razlik. Najprej robni problem:\n",
    "$$y''(t)=F_y/m-g,\\qquad y(0)=0,\\quad y(1)=10.$$\n",
    "zapisati s pomočjo centralne diferenčne sheme ($F_y=-c\\,y'$):\n",
    "\n",
    "$$\\frac{1}{h^2}\\left(y_{i-1}-2\\,y_{i}+y_{i+1}\\right)= -c\\left(\\frac{1}{2h}\\left(-y_{i-1}+y_{i+1}\\right)\\right)/m-g.$$\n",
    "\n",
    "(Tukaj smo predpostavili, da je zračni upor linearno odvisen od hitrost. V nasprotnem primeru bi imeli nelinearni robni problem, in posledično razvili sistem nelinearnih enačb.)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Zgornji izraz preoblikujemo:\n",
    "$$(2-c\\,h/m)\\,y_{i-1}-4\\,y_{i}+(2+c\\,h/m)\\,y_{i+1}=-2\\,g\\,h^2,\\qquad i=1,2,\\dots,n-2.$$\n",
    "\n",
    "Robna pogoja sta:\n",
    "\n",
    "$$y_0=0,\\qquad y_{n-1}=10$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Robni problem smo torej preoblikovali na sistem $n$ linearnih enačb. Poglejmo si sedaj konkreten izračun za $n=11$; najprej definirajmo konstante, časovni vektor `t` in korak `h`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [],
   "source": [
    "n = 11 # liho število\n",
    "c = 0.5\n",
    "m = 1.0\n",
    "g = 9.81\n",
    "t = np.linspace(0, 1, n)\n",
    "h = t[1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Nato nadaljujemo z izračunom tridiagonalne matrike koeficientov `A`. \n",
    "\n",
    "Pomagamo si s funkcijo `numpy.diag()` ([dokumentacija](https://docs.scipy.org/doc/numpy/reference/generated/numpy.diag.html)):\n",
    "\n",
    "```python\n",
    "numpy.diag(v, k=0)\n",
    "```\n",
    "s parametroma:\n",
    "\n",
    "* `v` vektor, ki bo prirejen diagonali,\n",
    "* `k` diagonala, kateri se priredi `v`. `k=0` uporabimo za glavno diagonalo, `k<0` oz. `k>0` uporabimo za  diagonale pod oz. nad glavno diagonalo."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-4.  ,  2.05,  0.  ,  0.  ],\n",
       "       [ 1.95, -4.  ,  2.05,  0.  ],\n",
       "       [ 0.  ,  1.95, -4.  ,  2.05],\n",
       "       [ 0.  ,  0.  ,  1.95, -4.  ]])"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = np.diag(-4*np.ones(n), 0) + \\\n",
    "    (2-c*h/m)*np.diag(np.ones(n-1), -1) + \\\n",
    "    (2+c*h/m)*np.diag(np.ones(n-1), 1)\n",
    "A[:4,:4]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Definirajmo še vektor konstant:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [],
   "source": [
    "b = -2*g*h**2 * np.ones(n)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Sedaj *popravimo* matriko koeficientov `A` in vektor konstant `b`, da zadostimo robnim pogojem:\n",
    "\n",
    "$$y_0=0,\\quad y_{n-1}=10.$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [],
   "source": [
    "A[0,0] = 1\n",
    "A[0,1] = 0\n",
    "A[-1,-2] = 0\n",
    "A[-1,-1] = 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [],
   "source": [
    "b[0] = 0.\n",
    "b[-1] = 10."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Rešimo sistem linearnih enačb:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [],
   "source": [
    "y_mkr = np.linalg.solve(A, b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Prikažemo rezultat:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.title('Vertikalni met (upor zraka: $-c\\,\\dot y$)')\n",
    "plt.hlines(10, 0, 1, 'C3', label='Ciljna lega', linestyles='dashed', alpha=0.5)\n",
    "plt.plot(t, y_mkr, '.', label='Metoda končnih razlik')\n",
    "plt.xlabel('$t$ [s]')\n",
    "plt.ylabel('$y$ [m]')\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "S pomočjo funkcije [`numpy.gradient()`](https://docs.scipy.org/doc/numpy/reference/generated/numpy.gradient.html) izračunamo še hitrost in pospešek:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([17.99109496, 16.20009044, 14.45276895, 12.79068266])"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v_mkr = np.gradient(y_mkr,h, edge_order=2)\n",
