\n",
" \n",
" $F_y(x,y,y')\\geqslant 0$ za svaki $(x,y,y')\\in D$; \n",
" postoji konstanta $C,$ takva da je $\\left| F_{y'}(x,y,y')\\right|\\leqslant C,$ za svaki $(x,y,y')\\in D,$ \n",
" \n",
"tada dati rubni problem ima jedinstveno rješenje. \n",
"\n",
"\n",
"Primjer Dat je rubni problem $$ y''+e^{-xy}+\\sin y'=0,\\:x\\in[1,2],\\:y(1)=y(2)=0.$$ Vrijedi \n",
"$$F(x,y,y')=-e^{-xy}-\\sin y', $$ te kako je\n",
"$$F_y(x,y,y')=xe^{-xy}>0 \\text{ i } \\left| F_{y'}(x,y,y')\\right|=|-\\cos y'|\\leqslant 1, $$\n",
"ovaj rubni problem ima jedinstveno rješenje. \n",
"
\n",
"\n",
"Ako funkcija $F$ ima oblik $$F(x,y,y')=b(x)y'+c(x)y+f(x), $$ tada je diferencijalna jednačina $$y''=F(x,y,y')$$\n",
"\n",
"linearna. Sada se uslovi pod kojima postoji jedinstveno rješenje mogu oslabiti. \n",
"\n",
"Ako linearni rubni problem $$y''=b(x)y'+c(x)y+f(x),\\:x\\in[a,b],\\:y(a)=\\alpha,\\,y(b)=\\beta,$$ zadovoljava\n",
"\n",
" funkcije $b,c$ i $f$ su neprekidne na $[a,b];$ \n",
" $c(x)\\geqslant0$ na $[a,b];$ \n",
" \n",
"tada rubni problem ima jedinstveno rješenje."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
" \n",
"\n",
"# Metoda centralnih razlika\n",
"## Rješavanje linearnih rubnih problema\n",
"\n",
"Bez umanjenja opštosti nastavićemo analizu na $[0,1]$ umjesto $[a,b].$ Ovo se lako izvrši funkcijom $x\\mapsto \\frac{x-a}{b-a}.$\n",
"\n",
"\n",
"\n",
"\n",
"Dat je rubni problem \n",
"\\begin{align}\n",
"-\\varepsilon^2 y''+b(x)y'+c(x)y=f(x),\\:y(0)=\\alpha,\\,y(1)=\\beta,\\:x\\in[0,1],\\nonumber\n",
"\\label{rubni1}\n",
"\\end{align}\n",
"i neka je \n",
"\\begin{align}\n",
"c(x)\\geqslant 0,\\:x\\in[0,1],\\nonumber\n",
"\\label{rubni2}\n",
"\\end{align}\n",
"gdje je $0<\\varepsilon\\leqslant 1$ i neka su funkcije $b,c$ i $f$ su dovoljno glatke na $[0,1].$ \n",
"Dati rubni problem ima jedinstveno rješenje. \n",
" \n",
"\n",
" Napomena 1: Za rješavanje klasičnih linearnih rubnih problema koristiti vrijednost parametra $\\varepsilon=1.$ Dobijena šema može se koristiti i za testiranje numeričkog rješavanja odgovarajućeg singularno-perturbacionog rubnog problema. \n",
"
\n",
"\n",
"Ovaj problem rješavaćemo na ekvidistatnoj mreži, čiji je parametar mreže (dužina koraka) $h=\\frac{1}{N}$ (u opštem slučaju je $h=\\frac{b-a}{n}$), i $N$ je broj podsegmenata segmenta $[0,1],$ dakle tačke mreže dobijamo na sljedeći način\n",
"$$x_i=ih,\\:i=0,1,\\ldots,N, \\text{ gde je } x_0=0,\\:x_N=1. $$\n",
"\n",
"\n",
"\n",
"Metoda konačnih razlika sastoji se u diskretizaciji diferencijalne jednačine koristeći tačke mreže $x_i,$ gdje s nepoznate $y_i,\\,i=1,\\ldots,N-1,$ približne vrijednost tačnog rješenja u tačkama mreža $y(x_i),$ tj. ($y(x_i)\\approx y_i $).\n",
"\n",
"\n",
"Prvi i drugi izvod aproksimiramo centralnim razlikama \n",
"$$ y'(x)\\approx (D^0y)(x):=\\frac{y(x+h)-y(x-h)}{2h},$$\n",
"i\n",
"$$y''(x)\\approx (D^{+}D^{-}y)(x)=\\frac{y(x+h)-2y(x)+y(x-h)}{h^2},$$\n",
"gdje su $(D^{+}y)(x):=\\frac{y(x+h)-y(x)}{h},\\,(D^{-}y)(x):=\\frac{y(x)-y(x-h)}{h}.$ \n",
"\n",
"Uzimajući $x=x_i,$ dobijamo aproksimacije prvog i drugog izvod u tačkama mreže \n",
"$$y'(x_i)\\approx\\frac{y_{i+1}-y_{i-1}}{2h} \\text{ i } y''(x_i)\\approx\\frac{y_{i+1}-2y_i+y_{i-1}}{h^2}.$$ \n",
"\n",
"Uvrštajući prethodne aproksimacije dobijamo diferencnu šemu\n",
"\n",
"\n",
"$$\\left(-\\frac{\\varepsilon^2}{h^2}-\\frac{b_i}{2h} \\right)y_{i-1} +\\left(c_i+\\frac{2\\varepsilon^2}{h^2} \\right)y_i+\\left(\\frac{b_i}{2h}-\\frac{\\varepsilon^2}{h^2} \\right)y_{i+1}=f_i,\\,i=1,2,\\ldots,N-1,$$\n",
"gdje su $b_i=b(x_i),\\,c_i=c(x_i),\\,f_i=f(x_i).$ \n",
"\n",
"\n",
"\n",
"Sada uvrštavajući vrijednosti indeksa $i=1,2,\\ldots,N-1$ u diferencnu šemu dobijamo $N-1$ linearnu algebarsku jednačinu sa $N+1$ nepoznatom, te uvrštavajući $y_0=\\alpha$ i $y_N=\\beta$ na kraju dobijamo $(N-1)\\times (N-1)$ sljedeći sistem algebarskih linearnih jednačina \n",
"\n",
"\n",
"\n",
"\n",
"\n",
"\n",
"\n",
"\\begin{align}\n",
"\\left( c_1+\\frac{2\\varepsilon^2}{h^2}\\right)y_1+\\left(\\frac{b_1}{2h}-\\frac{\\varepsilon^2}{h^2}\\right)y_2\\hspace{9.5cm}&=f_1-\\left(-\\frac{\\varepsilon^2}{h^2}-\\frac{b_1}{2h} \\right)\\alpha\\nonumber \\\\\n",
"\\left(-\\frac{\\varepsilon^2}{h^2}-\\frac{b_2}{2h} \\right)y_1+\\left( c_2+\\frac{2\\varepsilon^2}{h^2}\\right)y_2+\\left(\\frac{b_2}{2h}-\\frac{\\varepsilon^2}{h^2}\\right)y_3\\hspace{5.75cm}&=f_2\\nonumber\\\\\n",
"\\left(-\\frac{\\varepsilon^2}{h^2}-\\frac{b_3}{2h} \\right)y_2+\\left( c_3+\\frac{2\\varepsilon^2}{h^2}\\right)y_3+\\left(\\frac{b_3}{2h}-\\frac{\\varepsilon^2}{h^2}\\right)y_4\\hspace{2cm}&=f_3\\nonumber\\\\\n",
"\\vdots\\hspace{3cm}&\\nonumber\\\\\n",
"%\\vdots&\\nonumber\\\\\n",
"\\left(-\\frac{\\varepsilon^2}{h^2}-\\frac{b_{N-2}}{2h} \\right)y_{N-3}+\\left( c_{N-2}+\\frac{2\\varepsilon^2}{h^2}\\right)y_{N-2}+\\left(\\frac{b_{N-2}}{2h}-\\frac{\\varepsilon^2}{h^2}\\right)y_{N-1}&=f_{N-2}\\nonumber\\\\\n",