    "a_mkr = np.gradient(v_mkr,h, edge_order=2)\n",
    "v_mkr[:4]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Izračunajmo sedaj še rezultat z dvojnim korakom:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [],
   "source": [
    "n2h = n//2+1\n",
    "t2h = np.linspace(0, 1, n2h)\n",
    "h2h = t[1]\n",
    "A2h = np.diag(-4*np.ones(n2h), 0) + \\\n",
    "    (2-c*h2h/(m))*np.diag(np.ones(n2h-1), -1) + \\\n",
    "    (2+c*h2h/(m))*np.diag(np.ones(n2h-1), 1)\n",
    "b2h = -2*g*h2h**2 * np.ones(n2h)\n",
    "A2h[0,0] = 1\n",
    "A2h[0,1] = 0\n",
    "A2h[-1,-1] = 1\n",
    "A2h[-1,-2] = 0\n",
    "b2h[0] = 0.\n",
    "b2h[-1] = 10.\n",
    "y2h_mkr = np.linalg.solve(A2h, b2h)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Primerjajmo prvih šest rezultatov pri koraku $2h$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.        ,  2.40584652,  4.59862736,  6.58873597,  8.38605878,\n",
       "       10.        ])"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y2h_mkr[:6]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "z vsakim drugim rezultatom pri koraku $h$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.        ,  3.24001809,  5.79815462,  7.73931207,  9.12221538,\n",
       "       10.        ])"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y_mkr[:12:2]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Sedaj lahko ocenimo napako:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.        , 0.27805719, 0.39984242, 0.38352537, 0.24538553,\n",
       "       0.        ])"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(y_mkr[::2]-y2h_mkr)/3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## Numerični zgled: nosilec z obremenitvijo"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Vrnimo se k robnemu problemu nosilca s polsinusno obremenitvijo ($q(x)=-F_0\\,\\sin(\\pi\\,x/l))$), ki smo ga že obravnavali s strelsko metodo. \n",
    "\n",
    "Diferencialno enačbo četrtega reda zapišemo s pomočjo centralne diferenčne sheme (za $i$-to točko):\n",
    "\n",
    "$$-\\frac{E\\,I}{h^4}\\left(1\\,w_{i-2} - 4\\,w_{i-1} + 6\\,w_{i} - 4\\,w_{i+1} + 1\\, w_{i+2}\\right) - F_0\\,\\sin(\\pi\\,x_i/l)= 0.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Robni pogoji so štirje, najprej poves na robovih: \n",
    "\n",
    "$$w_0=w_{n-1}=0$$\n",
    "\n",
    "Ker na robovih ni momenta, je drugi odvod nič. S centralno diferenčno shemo torej zapišemo dodatne enačbe: \n",
    "\n",
    "$$w_{-1}-2\\,w_{0}+w_{1}=0\\qquad w_{n-2}-2\\,w_{n-1}+w_{n}=0.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Če za neodvisno spremenljivko $x$ uporabimo $n$ ekvidistantnih točk, potem diferencialno enačbo četrtega reda zapišemo za $n-2$ notranje točke. S tem pridobimo dodatni nefizikalni točki $w_{-1}$ in $w_n$. Če dodamo še štiri robne pogoje, imamo rešljiv sistem linearnih enačb z $n+2$ neznakami in $n+2$ enačbami."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Najprej pripravimo podatke, neodvisno spremenljivko `x` in korak `h`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [],
   "source": [
    "l = 10.\n",
    "E = 2.1e11\n",
    "I = 2.1e-5\n",
    "F_0= 1e3\n",
    "n = 100\n",
    "x = np.linspace(0, l, n)\n",
    "h = x[1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Nato pripravimo matriko koeficientov (matrika je dimenzije `(n+2, n+2)`, v prvi in v zadnji dve vrstici bi lahko zapisali tudi vrednosti 0, to bomo pozneje popravili):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 6., -4.,  1.,  0.],\n",