"\\left(-\\frac{\\varepsilon^2}{h^2}-\\frac{b_{N-1}}{2h} \\right)y_{N-2}+\\left( c_{N-1}+\\frac{2\\varepsilon^2}{h^2}\\right)y_{N-1}&=f_{N-1}-\\left(\\frac{b_{N-1}}{2h}-\\frac{\\varepsilon^2}{h^2}\\right)\\beta.\\nonumber\n",
"\\end{align}\n",
"\n",
"\n",
"\n",
"\n",
"\n",
"\n",
"\n",
"\n",
"\n",
"\n",
"\n",
"\n",
"\n",
"Ovo trodijagonalni sistem $$AX=B,$$\n",
"gdje su \n",
"\\begin{gather}\n",
"A=\n",
"\\left(\\begin{array}{ccccccccc}\n",
"c_1+\\frac{2\\varepsilon^2}{h^2}& \\frac{b_1}{2h}-\\frac{\\varepsilon^2}{h^2}&0&0&\\cdots&0&0&0&0\\\\\n",
"-\\frac{\\varepsilon^2}{h^2}-\\frac{b_2}{2h}&c_2+\\frac{2\\varepsilon^2}{h^2}&\\frac{b_2}{2h}-\\frac{\\varepsilon^2}{h^2}&0&\\cdots&0&0&0&0\\\\\n",
"0&-\\frac{\\varepsilon^2}{h^2}-\\frac{b_3}{2h}&c_3+\\frac{2\\varepsilon^2}{h^2}&\\frac{b_3}{2h}-\\frac{\\varepsilon^2}{h^2}&\\cdots&0&0&0&0\\\\\n",
"&&&&\\vdots&&&&\\\\\n",
"0&0&0&0&\\cdots&-\\frac{\\varepsilon^2}{h^2}-\\frac{b_{N-3}}{2h}& c_{N-3}+\\frac{2\\varepsilon^2}{h^2}&\\frac{b_{N-3}}{2h}-\\frac{\\varepsilon^2}{h^2}&0\\\\\n",
"0&0&0&0&\\cdots&0&-\\frac{\\varepsilon^2}{h^2}-\\frac{b_{N-2}}{2h}& c_{N-2}+\\frac{2\\varepsilon^2}{h^2}&\\frac{b_{N-2}}{2h}-\\frac{\\varepsilon^2}{h^2}\\\\\n",
"0&0&0&0&\\cdots&0&0&-\\frac{\\varepsilon^2}{h^2}-\\frac{b_{N-1}}{2h}&c_{N-1}+\\frac{2\\varepsilon^2}{h^2} \n",
"\\end{array} \\right)\\nonumber\\\\\n",
"X=\\left(\\begin{array}{c}y_1\\\\y_2\\\\\\vdots\\\\y_{N-2}\\\\y_{N-1}\\end{array} \\right)\\nonumber\\\\\n",
"B=\\left(\\begin{array}{c}f_1-\\left(-\\frac{\\varepsilon^2}{h^2}-\\frac{b_1}{2h} \\right)\\alpha\\\\f_2\\\\\\vdots\\\\f_{N-2}\\\\f_{N-1}-\\left(\\frac{b_{N-1}}{2h}-\\frac{\\varepsilon^2}{h^2}\\right)\\beta\\end{array} \\right)\\nonumber\n",
"\\end{gather}\n",
"\n",
"\n",
"\n",
"\n",
"\n",
"\n",
"\n",
"\n",
"\n",
"\n",
"\n",
"\n",
"\n",
"\n",
"\n",
"\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
" Primjer. \n",
"Riješiti rubni problem $-\\varepsilon^2 y''-x y+\\sin x\\cdot y=-5x\\cos 5x+25\\varepsilon^2\\sin 5x+\\sin x\\sin 5x,\\quad$$y(0)=0,\\,y(1)=\\sin 5,\\,x\\in(0,1),$ (tačno rješenje ovog problema je $y(x)=\\sin 5x,$ i vrijednost argumenta $\\sin 5$ je u radijanima).\n",
"
"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"data": {
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cNGgA9eu7VlnIdqOVSDMRa101dNgvCSyYepKFm/Jy+rShkscGmhTZyUPtAnigfzU87kidD7c01b8kw1H/ktSUmtcIli5d2u3HlfTHqWsEb9QHjTHrrbUh122QxK1nBK21Z4Gvk75SWgMEp2gXDzz/D8dZBaxKujvenRlFJBW9/fblRWCSow07M6foc8zpBL/8Ap6eBWjcGN591zVyNCjIgazphDGuSx6L9PCkW49AEhJg7cqL/PRJDkIXPMD//leE4P8domWh5TzazpuaL1bB845bnwRHREREMja3LigvIpnQyZPw44+u7zt2TB6ueJqcfMoT1PdYTP73+jFsmGtd9Vmz4MQJ+O476NpVReC/5ekJVWt6MWzK3aw9XoRDe+MZ2DuaPy4Uo/47DbkzzwV6VQpn0SLXSF0RERGRa1EhKCK35scfYcgQSEjg7ONP8k2RQTT3nEcQEQzkHUrmiyLsl0QOHYIJE+Chh1yXCYp7BBf25plxpVl8pDTHDiXwVr/jHPQqyoMPQsHc0bx0/xK2bHE6pYiIiKQ1KgRF5N9btMhV0cXGYjt0ZN3YNTzZ25M783vSc/8r5KhVntmdZnH0+9WM3/8QNet44qGfNqkub7APPT4oS+ia3Bw5AgO7RrDkTAXuvRcqFjzJqHsnEfHRd+DA4rciIiKStuhXMxG5ebt3Q/Pm0KQJpwNLMnYsVKzkQeXq3uzcCR99BBERhi8X56fpl+3xbvFQmlsPKbPImxee/bAY63bl4o8/oEntWN7f04L8vZvRNucCltR8DTvtG0gxJbiIiIhkHioEReTGoqLg5ZehTBnWHcxHp6aR3Dntfd4c6UfTprBzJyxZctklgpKGlCkDw6fkZ1/UHcydbTkfUp0GK/6P0o/fxwe5hnH6kS6wf7/TMUVEROQ2UiEoIteXmAiffUZ88TJM/+ws1YocpfLGCURcyM20aa7F0IcPh+LFnQ4qN8PTE5o84sPsNXeyb78HbV8uyoiAN7jrx4l0f+VONv9uoXdv2L7d6agiIiKSylQIish1ndx1irefO0bRuK10OzuGcnXv4I8/YOFCaNkSvL2dTij/VYEC8PpbXuyP8OWr6T78edCHcuUNjb9/kkXhubAWGDKEvEuXwrlzTscVERERN1MhKCKXS0jgYMu+9Gt9mAIV8zA+xyCefSU7hw4ZPv7YNcxQMg5vb3j0UVi6FNatg9x1ytO0axAVylsmL7qLosNHQmAgtG8P338PsbFORxYRERE3UCEoIi7nzrFj9Wm69/Sk6JxRhP2ei0mTYM9ew0svQe7cTgeU1BYSAtOmwZ9/Qv0Ghme29qZgtuN82GY55857ui4CDQyEDh3+XjsypYQEmDcP3njDdauFDEVE0qTo6GhWr17tdIxM4+TJk2zcuNHpGFdRISiS2VnLhrcX0ibPEkpXzcGOHfDDHE827fCnfXvw8nI6oNxuhQrB++/DoUPQouUR3pxXgaIrv+K9Iac5O+4LuHABZs50Nd62DebOdRV9jRu7zhwOHeq6bdxYxaCISBqzatUqqlSpwsCBA5O3vf/++7Ro0cLBVFfbv38/BQoUYP91JjNbtWoVRYoUITaNj1QJDQ2lXLlyfPjhh05HuYoKQZFM7Pfp22iRZzmVBjUiOl9xwkLjWL7ctUSgMU6nE6flyAEdOx5g71544QV498MsFH7+Ud4O+ZboDz9zNVq5EsaPh9BQWLXKtRyFta7bNWtc20VEJM2oWrUqjz766GXbatWqxeOPP+5QomsLDAzk2WefJSgo6JqP33333fTt2xc/P7+bOt7+/fuZP3++OyPelKZNm1K/fv2bbj9u3LhUTHM5FYIimdD2FSdpd/dayrcrTaxXNtbMOsRPe0pQq4m/CkC5SrZsrkJw714YMgRGj4bCRQzvvguxHZ5wDRPduPHqSWXOnoVNm5wJLSIi1+XhcXkJEBISQuvWrR1Kc21+fn689NJL+Pr6XvPxwMBA+vfvf1PH+uuvv2jdujWRkZHujHjTrny/rycsLIwBAwakcpq/qRAUyUT27IGuNXdzT42cHDnmwdKR4SyIqEjlVvmdjibpgL8/9Ovn6kevvQbvvutaOmTip4aL5e5zVYwpZckCFSo4E1ZE5HY7cwb27bv+15kzf7c7dsz1fVzcP+9zZbtLrnXcW7Bu3TreffddACIiInj++eepX78+kydP5r777qNw4cL89ttvye2//fZbXnrpJdq3b0+jRo04lpRz37599O7dm08++YRHH32UmUmXESxfvpwuXbrw4Ycf0rZtW/Lnz8/mzZtp06YNAwYMYODAgRQpUoSaNWsSHR0NwLFjxxg3bhzbr7Ok0b59+xgxYgQnTpwgPj6eUaNGUapUKUJDQ2nQoAF58uRh8uTJAKxYsYLdu3czd+5cRo8eDcCBAwcYMGAA/fv3p1KlSsyZMwdrLVOnTqVx48bMnj2b8uXL06JFCyZNmkStWrX45JNPqFevHgEBAbRp04aoqCjAdc3loEGD+Oijj3juuedo3749ERER132/p02bxiuvvMKYMWNo2rQphw4d4uLFi8ycOZMLFy7w5ptvsnTp0lv5J7051toM81WpUiWbFi1ZssTpCJKB3Uz/ijyeaPt2j7ZeXtZWLnnGLuzzg028EJ/64STd+6f+FRVl7bBh1gYEWFuiRKKdWfb/bGLWbNaCtd7ertu+fa09f/72BZZ0RZ+PkppSq39t3br12g8MHer6uXe9r6FD/25Xu/alkP+8z5XtLrnWcf+FoUOH2tpJx961a5d9/PHHbaFChZIfHz9+vM2dO7ddu3attdba5s2b286dO1trrd28ebPt2rVrctvq1avbNm3aWGut7datm+3bt6+11tr58+fb3LlzW2utPX/+vC1fvrxt1KiRPXbsmJ0wYYJNTEy0bdu2tSEhIfbYsWP23LlzNl++fPazzz6zcXFx9osvvrDANf8dT58+bd955x0L2L1791prXf8ugJ00aZK11tr333/fFixYMHmfQoUK2c8//zz5fqNGjeypU6estdZOnDjR+vv728jISBsREWEBO2TIEPvnn3/aWbNmJW8bNGiQjY+Ptxs3brQ5c+a0Tz31lLXW2kcffdR+8cUXycfu27evrVy5sk1ISLDWWtulSxfbpUsXa6218fHx1sfHx27YsMFaa22bNm3sgAEDrLWuPkvKf+fruG4fTAKE25uonTQNhEgGFhcHY8bA//4vnsDzEcyYmoUWbXJgTHOno0kGEBDgOjP49NMwfLih47jXeKdgP0bWnkOtFrldMw116eK6jnD6dCha1OnIIiKp57nnoGvX6z+eM+ff7Z56yvV9lSqucffXc2lY5JXtUn5/6bj/UbFixahfvz7Lli1L3ubn50dAQAD3338/AOXKlWP58uUAfPzxx0RFRfH2228n738u6dKAV199FV9fX6KioggPD+fUqVMA+Pj4kDNnTqpUqUJQUBC9evVKfp577rkn+TrA4sWLc/DgQbJkyUKXLl3oep33M2fOnLRt25aXX375sswA9erVA6B8+fIcPHjwmvuHh4ezY8cOJkyYAEBUVBRVq1Zl3759hISEJB+naNGiFE3x2dWoUSO8vLyoUKECnTt3Zvbs2URERDBr1ixGjBiR3K5Pnz6MHj2atWvXUqVKlcue28vLi19//ZVy5cqxbds2jh07RtasWa+ZM7WpEBTJgKyF6ZNiGPTSRaI9cvDm/wxP1onDu4JWgBf3y5vXNctov36GV1/NSZ3POtPqDIwYAUU3bYJOnWDnThWCIpKx5cx5c0VZyja+vlC48I33ubLdzezjRh4eHiQmJgKwZ88eHn74YXr37n1Vu6CgIN555x0KFy6cXESmZG4wEUHK53FHZtfJsavt2bOHwMDAy2ZOvdKNspYqVYpp06Zx4MABAM6ePZv8WJEiRTDGcOjQoWvu6+fnx8svv0yrVq0oWrTodXOmNl0jKJLBrF6RQJWiEXTt6cVjdjp/LjnIM895413hHqejSQZXqBB8+aVrstCjR6F0aRg0+k6iZi2CJk0gMhIGD3adqhYRkXQpb968l509BFdhdfLkSR577DGyZMlCt27dbno2TyfkzZuXP/74g9OnTydvs9aybt26mz5GTEwMZcqUST5jmPJaxvj4eKy1FCtW7Kr99u/fT/Xq1enfvz/VqlW7hVdx61QIimQQERHQrclRqtUwFD30KzsGfsHbx7qS496CTkeTTOb++2H5cpg8Gb7+GkqUNEyaBAl7D2gWURGRNCAxMfGys1CJiYmXnYm78qxcQoo1YVu3bs306dMZPnw4R44cYeXKlXzwwQfccccd/PLLL8n7Xlqw/tSpUyQkJCRfl3Zljus9z6XHrneG8MrH/+lYANmyZSMyMpJz584REhJCrly5aNmyJX/88QcHDhygf//+5MqVKznjtc7SXZrIxlrLDz/8QP/+/bnjjjvo3Lkz48ePT253acKaCkkTpqV8v9esWUNMTAyJiYnExMSwefNmzpw5w8WLF8mWNOlaZGRk8gQ8qUmFoEg6l5BgGD30JCULnCV8wQmWNB3BtIM1KTT8KdesjSIOMAbatYMdO+CZZ6BvX6j6TCXWv/Gja4jTpEkwbZrTMUVEMp1NmzYRGhrK9u3b+fnnn9m5cyfz5s3j6NGjfPvtt5w4cYK5c+cSERHBTz/9xIEDB1iwYAHbt28nPDycZs2aMWLECD766CPKlCnDuHHjGDZsGOC6Nm748OE0bNiQSpUqERQUxBNPPMH8+fPZtm0bCxYsYNWqVYDrDNrq1asJDw/njz/+YNWqVWzevJmwsDD279+ffP3e9OnTOX78+GWv4eTJk3z55ZcATJ48maioqOT706ZN4/Tp00ydOhWAKVOmANC5c2dGjBjBm2++SdasWZk9ezaxsbFUrlyZli1b0rJlS4oUKcLYsWMB+Oqrr9i9e/dlzztz5kyGDRtGjx496NatW/L1iB9//DH33nsvTz75JCNGjGDx4sV88803ACxbtozly5ezbt061qxZQ7169ShfvjyVKlVi4MCBdO/enaVLlzJ58mTKly9PSEgIjRs3ZsOGDe79h78G49SY1NQQEhJiw8PDnY5xlbCwMOrUqeN0DMmAli6Fbl2iOLnf8kaRz+g9tSZeVUKcjiUZiLt+fh06BP37w3ffuSaXeTNoDDmH9Yfu3eHDDyENDyGS1KPPR0lNqdW/tm3bRunSpd1+XEnbjDEsWbLksj4VHR1NQEDAbc9yoz5ojFlvrb3hL4Q6IyiSDp2ItHSp8Sd161pKlj7Lzp8P0vfP51QESpqVPz/MnAnz58NPP0HJsc/y1aAt2Pk/wgMPwHXWiRIREXHaPw0XTc9UCIqkI9bCV1MspctA+O/eLBuykJdf3kFQ/bKusXgiaVyTJrBlC/TuDT3eLUXdwnvZnqsqhIRA0vAdERGRtCIhIYFx48YB8PXXX7N161aHE7mPCkGRdGLPqggaB2/hiW4J9Otn2BhZgOqvN3Y6lsi/5usLQ4e6CsIs2bwpv/pj3qoZSny3XtCtGyStRyUiIuI0T09PnnnmGay1fPrpp5QpU8bpSG6jQlAkjYuPjmNE08WUrRbA+ajz/P7FRoYMAZ8sOgMo6VuxYq5hohMnGt5bU5PKRU+wYU9O8NBHk4iISGrTp61IWmUtm95fTOU8fzJ8wX2M6RzOktMVKPn41Qu0iqRXxkDnzrBtGxQrn5XKK0YxaJgvseM+c60/ISIiIqlChaBIGhS/YTOv3z2Z+5+vyd2FEti23YMnJtfCw9vT6WgiqSIoyDWZzIwZ8MUXUOH1lizfmtvpWCIiIhmWCkGRtMRatmyBKq3u5MNDrZgyMpKZO8qRr0R2p5OJ3BatWsHWrVDj4VzUercZ/Z+9SGyrjvDbb05HExERyVBUCIqkERcX/8rbBcZRqZIluNwd/HEgO+0G3KXJQCXTyZXLtd58aCjMmOVBpZ/fZv39T8GECa6pc0VEROSWqRAUcdq6dezYATVersbbp3ryyfgEZs825MvndDARZzVuDJu3eFD+ofxUSVjOG32OcvGxjhAV5XQ0ERGRdE+FoIhTdu3CPtyM8Q98QYXyieS8w4stO7PQ5QkvnQUUSZI7N0ybZpjytSejsg6h+pyX2Vm2FWzY4HSNmAFTAAAgAElEQVQ0ERGRdE2FoMjt9tdf8OKLRJapTfPVA3neZwwj3/cgNBTy53c6nEja1K4dbN7qRa4a91DhyHzGVZ6M/XiC07FERDK0V199lWXLlgEwcuRI5syZ43Aicad0UQgaY3IbY7I6nUPkliQkuC58KlGCRVOOUS7bn+wPrkb4eg9690ZnAUVuIDgYQhd5MXK0Dy96juSRj5py4oTTqURE0p/w8HDWrl17w3avvvoqgwcPplatWkRFRfHII4/chnRyu7i1EDTG5DHGTDDG9Em6bX+NNsYYs8sYY6/4Wp2ijb8x5sSlx4CVwDl3ZhVJVQkJMG8evPGG6zYmBh54gPPPvsDzxWfT+PiXtOvqx5o1hnvucTqsSPphDDzd27B+kxcHKEi5ey1Lqg+BLVucjiYiki4cPnyYNm3acO7cjX+19vHxYebMmfTr14//+7//S/1wclt5ufl43wEzrLXjjDG+wG5jzB5r7ZoUbRoAo4H5QGLStl7AXyna9AD6AjuT7h+0VlPFSTqRkOCa5WLNGlcB6O8PVauy7cHn6RD3KEd3+/Djj9CkidNBRdKv0qVd/8Ve6HeRBhNfZ9BncfzfCPBy96eaiMhNuHgRDh1y7vnz57/5n39Lly5l3759TJkyhQMHDuDt7c2WLVvIly8fP/74IxMnTiR/0rUqCxYsYM2aNXh4eFC3bl1Gjx5N4cKFGTVqFDNmzOC9997jzTffZPfu3Xz11Vc0bNgQgA0bNjBt2jSCg4NZvnw51atX57nnnsNo+FOa4raPTGNMbaAm0AXAWhtnjPkJeBFonaLpVmvtomvs2yXpey9cheD7wFFr7WF3ZRS5LUJD/y4CAc6dY/Lyu+m9vA11G3ixYDEEBjobUSQj8PWFsRO8adAUunf3Z3HoaaYWeJnC37ztmmVGROQ2OXQIihRx7vn37oXChW+ubYcOHejYsSOdOnWiRo0aZM2aldWrV1OxYkWWLVvGqFGjGDlyJGFhYUydOpXJkycDsHXrVoYPH860adOoXLkyr7/+Ol5eXqxYsYJ+/frx5ptv0rBhQ44cOUKLFi3YsmUL2bNnp2fPnhQvXhwvLy+effbZ1HsT5F9z599OawNx1tq9KbbtAF5N2ejKws4Ykx/wt9buTtpUFtfZwQ+BbMaYicAz1tqLbswqknrWrUsuAs/hxzOMZcr5TrzTZAn95zbUtYAibtaiBVSqBI+38qHCL+/xaYmXaD2nM1Sr5nQ0Eckk8ud3FWNOPv9/4eXlxa+//kq5cuXYtm0bx44dI2tW17QcI0eOpHv37sltP//8cxISEgDw8/MjMTEx+Qxg+fLlmTt3LgBTpkyhcOHCZM+eHYCsWbPStWtXxo0bp0IwjXFnIXgXcOaKbaeBAGNMTmvtlY9d0gbXkFIArLWbgJrGmCxAT+C9pOMOvNbOxpheuIaWEhQURFhY2K28hlQRExOTJnOJ+3nGxlJh6lSyGcMOW4I2zCSK7CzxaUCOWk1YutTb7c+p/iWpKT31r9fehq+/COaxr8bRp/o4nn7iKyI6tAaPdDEvWqaUnvqXpD+p1b9y5MhBdHT0VdvvuMPtT3XTYmP//T7nzp0jOjqaxMRE+vfvzyOPPEKBAgW4cOEC0dHRbN++ndjY2Ktea3R0dPL1hZceu3DhAgkJCURHR7N7926io6Mv2y84OJhDhw5d831Lzy695tstLi7OLX3bnYXgBSD+im2XPn3/6RxIW+CJKzdaa88DY5PGEg/hOoWgtfYT4BOAkJAQW6dOnX8V+nYICwsjLeYSNzt5Eh58EKzl62JDeXLX89RlCZOz9iF3lRLw0kvg6en2p1X/ktSU3vpX/frQsSu0b9WLdZ//xow/x1FgxkjIm9fpaHIN6a1/SfqSWv1r27ZtBAQEuP24t5u/vz+nTp2iUaNGbN++neDgYL766isAAgICuOuuu1i+fDkdOnRI3mfXrl0UL14cf3//5HYAvr6+GGMICAigdOnSfPvtt/j6+uLt/fcfwIsVK5Yh3reUoqOjHXlNvr6+VKxY8ZaP484/kx4GclyxLRdw1lp7+lo7GGMKANmttVv/4bgzrnFckbTl8GGoVYvYWOhV5Xe67HmNod0OMHvYb+T+ZjwsWJAqRaCIXK1+fdi41RfPCvdScdmHLCzVFzZudDqWiEiakTVrViIjI1mzZg0xMTEkJiYSExPD5s2bOXPmDBcvXqRDhw5MnDiRkSNHsnv3bn744YfkJScSExMvO96lIaMAnTt3xsPDg6+//jp5208//cSLL754e16c3DR3FoILgezGmJTTYJRK2n49lw0LvQ4v4J8KRRFn7doF1auz1/8eqrGCH8P8WbrU8OJnZfB4bQg8/LCKQJHbLDgYlqz2p+sz2Wh6+muGfVOCxERAE1CLiNCtWzf69+/Pvn37KF++PJUqVWLgwIF0796dpUuXMnnyZHr27MngwYN57733qF69Olu3bqVjx46cPXuWGTNmADB58mROnDjB999/z7Fjx/j555/JkycPYWFhzJw5kzfeeIPBgwfTsGFDOnbs6PCrlisZd67KkLQW4OfW2gnGGB9gL9AVOAL0sNb2v0b7p5KuC7y0rRNw0lr7Y9L994Awa+28Gz1/SEiIDQ8Pd9vrcRcNfcnAfvsNGjViYaGetNv9BhUrGr755vaOQlP/ktSUEfrXd99Bt25QtdhxvsrxDHkWTtU6E2lERuhfknal5tDQ0qVLu/24kv44NTT0Rn3QGLPeWhtyo+O4+wr6lkANY0w/4ANgaNJSESWBR5LWFrwUsCCQJ2URmKQCMN0YM8MYMwj46WaKQBEn2Ow5GF5sEk3Xv8ETTxgWLNClSCJpTatWEB4OR8/l4L5Nk9jwuxdcMaxJREQks3Hrn0SttUeBTtfY/h1XDAG11h4Ail2j7fPA8+7MJeJ28+YRnSUPXT+uwk+bCjN1Kjz2mNOhROR6iheHVeuz0LNnFqpXt3xaciQdm8fAa69p6LaIiGRKmlNb5D/YMXU9DzxejN9+c60dryJQJO3z94evvoI33zR03vwCL7ydh4v1GsGRI05HExERue1UCIr8G7/8wpzZlsrzX6NISB7Cw6FsWadDicjNMgaefx5++snwmV9vmm56i5Pl6sLCf5rXTEREJONRIShyM6zFvjaU1xsuo2UreO45w9y5kDOn08FE5L9o2BDCN3hyrMD93H9xJb83eQleeQUuXnQ6moikA+6cbFHk33Bn31MhKHIjiYnEPj2A9v8ry4gsQ/j2W8OwYeCh/z0i6VrRorBqtQf3NbiDqj7hfPvhYdcihPHxTkcTkTTM29ub2NhYp2NIJhUbG4u3t7dbjqVfZUX+SXw8R1o9Q62JnViZpxkrVnvRsqXToUTEXbJlg5kz4ZXXvHgs9nNeC3ifRE9vnRkUkesKDAzk8OHDnDt3TmcG5bax1nLu3DkOHz5MYGDgjXe4CVpISeR6zp0jvOEgmq9+lUJlA1i30JegIKdDiYi7GQODB8O99xrat6/EjvaWL/5qg1+zBtCnj9PxRCSNyZ49OwBHjhwhXiMIMrW4uDh8fX1v3NBNvL29CQoKSu6Dt0qFoMi1REUxvfJIuu58hzaPXOCTb7JxG/+fi4gDmjWDFSugWTNDHb/PmV3xFPkA4uLQDwARSSl79uxu+2Vc0q+wsDAqVqzodIz/TENDRa6QmAhDR2Slw86h/N/A80z+Prt+BxTJJMqXh7VrgZw5qdyuKL9//ycUKQKzZzsdTURExK1UCIqkELttH+0q/8nIDzz5/gcPXn4rB8Y4nUpEbqd8+SAsDKpWheqdizKv9gho1Qqeew4uXHA6noiIiFuoEBRJEhkJ9VpkZ/UfAaxclsAjjzidSESc4ucH06ZB//6G5jM7MarnVuzUaVCjBuzd63Q8ERGRW6ZCUATY+e3vVKmcwPmsuVm9Oy/lKno6HUlEHObhAa+/Dl9+CQM/L8lTjfcQnzUnVKwIs2Y5HU9EROSWqBCUTG/5yDVUbRNMKd/9/Por3BWssaAi8reOHWHxYvjup6w8kmUBMX1ehrZtYeRIp6OJiIj8ZyoEJVOb3m8l9V+oQNuKu5j9exGyZXM6kYikRdWrw6pVsGu3ofZPgzg2c5lrmlGAs2edDSciIvIfqBCUTMlaeKflajqMfoA3H1zF+PVV8PLWmUARub5ixVzFoJcXVBlQjW0JJWD7dihcGHbuhHnz4I03XLcJCU7HFRER+UdaR1AynYvxlj7VNjI5vALfPLWUNh/VczqSiKQTefPCkiXQvr3rLOHs70tQ8+OP4emnYc0a19nBbNnggQdgwQLw1PXGIiKSNumMoGQqZ2MszUtuY1Z4QRb/b7WKQBH51/z94bvvXMVgw8YezFxX2LX44KUhojExrqIwNNTRnCIiIv9EhaBkGidPQv0Ghq1/BbPyi51UG1zH6Ugikk55esLYsa5ZRduNqMj7Mb2wKRvExMCmTU7FExERuSENDZVM4cDOOBrXiMEnX25WbsnBnXdWczqSiKRzxsBLL0GByE10fe8tDlCA9xmAB9a19kSJEk5HFBERuS6dEZQMb8sWqFbbi6DY/fz61UHuvNPpRCKSkbR/uzwL7hvMZ3SnE1O44JcDsmSBMWM0o6iIiKRZKgQlQ1s+9zQ1q12kSnUvfjp+HznKFXI6kohkNJ6e1Fk7gqUfbOLnrM1pXnonZ8O3weHD8MgjEBvrdEIREZGrqBCUDGvOpEgaNvejfY5Qpk8HXz8tDyEiqcTTk4r9arFiUzZ2nA6kwROFOPVdGOzeDR995HQ6ERGRq6gQlAzp0zeO0qpHLgYX+ppxW2prBncRuS2KFYMVK1wjQmt2LMih79ZCv36uxUsvXnQ6noiISDIVgpKhWAtv9TnEk68FMr7cBF7d2gGTI7vTsUQkE7nzTli6FHLlguqtgtix2xPeegu6dHE6moiISDIVgpJhWAsvPnaAYePz8m2dcfRa/yT4+TkdS0QyoVy5YOFCKFcOatSA8OLtoXt3p2OJiIgkUyEoGUJCAjzZeB8fz8zNj60m0fKXZ8BLq6OIiHMuLTz/4INQ94mi/Gzrw5EjMGSI64eWiIiIg1QISroXHw8dO8LMVflZ9NR31P/2adcaXiIiDvP2hs8/hyefhIcegu+nxblWon/ySUhMdDqeiIhkYjplIulabCy0qbyfdcfyE7bci/LlOzsdSUTkMh4e8O67ruGibV4uyuRXw+k48j7w8YFx41wr04uIiNxmKgQl3YqOhkeaJfLnbj+WvR9GifL1nY4kInJNxsArr0C2bNB5QDHOPr+eXuMruorBUaNUDIqIyG2nQlDSpVORCTStFcPphBws356XgoVUBIpI2tevn6sY7NWrODFPbmLAJ+UgSxZ4+20VgyIiclupEJR05+j+CzQqfwyPmCiWbU4gqFBupyOJiNy0J55wTSTTuXMxYjps5tUP7sFkyQKvv+50NBERyURUCEq6su+PszS4/y/yXIwkdKkvuUqrCBSR9Kd9e1cx2Lbt3UQ/vJURhZei84EiInI7pZmpFY0xBZzOIGnbrnVnqHlfDIXsPhaF5yZX9XucjiQi8p81bw7z5sH4n4rSe103EqNi4JtvnI4lIiKZhFsLQWNMHmPMBGNMn6Tb9v/QdqAxxl76AmqkeKyeMWa8MeZZY8xUY0xhd+aU9Gfb0uPUrnaB8lm2M39rEQLKFXE6kojILWvYEBYsgKlToeujUVwc+gbExDgdS0REMgF3Dw39DphhrR1njPEFdhtj9lhr16RsZIzJApQD7k/aFA/8nvRYWWAWUNxae8IYcwCYY4wpb621bs4r6cCW0IPUf9iPajn/YPrWcvgE5XI6koiI29SoAYsXQ+PGd9G+9mamZvHAOyICgoKcjiYiIhmY284IGmNqAzWB+QDW2jjgJ+DFazTvDHgC2YCN1trfUhR5zwO/W2tPJN3/ESgOPOSurJJ+bNoEdR4PplaRA8z4M0RFoIhkSJUqwZIlsHSZB4+1iONCsTIwaZLTsUREJANz59DQ2kCctXZvim07gEbXaFsa1xnBJbjOGla74jhbL92x1sYDe65zHMnA1k/aSL2aF2jU2INp2+/DO2dWpyOJiKSae++FsDBYud6XNnev53zPZ2DKFKdjiYhIBmXcNdrSGPMx0Nxae2eKbT2AiUAua+2Za+xTBvgYV1FY0lobYYyJA0ZZawelaLccOGGtbXGNY/QCegEEBQVV+