       "       [-4.,  6., -4.,  1.],\n",
       "       [ 1., -4.,  6., -4.],\n",
       "       [ 0.,  1., -4.,  6.]])"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = +1*np.diag(np.ones(n-2+2), -2) \\\n",
    "    -4*np.diag(np.ones(n-1+2), -1) \\\n",
    "    +6*np.diag(np.ones(n+2), 0)  \\\n",
    "    -4*np.diag(np.ones(n-1+2), +1)  \\\n",
    "    +1*np.diag(np.ones(n-2+2), +2)\n",
    "A[:4,:4]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Definirajmo še vektor konstant (dodamo en element na koncu in en na začetku):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.00000000e+00,  0.00000000e+00, -7.48966545e-10, -1.49717894e-09])"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b = np.zeros(n+2)\n",
    "b[2:-2] = - h**4 * F_0*np.sin(np.pi*x[1:-1]/l)/(E*I)\n",
    "b[:4]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Sedaj *popravimo* matriko koeficientov `A` in vektor konstant `b`, da zadostimo\n",
    "robnim pogojem. \n",
    "\n",
    "Najprej $w_0=w_{n-1}=0$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [],
   "source": [
    "A[0,:3]= np.array([0, 1, 0])\n",
    "b[0] = 0\n",
    "A[-1,-3:]= np.array([0, 1, 0])\n",
    "b[-1] = 0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Nato $w_{-1}-2\\,w_{0}+w_{1}=0$ in $w_{n-2}-2\\,w_{n-1}+w_{n}=0$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [],
   "source": [
    "A[1,:4] = np.array([1, -2, 1,0])\n",
    "b[1] = 0\n",
    "A[-2,-4:] = np.array([0, 1, -2, 1])\n",
    "b[-2] = 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.,  1.,  0.,  0.,  0.],\n",
       "       [ 1., -2.,  1.,  0.,  0.],\n",
       "       [ 1., -4.,  6., -4.,  1.],\n",
       "       [ 0.,  1., -4.,  6., -4.],\n",
       "       [ 0.,  0.,  1., -4.,  6.]])"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A[:5,:5]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 6., -4.,  1.,  0.,  0.],\n",
       "       [-4.,  6., -4.,  1.,  0.],\n",
       "       [ 1., -4.,  6., -4.,  1.],\n",
       "       [ 0.,  0.,  1., -2.,  1.],\n",
       "       [ 0.,  0.,  0.,  1.,  0.]])"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A[-5:,-5:]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.00000000e+00,  0.00000000e+00, -7.48966545e-10, -1.49717894e-09,\n",
       "       -2.24388381e-09])"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b[:5]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-2.24388381e-09, -1.49717894e-09, -7.48966545e-10,  0.00000000e+00,\n",
       "        0.00000000e+00])"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b[-5:]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Rešimo sistem linearnih enačb:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [],
   "source": [
    "y_mkr = np.linalg.solve(A, b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Prikažemo rezultat:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.title('Poves ob polsinusni obremenitvi')\n",
    "plt.plot(sol_pol_sin.x, sol_pol_sin.y[0], 'o', label='Strelska metoda')\n",
    "plt.plot(x, y_mkr[1:-1], '.', label='Metoda končnih razlik')\n",
    "plt.xlabel('$x$ [m]')\n",
    "plt.ylabel('$w$ [m]')\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Dodatno: simbolna rešitev nosilca"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Tukaj si bomo pogledali simbolno reševanje robnega problema. Poudariti je treba, da gre tukaj zgolj za zgled, ki ga lahko naredimo za obravnavani nosilec z relativno enostavno polsinusno obremenitvijo. **V praksi so seveda obremenitve in tudi oblike nosilca lahko bistveno bolj zahtevne in takrat druge poti kot numeričnega reševanja skoraj nimamo na voljo.**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Najprej uvozimo `sympy`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy as sym\n",
    "sym.init_printing()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Definirajmo spremenljivke:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [],
   "source": [
    "w = sym.Function('w')\n",
    "x, E, I, F_0, l = sym.symbols('x, E, I, F_0, l')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Definirajmo differencialno enačbo (robne pogoje dodamo pozneje):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - E I \\frac{d^{4}}{d x^{4}} w{\\left(x \\right)} - F_{0} \\sin{\\left(\\frac{\\pi x}{l} \\right)} = 0$"
      ],
      "text/plain": [
       "        4                        \n",
       "       d                ⎛π⋅x⎞    \n",
       "- E⋅I⋅───(w(x)) - F₀⋅sin⎜───⎟ = 0\n",
       "        4               ⎝ l ⎠    \n",
       "      dx                         "
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq = sym.Eq(-E*I*w(x).diff(x,4)-F_0*sym.sin(sym.pi*x/l),0)\n",
    "eq"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Rešimo:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle w{\\left(x \\right)} = - \\frac{F_{0} l^{4} \\sin{\\left(\\frac{\\pi x}{l} \\right)}}{\\pi^{4} E I}$"
      ],
      "text/plain": [
       "            4    ⎛π⋅x⎞ \n",
       "       -F₀⋅l ⋅sin⎜───⎟ \n",
       "                 ⎝ l ⎠ \n",
       "w(x) = ────────────────\n",
       "             4         \n",
       "            π ⋅E⋅I     "
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sol = sym.dsolve(eq, func=w(x), hint='nth_linear_constant_coeff_undetermined_coefficients',\n",
    "                 ics={\n",
    "                     w(0):0,                      #poves pri x=0 je nič\n",
    "                     w(x).diff(x,2).subs(x, 0): 0,#moment pri x=0 je nič\n",
    "                     w(x).subs(x, l): 0,                     #poves pri x=l je nič\n",
    "                     w(x).diff(x, 2).subs(x, l): 0 #moment pri x=l je nič                 \n",
    "                 })\n",
    "sol"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [],
   "source": [
    "podatki = {E: 2.1e11, I: 2.1e-5, l: 10, F_0:1e3}\n",
    "resitev_np = sym.lambdify(x, sol.rhs.subs(podatki), modules='numpy')\n",
    "x_ana = np.linspace(0, podatki[l], 100)\n",
    "y_ana = resitev_np(x_ana)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.title('Upogib ob polsinusni obremenitvi')\n",
    "plt.plot(x_ana, y_ana, '-', label='Analitična rešitev')\n",
    "plt.xlabel('$x$ [m]')\n",
    "plt.ylabel('$w$ [m]')\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Primerjajmo sedaj analitično rešitev, z rešitivijo z metodo končnih razlik in strelsko metodo:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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em1f7Kb0uTEV3beIKhII9rlgN9J64cseL9PP0/70qsWclG1+iL64FEyC6zeyUrF38cdroxkfyXGsQjDy6A12PreO3XdflntWyHqr3AouXMNfV2NK6KcrUOOul5XrSen/z2FiNgxY7g89AhFxK6frk8ifIfLlbone0s392BD4uBBQ47ATzYHL5cXnWvekInBeBHlvnxb9r//gQsETv5sfnWveoI7ArArwP2pO8XSHuwj9RBHpsfaIT393eF4Ge6O2Lb5f+ESGgBI8tdV5D6KUj0BHYEIEeWxuC2UV1BCYI9ERvAkg/7QgUEHiiG9LsP64U6PtQR6Aj4EOgx5YPp07VEWhGwN7R470jymPdyEY/ljh298+OQEegI9AR6Ah0BDoCHYHrjoDyOP6yBz9w5YeSj/8P7dOQhEiBxnoAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle \\left[ -0.0232759411601314, \\  -0.0232798479987807, \\  -0.0232786462140291\\right]$"
      ],
      "text/plain": [
       "[-0.02327594116013143, -0.023279847998780707, -0.023278646214029086]"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[np.min(y_ana), np.min(y_mkr), np.min(sol_pol_sin.y[0])]"
   ]
  }
 ],
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