iYNXmgfExNDtmzZnI6RrmzdGsDA/qVpmG8dvT+zeHo6nSjtUv+S1KT+dfsdOODPgAHluTdgNz/uv589Q14gsl49p2OlCvUvSU3qX5La0mofq1u37nprbciN2rnzGsELuK71S+nSGcdrzoptrd1qjHkI1/WBLXEVhdc7zvWO8QnwCUBISIitU6fOf8meqsLCwkiLudKqFaPXM3DIfbR9HCZMqIWnlyZV/yfqX5Ka1L+cUaUK1K17D61K/sH3/yvLPRUqQKtWTsdyO/UvSU3qX5La0nsfc+fQ0MNAjiu25QLOWmtPX28na200EJpi3+sd56Cbckoa9utz39G4X0k61jrIJxONikARyZRKlIClS2FLTBEeKbqF2Me6utaaEBERcRN3FoILgezGmMAU20olbb8RL/6+LnAhrslhAEiafbTwTR5H0itr+aXLlzT5sAndGx5i/NyCeKSZVS5FRG6/YsVcxeCOC0V4uMAmzvXsB7GxTscSEZEMwm2/altrNwJrcA3xxBjjg2uCl4+MMfcYY0YlbS9gjHnRGJMn6X5BoACu2UEBPgWqGWNyJ91vABzFddZQMqLERH5uOY6Hv2zD082P8uGCUhidCBQRoWhRVzG4xxbhoaJbiUnwg/grr54QERH599x9zqUlUMMY0w/4ABhqrV0ElAQeSTq7FwD0ADYZY4YCrYHHrLUJANbazUAnYIQx5hmgI/Bg0uyhktHEx/NLo3doNvsJnm13gve+v1tFoIhICoULw9KlhoMRWXiwwXmiS4a41tYRERG5BW5dUN5aexRXEXfl9u9wLTYPriGgJW9wnLnA3KS7Y92ZUdKQ8+dZUvv/aLbmVXp3jOKdKQVUBIqIXEPBgq6lJerV86Gp50J+KpiLtDdPnYiIpCe6Ckscs3SlNw+vH8aTXc7z3pQgFYEiIv8gf35YssRwzAbxUEsfzr49BtatczqWiIikUyoE5faLiGDZM9N58GFDj94+vP95LhWBIiI3ITgYliyBgwctzT5swLmGzWHjRqdjiYhIOqRCUG67FbOO0fSjZnTrdJEPPkBFoIjIv1CggOvM4J4spWjut4DYBs1g82anY4mISDqjQlBun61bWTX7OE1eLk+nnn6M+chbRaCIyH9QqJCrGNzhXZaW3vOIq/8QbN/udCwREUlHVAjK7bFmDaur9qdx2+y0bw/jxhsVgSIit6BIEVcxuMWrPK09ZnG+bhPYtcvpWCIikk64ddZQkWQJCRAamnztytq3fqZxQihtH/fh44/RYvEiIm5w992weLGhTp0Q2iZ8w8y6jfFZvti15oSIiMg/UCEo7peQAI0bw5o1EBNDOJVoZH7h0c6+fPKph4pAERE3KlHiUjH4AO38ZjM9a068nQ4lIiJpnn4lF/cLDU0uAjdSgYYsornnfCa2/FFFoIhIKihVylUMLv/rXjr0zsnF7+fC4cNOxxIRkTRMv5aL+23cCDExbKU0jVhIU0L57GJnPDdvcjqZiEiGVaYMLF4MYWGWTk/6c3H+AqcjiYhIGqZCUNyvYkV2eZehPr9Qk2VMpgue2fygQgWnk4mIZGhly8LPPxsWJtSj27LuJMaeh5MnnY4lIiJpkApBca99+9hfsCb1PZZQ0fN3ptEB72y+8MAD0LSp0+lERDK88uVh4ULDnDnQu8oGbN16KgZFROQqKgTFrY50GUT9WvEUq5qXWdMTyPLGEJg2DRYsAE9Pp+OJiGQKlSrBjz/ClN1VeOHo89iGjeDMGadjiYhIGqJZQ8VtIiOhwbEpBJYyzJlr8Mv2IPCg07FERDKl6tVhzhzDQw91ItvBGIY1aQILF0L27E5HExGRNEBnBOXWXbjA6Ycep2GVaPwDvAhd4Em2bE6HEhGR+vVh1izDW2eeZsSfj8KDD0JMjNOxREQkDVAhKLcmMZGoTn1osnAACV5ZWLAAcuRwOpSIiFzy0EMwdaph8OkXGLe9PjRrBufOOR1LREQcpkJQbsnZ/kN4+NuunAkuw6KlPtxxh9OJRETkSm3awKRJhr6n/48vfr8PPvvM6UgiIuIwXSMo/1nciNG0HFOXg4H3sWy5L/nyOZ1IRESup0sXOHfO8MQz7+GXBx6zFi5eBG9vp6OJiIgDVAjKfxL/1XTavlyErbmq8+tKP/LndzqRiIjcyNNPw9mzhsc7gf93U2jmswC++srpWCIi4gAVgvKvJSxazOOdPViTrR6/rvKnaFGnE4mIyM164QU4exZa/68j80dUpIHTgURExBEqBOVfsRaemlCRRb6+hK3wo2RJpxOJiMi/9dprEBPjQfNXyrIg+Cg11o2C4cO13quISCaiQlBumt27j5df92faT4H8vBjKlXM6kYiI/BfGwIgRrslDH+wWyBKzlkonesKnn4KH5pETEckM9NNebtrbXbbx4Ze5+OEHqFLF6TQiInIrjIExY6BFK0+aeC1i+4zfoXdv19APERHJ8FQIyo1dvMhHH8GrK5swbUoCDXRBiYhIhuDhAZMmQbVa3jTKtoIDX4ZBv34qBkVEMgEVgvLPLl5kWuVRPNMnkYkTDa06+DqdSERE3MjbG6ZPh7tLZ6Fhng0cnzgbXnxRxaCISAanQlCuz1rmPziOzhuf472BJ+jWzelAIiKSGnx9YfZsCAj0p0n+zfw1doprRhkREcmwVAjKdS3t+jmtF/ViUPcI+r8V6HQcERFJRdmzQ2goxHplp1nxbcSWKO90JBERSUUqBOWa1g/6lmZftqZns2MM+1SrxYuIZAZ588LChbA/KjdtprcmPvIMfPml07FERCQVqBCUq2wfs4gmb9emRdXjfPBDEYxxOpGIiNwuBQrAokWwdi10fewciSNHudaZEBGRDEWFoFxm/5zfaNi3NNVKnWLS0mJaTkpEJBMqUQIWLIB56++ib/X1WD9/OHzY6VgiIuJG+jVfkkVEQMPn76V4ScP0DSXw9nY6kYiIOKViRZg3DyZ97sHQAdGu6nDiRKdjiYiIm3g5HUDShjN/HKbxg37kDMrN7F+C8fVzOpGIiDitZk349lto0SKAXK1+of9T1cHHB7p0cTqaiIjconRRCBpjcgPnrbVnnc6SEZ07Bw8/moX4E1H8si4HAQGeTkcSEZE04qGHXPPFPP54FXJ1WEjX7o1cxWD79k5HExGRW+DWoaHGmDzGmAnGmD5Jt9f8lDDGlDXGhBljoo0xO40xj13xuL8x5oQxxhpjLLAS0JXqqSA+Oo62jyZwKC4PC7cX5I5AFYEiInK59u1h7FjoMa0+P3T9ATp1cp0qFBGRdMvdZwS/A2ZYa8cZY3yB3caYPdbaNZcaGGOyAq8Bg4DzwBDga2PMdmvtb0nNegB9gZ1J9w9aa62bs2Z69mICPcuuYs2x+1ixOQfBBXTJqIiIXNvTT8OpU/DY680I7f4V9dq3B29vaN7c6WgiIvIfuO03f2NMbaAmMB/AWhsH/AS8eEXThsCL1tpV1toNwONAQtK+GGO8cBWCPsBRa224tTbCXTklibW8XHkxsw6EEPrlCUqUcDqQiIikdYMHQ58+0OKbdmzsOR769oW4OKdjiYjIf+DOU0C1gThr7d4U23YAjVI2stb+YK3dn+L+OeAvYF/SprJJ9z8EDhhjPk4qDsWNRj70Cx9srM337+8j5LG7nY4jIiLpgDHw3nvwyCPQZFZPds/cCL6+rovN582DN95w3SYkOB1VRERuwLhrxKUx5mOgubX2zhTbegATgVzW2jPX2a8wEArca629mGJ7FqAn8B7wgbV24HX27wX0AggKCqr0zTffuOX1uFNMTAzZsmVzOkay1e8d55X5j/JOm+8J6f3/7d13lFXV3YfxZ88MfaQICoKiaFAsEAQUawBRmhRBBBtRk1ijscUSexeNQfMGNaJRgwViAQvdhgoqKkVQbNhQFAURdZDOfv84QzKZUOXOnLkzz2etu+7cc/c5fIe11535zd5n73ppx9EWKmv9S+WL/UvrsnJl4NJLmzNvXjXuumwcXc/uT8zJIXfFClZXrcoPu+/OzJtvhtwN33du/1JJsn+ppJXVPtahQ4epMcY2G2uXyULw/4AjYoyNixw7BbgL2DrG+N0GznsgxvjGet4/E7gsxthgYxnatGkT33zzzZ+VvyRNnDiR9u3bpx0DgNFXv0mvq1pyW79XOfNfB6cdRxlQlvqXyh/7l9anoAA6doTlC77nxXlNqbViwX/ezM+HYcOge/cNXsP+pZJk/1JJK6t9LISwSYVgJqeGzgNqFTtWB1iygSLwGOCZ9RWBhR5Zx3X1M7wyYj5HXbUHlxz0skWgJGmL5OfD6NGw7MeVHLHiXyyjyn/eXLIEZsxIL5wkaaMyWQhOAGqGELYtcqxZ4fH/EULoCvwUY3y6yLFK62iaB8zOYM4K6Z13oPvvGjCg67dc/WL7tONIksqBevVgwk0z+DDsynE8xOq1v1ZUrw4tW6YbTpK0QRkrBGOM04EpQG+AEEJlkoVi7gwh7BlCuHVt2xDCYUAr4P0QQrMQQvMQwuVA9RDCgBBCtyKXPg+4MlM5K6K5b35D5/0W06Hdau54egdCTkg7kiSpnGh8QgfG7Xslz3MIv+d2YshJ7g1s1y7taJKkDcj0xnG9gYNCCGcDtwFXxhifAXYDeoYQqoYQ9geeAK4D3i18zASaxBi/B1oC/wohPBJC+BMwLsY4KsM5K4yFC6FTv1o0rfo5D92zbGP37UuStHlyc9lr8l2Mumk2/8z7LVf3mgZbbw3HH+/qoZJUhlDdTKsAACAASURBVGV0W4YY41fAgHUcH0Gy2TzAq0CNDVzjfOD8TOaqqAoWreDwjmuoVqsqT85oTtWaaSeSJJVLubkceOGBPLI79O79S+pf+iqnD94TzjoLbr892XdCklSmuD9fObVi2Rr67vUuCxfWYfInjahZ06FASVLJ6tED7r4bfve7BtS7ajJHXd8SdtwRLroo7WiSpGIyPTVUZcCaNXDS3jOYMb8BEx77kQaNLAIlSaXjpJPghhvg+Oua8fzFE+CFF5wiKkllkCOC5UyMcF77aYx6bxcm3vkeu/Rsm3YkSVIFc+GF8PXXcMSgX/HixIPZOzfAggWwzTZpR5MkFXJEsJwZeMxb/P3lPXjysjfZ+zSLQElS6QsBbrkFevaELl0Dcx58DXbbDRYtSjuaJKmQhWA5cu8fZ3PZv/bi4RPG0/7ajmnHkSRVYDk5cO+9sPfe0PnKtsy/f1yymqgkqUywECwnRo9Yzil/2ZU7Dh1Bn/t7pR1HkiQqV4bHHoN69QJdrtiXH+YuhlNPhcWL044mSRWehWA58Prr0G9AFS45YzGnjj8y7TiSJP1bfj6MHg3LlkGfATVY8dJr0Ls3LF+edjRJqtAsBLPcnNcXcfjB39P/8AKuHlwvmYsjSVIZUq8ejBsH73xQiRN3fYU1776fLC+6Zk3a0SSpwrJqyGJffw2d+9dmn+3mcdeda9yvV5JUZu20E4wdC6NeqMFFnabD00+z8913px1LkiosC8EsVbB4FYcf/AN1t8nh0Xf2oFLdmmlHkiRpg1q2hBEj4K/D63PbcW+w/aOPwuDBaceSpArJfQSz0MoVkaP2epfFX+XzyvtQo4ZFoCQpOxx6KNx/PwwY0Iw13Ydy3tkDoFGj5L5BSVKpsRDMMjHCyftMZ+q8HXj18S/Z9hdN0o4kSdJmOfZY+PJLuPiSo9n7hNV0+OCDtCNJUoVjIZhlLu82lcdm7sbE295ilz4Hph1HkqSf5fzzYcqUeRzx+ABefhlarFyZ3Py+/fZpR5OkCsF7BLPInadMZ+C4X/Loua/S5myLQElS9goBTj/9I7p0ga5dYe7Fd0DfvsnUF0lSibMQzBJP3DCbM+9uwd19xtF10GFpx5EkaYvl5MDQobDrrtBl9FksumM4LoEtSaXDQjALTJ4UOebyXbh63zGc9NjhaceRJCljqlSBkSMhr3IOPc7aiaWTp8GJJ8KqVWlHk6RyzUKwjHv3nTX06Bk48US4dHI3/1IqSSp3atdO9hj8/HM49qpdWf30GDjjDKeJSlIJshAsw76c8xNdWn3Nr3ZfwOAhVQh5uWlHkiSpRDRqBOPGwYtT8zmr3Uzi0Afg+uvTjiVJ5ZarhpZR338PXY+sRqPtf+DhR/LItQaUJJVze+wBTz0Fhx3WgEZHTuXSK5snq4ieeGLa0SSp3LEQLINWLF1Nn32/ZDnb8/zrDaheN+1EkiSVjoMOgocfhr5996DhMRM46eQu0LAhdOqUdjRJKlecGlrGrFkdObHFNGZ/WIlxd31GXYtASVIF07s3/O1vcPLwjow94i448kiYMSPtWJJUrjgiWMZc1P41Rs3Zk5fu/Yid2u+ddhxJklJxxhkwbx70ve0kJnb5hn1+/DHtSJJUrjgiWIbcdsxr/HVSa0ZeNZOWJ1kESpIqtuuug6OOChz+8sXM2e7gZFnR775LO5YklQsWgmXEIxdP4/zh+3DfSS/T8cqD0o4jSVLqQoC774bWraFzZ/j61xck1aEkaYtZCJYBE++YzYCb9mRgx2c47t6OaceRJKnMqFQJHn0Utt4aDl/0AAV/cksJScoEC8GUzZoFR5zbhNP3eIk/TuicdhxJksqc/HwYPRoWL6lE3+OrsvL2IXDhhWnHkqSsZiGYormzC+jSJdKpV1UGTe9AyAlpR5IkqUzadttkw/lp0+Dkp7oTb70N/vrXtGNJUtayEEzJd4siXdt8Q9NqXzB0aCCnsgu4SpK0Ib/4RTIy+OikhlzWdSqcdx48/njasSQpK1kIpmDZMuh1RCBs14AnxlenatW0E0mSlB322QceewxuGtOcO7qPgeOOg0mT0o4lSVnHQrCUrV65huNbzuKTOasY92J1au/ijvGSJG2Orl2T1UTPGtWJke1vg5494b330o4lSVnF+YilKEY4Z99XePb9vZj06By2375Z2pEkScpKJ50E8+YFjr3+VJ5t/QkHdukCr70GDRqkHU2SsoKFYCm6uftLDJnRlgmD3mGvvq3SjiNJUla79FL44otAj0cGMvmwauye40QnSdpUGS0EQwj1gOuBmUALYGKMcdg62lUDBgKfADsCc2OMtxZ5/xCgL/AusD9wSYzx00xmLW0PnP4Kl4w5kOFnv0a7cw9MO44kSVkvBLj9dpg/P9Dltat4dRU0fPVVaNMm2YBQkrRemR4RHAE8EmO8PYRQFZgTQvg4xjilWLshwNcxxttCCAGYGkL4PMb4WAhhL+BxoGmMcWEIYS7wVAjhlzHGmOG8JWv1ahg7lvlXTeY3U6/h1l4vctRth6SdSpKkciM3Fx5+GA49FLp2XsNLC06k1t9vgiOOSDuaJJVpGZtDEUJoBxwMjAaIMS4DxgEXFGvXBDiuSLtIUkBeXNjkfGBmjHFh4esxQFPg8ExlLRWrV0PnzkzrewMnT72Ec3P+jz8U3JAclyRJGVO9Ojz9NCxfmUOfXWexotsRyY35kqT1yuRk+nbAshjjJ0WOvQ90KtbuYCAAs4u1ax1C2LrwOv9+L8a4Evh4Hdcp28aOZeVrU+m7/EF68SQD11wAU6bA2LFpJ5MkqdypWzfZcH72h5U58cTImiOPgnvvTTuWJJVZmZwa2hBYXOzYd8BWIYTaMcbFRdpRrO13hc+NN3Cdxuv6R0MIpwCnANSvX5+JEyf+rPCZtuOIEez00/eMoA97MJscInHJEj4dOZLP8vPTjqdypKCgoMz0e5U/9i+VpJLoX9dem8/ZZ7fk5N1/zz0nH8qshQtZtO++Gf03lB38/FJJy/Y+lslCcAWwstixtSOOoVg7irUt2m591wmsQ4xxCMk9h7Rp0ya2b99+s0KXmIICePRRWha89e9DoUYNmvTuTZOyklHlwsSJEykz/V7ljv1LJakk+lf79rDTTtCtWweaH/IU51zTH156CVq5WndF4+eXSlq297FMTg2dB9QqdqwOsCTG+F2xdhRrW6fw+fMNXOfzDOUsHV27Qtu2kJ9PDAHy85PXXbumnUySpHLt0EPhvvvg/Oe78UirgdCtG3zyycZPlKQKJJOF4ASgZghh2yLHmhUeL+p5YDXJAjBF200vXCBmQtH3Clcf3Wkd1ynbcnNh/HgYNoxPTzoJhg1LXufmpp1MkqRy77jjYODAwIApv2dio+OSP8R++23asSSpzMhYIRhjnA5MAXoDhBAqkyzwcmcIYc8Qwq2F7RYAj61tV6gHcEfh1/cABxQuHANwKPAVkH2rrOTmQvfufDZgAHTvbhEoSVIp+uMf4bTTAkfMuYVZa/aEnj1h2bK0Y0lSmZDJEUFIiruDQghnA7cBV8YYnwF2A3oWju4BnAzUCyFcEEK4hWT7iH8AxBhnAQOAm0MIZ5JsNdGtcPVQSZKkTRICDBoEh3UKdP3xET7f7yioXDntWJJUJmR0Q/kY41ckRVzx4yNIir21r38kKQbXd52ngacLXw7OZEZJklRx5ObCAw9A5865dBl3DpMuWUOdSU8mo4NhnevQSVKFkOkRQUmSpDKlalV44omk7uvVaSnLfvt7mDs37ViSlCoLQUmSVO7VqQNjx8LHX9dgwMGfsnr7HWHVqrRjSVJqLAQlSVKFsMMOMG4cPPNCHueevpS4x57JHoOSVAFZCEqSpApjr73gySfhrn9W5c/b3Ay9esHs2WnHkqRSZyEoSZIqlHbt4IEHAn96rScP7nJlssfgl1+mHUuSSpWFoCRJqnD69YNBgwInvXU2z9Q4Arp1gx9+SDuWJJUaC0FJklQhnX02nHNOoM/cW5n+3U5w5JGwYkXasSSpVFgISpKkCuumm6Bnrxy6LX2MT6YvhtNOSzuSJJUKC0FJklRh5eTAfffBni3y6FLjZRZ27J92JEkqFRaCkiSpQqtcGUaMgGp1qtLz9s789NkCGDky7ViSVKIsBCVJUoVXsyaMGZMsHnpMn+WsuuFmN5yXVK5ZCEqSJAENGyYbzk/6dHt+/8tJxNw8+OmntGNJUomwEJQkSSrUrBk89RQMfSiX60/5DHbdFT76KO1YkpRxFoKSJElFHHggDBsGV97bmPtqnZNsOL9wYdqxJCmjLAQlSZKKOeIIuP32wMnvn8+Y1Z2hRw+niUoqVywEJUmS1uG00+DiiwNHffVX3vikHhx3HKxenXYsScoIC0FJkqT1uPZa6Nc/h8NXjmTOc5/B2WdDjGnHkqQtZiEoSZK0HiHAkCHQet88utSczDdDnoDBg9OOJUlbzEJQkiRpAypVgkcfhdoNqnH4Tm9TcOgRaUeSpC1mIShJkrQR+fkwejQsWl2bfufvwMqJk2HixLRjSdLPZiEoSZK0CerXTzacf+MNOPWsysTRY9KOJEk/W17aASRJkrJF06bJyGCHDm1oVKUN165aBY89Bh9+CHvvnew5mJubdkxJ2igLQUmSpM2w777wyCOBXr0ije66mtMW3QBr1iTzR9u2hfHjLQYllXlODZUkSdpMhx8Od50xk98vvIon13RPDhYUwJQpMHZsuuEkaRNYCEqSJP0Mv93mKa7kGo5mOK+yX3KwoABmzEg3mCRtAqeGSpIk/Rx7783lNY5h3pKGdGcUkzmQZrlzYK+90k4mSRvliKAkSdLP0bUrYb+23F7jIg5iEp3CM3wRdoB//QtWr047nSRtkIWgJEnSz5GbC+PHkzf8QYZf8S477l6dTjvM5tsxU+Dvf087nSRtkFNDJUmSfq7cXOjenWrdu/P0ufCrX0H3mu/w7HF51Eg7myRtgCOCkiRJGVC7drJzxPzvq9H3mEqsvGkQXHJJ2rEkaZ2yohAMIeSFELZLO4ckSdKGbLcdTJgA06bBSROOYU2rNmlHkqR1ylghGELICSFcG0K4OIRwXQjhxhDCOq8fQqgTQngghPBtCOGrEMINxduGEMaFEGIIIQLLcBqrJEnKAk2bJlsJPvXGdpw3qQ9x4bfwwANpx5Kk/5LJ4upqYPcYY1+AEMJI4DzglnW0vQF4FBgI9AWuAuYBtxee2wJ4G7issP3iGOPnGcwqSZJUYlq1giefhC5doP533/CnB0+EH3+EM85IO5okARkaEQwh1ADOBUYXOfw4cF4IoVKxtvWBF2KMT8UY34kxXg1MAjoUaXYR8COwPMb4ZoxxTiZySpIklZYOHWDYMLjswd2559cvwVlnwf33px1LkoDMTQ1tDdQAZhc59j6wHdC8aMMY49cxxkeKnf8V8ClACKEKUBc4BZgZQpjo/YGSJCkb9ekDd94Jpz1wIE/8bhT89rfJPoOSlLIQY9zyi4RwNDAMaBZjfL/w2C+AD4HeMcYnNnBuAN4BehYf+QshdAGGAPNijPuv5/xTSIpG6tev33r48OFb/P1kWkFBAfn5+WnHUDll/1JJsn+pJFWk/vXgg40ZOnQn7u96G0ePvpB3rr6abw88MO1Y5VpF6l9KR1ntYx06dJgaY9zoSlWZukdwReHzyiLH1o42ho2cexxwx7qmf8YYx4UQegDTQgjNYozvraPNEJJikTZt2sT27dtvbvYSN3HiRMpiLpUP9i+VJPuXSlJF6l/t2kGNGnD6/eexx++q0/Kas+Hpp6FTp7SjlVsVqX8pHdnexzZaCBau2rkhA0hG/gBqFTlep/B5vYu8hBD2AnaIMd64vjYxxrdCCO8Xu7YkSVLWCAFuvRUWLIAuT5zK5BO+Z5f33rMQlJSajd4jGGMMG3k8CEwFvgWaFjm1GbAAmLGu64YQdga6FS0CQwiV1xeD/xSbkiRJWScnJ1krpmXLQKfnLmJ+vz/A8uUwe/ZGz5WkTMvIYjExxlXAPUDvIod7AENijKtCCDVCCHeGEBoAhBB2IFkZ9KkQQrPCxwCgbQjhoBDCKWtXGw0h9AJGxxgXZSKrJElSWipXhscfh222SbaW+P7We6F/f1izJu1okiqYjG0oD1wOfBlCuCKEcBUwl2RvQYCtgZ7ADiGEesBzJAu8vFvkcRXJNhLbA9cCk0MIFwH1SIpGSZKkrFejBoweDStWQM8xp7H0ifHJcKHFoKRSlLEN5WOMK4Hz1/Pe50CjIod23cClhhc+JEmSyqW6dWH8eDjwwEC/cxoy4rrZVDquX7IL/S67pB1PUgWQyRFBSZIkbaIddoBnn4UpU+CEG3ZjdaPG0LEjzJ2bdjRJFYCFoCRJUkp23RUmTIAx43M5s/GTxB13SorBr75KO5qkcs5CUJIkKUUtWyb3DP7zoUpc0mY81KsHhx4KCxemHU1SOWYhKEmSlLIDD4SRI+Evf6vCwE7PJ8uLduoEixenHU1SOWUhKEmSVAZ07gwPPwyXXleNvx/7UrLHYL9+EGPa0SSVQxlbNVSSJElbpm9fuPtuOPnkraj1t0kc0+ZDCCHtWJLKIUcEJUmSypDf/Ab+8hf49dl1GPXNvvDxx3DKKckIoSRliCOCkiRJZcw55yS3Bx51FIy9pwrtly6F1avTjiWpHLEQlCRJKoOuvDIpBnuc1ojnn3+AfaqTbDzYoQPk5qYdT1KWc2qoJElSGRQCDBqU3DfYpQu8M3lxsnjMqafCmjVpx5OU5RwRlCRJKqNycpLFY77/Hjr1q82kfzxPk18fDNWrw1//6kIykn42RwQlSZLKsLw8GDYM9twTDv1jS768fwLccw9ccolbS0j62SwEJUmSyrgqVZIN5xs0gI6X7c83949J5o1ef33a0SRlKQtBSZKkLFCjBowZkzwfdn17Fv1jJFx9Ndx6a9rRJGUhC0FJkqQsUasWjB+fzAjt/NdufH/XcBg6FJYtSzuapCxjIShJkpRF6tZNdpEoKIBu/ziSghfegKpVYf78tKNJyiIWgpIkSVlm223huefg66+hZ588lr73Gey8M8yalXY0SVnCQlCSJCkLNWyYFIMffQS9z9mR5WOfh+bN044lKUtYCEqSJGWpHXeE559PBgL737ofK5ethhNOSOaOStIGWAhKkiRlsV12SUYGX3kFjj8hh9XbNIBeveDll9OOJqkMsxCUJEnKcs2aJYOAzz4b+M2Cgaw58Tdw+OHw+utpR5NURlkISpIklQMtWsCECfDEE4EzVv0fsc+R0KULzJyZdjRJZZCFoCRJUjnRujWMHQsPPhQ4t9Y/iId1gkMPhffeSzuapDLGQlCSJKkcOeAAGDUKhtydwx8bPkTc/wDo2BHmzUs7mqQyJC/tAJIkScqs9u2TYrB791zCqY/x5/aDCQ0apB1LUhniiKAkSVI5dMgh8PTTcPvf87jwy3OIIQcGDYL589OOJqkMcERQkiSpnOrYMSkGe/SAnDWrGDhtFKFVK3B0UKrwLAQlSZLKsUMPhaeegh49KhHOfo4b2wXC/PlQtSrUrp12PEkpsRCUJEkq5w47LCkGe/YMhBy4YfZphAXfJPtN5OenHU9SCrxHUJIkqQLo1AmefBJuvRUu2/EB4tffQM+esHRp2tEkpcBCUJIkqYLo3BmeeAJuuWsrLu82lfjhHDjySFi+PO1okkpZmSkEQwiN084gSZJU3nXpkhSDfx5Siyt6ziBOmw7HHgurVqUdTVIpylghGELICSFcG0K4OIRwXQjhxhDCeq8fQmgbQohrH8ClRd5rEkK4J4RwegjhnyGEQzOVU5IkqaLr2hVGjoSb79maK3vPJL74EpxwAqxenXY0SaUkk4vFXA3sHmPsCxBCGAmcB9yynvanAPsDa//8NLvwvKrABODcGOOoEMLjwLshhFYxxs8ymFeSJKnC6tYNRoyAPn22YeVxs7jhiT0JkyZBu3ZpR5NUCjIyIhhCqAGcC4wucvhx4LwQQqV1tN8NaAlsB7wXY3wzxvhT4dv9gR2AZwFijN8A04GzMpFVkiRJicMPT1YTvW1YA87rP4/4q3bJ/YIxph1NUgnL1NTQ1kANCkf1Cr1PUug1X0f7tsAa4F/AVyGEU4q81w74JMa4rNi1OmUoqyRJkgp17gyjR8OQoVX5/RmRNT2PgL/8Je1YkkpYiBn4i08I4WhgGNAsxvh+4bFfAB8CvWOMT6znvHrAtcBpQLcY49gQwjigVoxx/yLtrgPOjDH+z66nhUXkKQD169dvPXz48C3+fjKtoKCAfPfoUQmxf6kk2b9UkuxfZcusWbW4+OLmdNrrXc4/7z1W1d8m7UhbxP6lklZW+1iHDh2mxhjbbKxdpu4RXFH4vLLIsbWjjWF9J8UYFwKnhxC2AU4AxhZea2Wxpjnru06McQgwBKBNmzaxffv2m5u9xE2cOJGymEvlg/1LJcn+pZJk/ypb2reHffeFzp2bs9Xo5tx7wovkzZ4JZ2Xn3Tn2L5W0bO9jGy0EC1f03JABJCN/ALWKHK9T+Pz5JuR4BDip8Ot5wI7F3q+zideRJEnSz9S2LTz3XLL5/HEfNuHBKZ2oVKkSnHZa2tEkZdhG7xGMMYaNPB4EpgLfAk2LnNoMWADM2IQcefzn/sIJwM7Ftp5oVnhckiRJJah1a3jhBXjho8b0a/kBy884F4YOTTuWpAzLyGIxMcZVwD1A7yKHewBDYoyrQgg1Qgh3hhAahBByQwgXhhCaA4QQqgDHArcVnjeWpKg8rPD9ekCbwutLkiSphLVoARMnwmtf7UifPd5l2Umnw6OPph1LUgZlbEN54HLgyxDCFSGEq4C5JHsLAmwN9CTZFiIP6Aq8EkIYBJxPsmfg5wCFq4V2AX4dQjgT+DNwXIyx6IqkkiRJKkF77AEvvQQzv9+Jnru8zZJjfgdPP512LEkZkrEN5WOMK0mKunW99znQqMihDhu51nvAcZnKJkmSpM3XtGlSDHbs2IRO281i9JFtqD3mYTj00LSjSdpCmRwRlCRJUjnTpAlMmgSLa+5A+7oz+Zr6aUeSlAEWgpIkSdqghg3hpZcClbZvwMFnNGfu83PgzTfTjiVpC1gISpIkaaPq1k22lmjYEA46oi7vX/8YjBoF116bPK9enXZESZshY/cISpIkqXyrWRPGjoV+fWtx8OiLGD++O3svfQXy85NNCMePh9zctGNK2gSOCEqSJGmTVasGI04ey2FxAh2WjmYyB0BBAUyZklSJkrKChaAkSZI2S6VZ03hg1bEcwzAO4xnG0TkpBqdPTzuapE1kIShJkqTNs/fe5ORX5w7O4Gz+Sk+e4l/0S3ah/+mntNNJ2gQWgpIkSdo8XbtC27aE/HxuDJdyfeVrOJaH+b9pB8EBB8BHH6WdUNJGuFiMJEmSNk9ubrIwzNixMGMGF7RsSYNvA7/53VV89f3u3NCqNeH556B167STSloPC0FJkiRtvtxc6N49eQADgG3qQ9++/flqz19yd9OdqBQjrFnjSqJSGeTUUEmSJGVEly7wwguB0R/vzhHHVGPJ4PugVy+IMe1okoqxEJQkSVLG7LMPvPIKvPsudLz3WBYeeSqE4IbzUhljIShJkqSMato0KQaXU5WDburBpx+sSBaRuf12RwelMsJCUJIkSRnXoAG8+CI0agQHtK/MW10ugvPPh1//GpYsSTueVOFZCEqSJKlE1KwJY8ZAu3Zw8K19mHDLTHj5ZdhvP/jgg7TjSRWahaAkSZJKTJUq8NBDcMYZ0O2cXbn77Ldh++2TmwlHjkw7nlRhWQhKkiSpROXkwMCBcMcdcPoF+fzpl2NYc8550LcvXH552vGkCsl9BCVJklQqTjkFGjeGfv0Cn3S7kvtH7k/V775KO5ZUITkiKEmSpFLTpQtMmgSTJ0PHmzqx8PAT4KefoFs3mDEj7XhShWEhKEmSpFLVogW89lpS/+2/P3z4cS788pfJEqOSSoWFoCRJkkpdo0bw0kuw666wX7sqvNT1RthmGxg9Go49FgoK0o4olWsWgpIkSUrFVlvBk08mdV/HjnD33SQV4muvQdu28N57aUeUyi0LQUmSJKUmLw/+9jcYPDjZYuLs+1qyaspUaNIk2WLiscfSjiiVSxaCkiRJSt2pp8Izz8CDD0K34+rw3T+fgosvhv794fzzYeXKtCNK5YqFoCRJksqE9u3h9ddh3jxou38O7/e9FMaNg3/+M5k7On9+2hGlcsNCUJIkSWXGLrvAq6/CbrsltwlOiIfBtGnJQjK5uWnHk8oNC0FJkiSVKTVrwhNPJNNFu3WD20Y0Jj72eFIMjhkDgwalHVHKenlpB5AkSZKKy82Fm26CvfaCk0+GqVPhrrug+o8/wo8/ph1PynqOCEqSJKnMGjAAJk1K9hw84AD4eJ/+cOWVsGYNnHkmvPNO2hGlrGQhKEmSpDKtTZtkRLBePWjdGsaOBZYtgy++SG4kHD487YhS1rEQlCRJUplXr16ygOipp0L37nDtX6qz5rERcPnlcNxxcPbZsGJF2jGlrJEVhWAIIS+EsF3aOSRJkpSevDwYOBAeeQRuvhmO6JPD4lMvSjYgHDYMOnRI9p6QtFEZKwRDCDkhhGtDCBeHEK4LIdwYQljn9UMI/wghxGKPFSGEmkXajFv7HrAMF7aRJEkScOSRyX6DH34I++wDs7Y5BKZPhxihVSt44YW0I0plXiZHBK8Gdo8xDowxXgY0A84r3iiEUIuksGsGNCl89AKeizH+UNimBfA2sE/ho1mM8fMMZpUkSVIW2313mDIFWrRIbhO8d3wj4gsToX9/mDAh7XhSmZeRQjCEUAM4Fxhd5PDjwHkhhErFmlcCzosxvh9j/DTG+CmwX2H7tS4CfgSWxxjfjDHOyUROSZIklR81a8JjjyXTRU87DU44uTIFN/wf3HBDsqrojTfC99+nHVMqkzI1ItgaqAHMLnLsfWA7oHnRhjHGnyNKrgAAFhRJREFUhTHG5cXO7w08ARBCqALUBU4BZoYQJnp/oCRJktYlBPjDH2DyZHj55cKpom8HKn/3HYwcCd98k3ZEqUwKMcYtv0gIRwPDSKZwvl947BfAh0DvGOMTGzh3b+CWGGPHdbzXBRgCzIsx7r+e808hKRqpX79+6+FlcPnggoIC8vPz046hcsr+pZJk/1JJsn8p0woK8rj55t2YMmVrTj31bXofsYiQE8j/8ENqfPopXx92WNoRVY6U1c+wDh06TI0xttlYu0wtwLJ2rd6VRY6tHW0MGzm3H/89LfTfYozjQgg9gGkhhGYxxvfW0WYISbFImzZtYvv27Tcnd6mYOHEiZTGXygf7l0qS/Uslyf6lknD44TB4MJx3XnMWLcrhzjthq48+gptuYvfFi2HQIKhSJe2YKgey/TNso1ND17G6Z/HH8cDadXprFTm1TuHzxhZ5ORIYub43Y4xvkUwzrbW+NpIkSRIkU0XPOgsGD57OK68ki4i+3vy38Nxz8Pjj0K4dfO4ahNJGC8EYY9jI40FgKvAt0LTIqc2ABcCM9V07hNAK+DrG+NXGYpBMM5UkSZI2arfdfmT6dNh/fzjgALh+UjtWvzEt2YywVSt49tm0I0qpyshiMTHGVcA9JIu+rNUDGBJjXBVCqBFCuDOE0KDYqf8zLTSEcFAI4ZS1q42GEHoBo2OMizKRVZIkSRVDrVowdCg8+CD8+c/Q/tiGfHrfC3D88dC5839WF5UqoEzuI3g58GUI4YoQwlXAXJK9BQG2BnoCOxQ7py8wotix7YFrgckhhIuAeiTbSUiSJEmb7eijYeZMyMmBX7apxENtboVhw+D22+GLL9KOJ6UiU4vFEGNcCZy/nvc+Bxqt4/gv1nFsOFD2lv6UJElS1mrcGJ5/PhkZPPFEGNOvH7dP7UntBlXhrbcgRmjZMu2YUqnJ5IigJEmSVGbl5sLFF8Orr8Kbb8JebaoydizJ/NHbbks7nlSqLAQlSZJUobRpAzNmQP/+yXYTv1l0C9/ffFfy5sMPw/Ll6QaUSoGFoCRJkiqcatXgL3+Bl16ClycF9mpdhfH/Wgx//CMcdBB89lnaEaUSZSEoSZKkCuugg5JbBPv2ha7H1OZ3HebwfZVtky0mxo9PO55UYiwEJUmSVKFVrw633govvggTp1Sn+dxRjGr3Z+jWDa65xi0mVC5ZCEqSJEnAwQevHR0M9HryNxzV9jO+uvkB6N4dFrmltcoXC0FJkiSpUI0aMGgQvP46fLx8e5qF97hzxv6sufratKNJGWUhKEmSJBXTujVMmQJXX5vLBT9cxoGv3sKsWcDkyWlHkzLCQlCSJElah7w8OOccmD07sE2DXFq1ilzc/lWWzPgQVq+GUaPg2muT59Wr044rbZa8tANIkiRJZVnjxvDkkzByZOAPZ57Lg91zuKXGVfT//BbCsqXJfNK2bZNVRnNz044rbRJHBCVJkqSNCAH69IH3P8zlhAPmcMIHl9B+6Rjeis2hoCCZRzp2bNoxpU1mIShJkiRtoho14Prmw5nNHtTie1oxjd8zmEUFlWD4cFi5Mu2I0iaxEJQkSZI2x957s0v+NzxFL0bRnWc5lKZ8yF+HbcuKRk3gggsgxrRTShtkIShJkiRtjq5dk3sC8/PpGsYzq8b+XNr0Ea6ueQu7x3d45O09iAT45BMYPBhWrUo7sfQ/LAQlSZKkzZGbmywMM2wYXHMNlYcP5bx3T+Gjj3Poc2ItBjx/EvvtBy+PWJCsMpObC999l6wualGoMsJCUJIkSdpcubnQvTtcdlnynJtLnTrw5z/D++9D06bQ7oJ96VX9GWa9HZId6o88ErbfHv74R3j77bS/A1VwFoKSJElSBu20Ezz4ILz5JixdCr/8JRx9X2fenfg1XHEFvPQSNG8O++wDt98OixalHVkVkIWgJEmSVAJatYIJE2DiRJg/H/Y8sDbHv3IGHzz4ejIi2KEDXHcdbLcdPPdc2nFVwVgISpIkSSXoV79KisHnnoNPP4Xdd4cT/7wnH/zuZvj8cxg5MhkdjDFpbFGoUmAhKEmSJJWCDh3g5ZeTfefffx+aNYOjjsnjzW27Qc2ayUIyRx8Nu+2WnHD66U4dVYmxEJQkSZJKSQjQqRO88gq88AIUFCSDgYceCs++WIl4+hnJgjJr1sBWW/1n6uhRR8Ho0a46qoyxEJQkSZJKWQjQrl0yOjh9Omy7LXTunBSFw4bBilU5cHPh1NERI5KT+vSBHXZINqx/5510vwFlPQtBSZIkKUUtW8LDD8OHHyb71J98Muy4I1x9NcxfmAeHHw6PPgpffplsVzFxIlxzTXLyV185dVQ/i4WgJEmSVAbsvHNyS+AXX8BFF8EDD0DjxnD88TBlClC3Lvz+9/DGG/DPfyYnDRqUTBsF+Oknp45qk1kISpIkSWVI7dpwzjnwwQfJgqILF8J++yXbUQweXDgAWLVq0vjGG2Ho0OTrQYOSqaMXXgizZ6eWX9nBQlCSJEkqg3Jyklmh48Ylq4x26gTXXw8NG8Ixx8Azz8CanDxo1Cg54fTT4dJL4fnnYc89Yd994Y47nDqqdbIQlCRJksq4XXeFgQNh7lx45BFYuhS6doUmTZLbBt9+m2Tq6JlnwptvwsyZyZ6EV1+drDrarx98/XXa30b5sHo1jBrFjkOHwqhRyessZCEoSZIkZYlKlaBnT3jiieRewjPPTHaVaN48GQS85ppkSinNm8MttySNHn8cqlWDOnVg5Uq4/HJYsCDtbyU7rV6dLO96zDHsdP/9ydBs585ZWQxaCEqSJElZqEGDZCeJ6dPhvfegf38YPjzZj37vveGGG2DWe5WIh3dPFpepXDlZefS55yAvD2JM7i906uj6rS3wli+H8eNZ89gIXnkFLiy4nM9i42QjyClTkn1AsoyFoCRJkpTldtsNrrgi2V5w5kzo1g0eeghatEimj555ZnKv4bL6Oya72depk2w9ccEFydTR/v2TYqYirDq6ahXMn598vXw53HdfMu/23HPh2GOhY0fYay/YZhto0IBFi2D4Ayv5ddcFNDipCwctncBkDmQh9ZJrLFkCM2ak9/38THlpB5AkSZKUGSEks0KbN08Wlvn442Tq6NNPJ1NKK1eGQw5Jap1DDmnInnO/IGfCOLj/fujVC+rVgwED4IQTYI890v52Ns+SJfDuu8m9kEUf8+cnz7vtBn//Ozz7bPK9LluW/IddeSXUr//vx4o2B/DGsua8MH93xs1uzKvbQH5+DTodeRw3NXqLrncfSYOfPv7Pv1ujRrIZZJYpkUIwhLBdjPGrkri2JEmSpE2z885w1lnJ48cfYcKE5DF4cLJFxTbbVKJDhx50OKwHB/zhO/ac8RC5Q++Dm29O5pb+6U/JFNIQSj/8smX/W9StffzqV9C3b/KNjB2bVLszZsBBByVbaxQp7KhfH37xi6Q6BjjwwGTYFKByZZZ/OJdp02DiRHjhBZg8PFmMp0UL6NQVrh8MBxwQqFQJWN0c3m4CU74hLllCqFED2rZNVu7JMhktBEMIuwFXAC2BPTfSth+wHzAPaAGcG2NcVPhePeB6YGbhexNjjMMymVWSJEmqSLbaCo48MnlAsgLpCy8ku03ceCPMnVuH/Pwz2WefM9nvt9+wX901tPoCGl16EqFFczj//OTE1auT4mv69ORmxK5dITd300L89NP/FnWHHJIUagMHQs2acMYZ8Le/wR/+8J/zqlVLbopcW9gtX54cb98emjZNvt53X/j+++QbXU/humYNfPjlVrz++m5M+Ru8/npSP65cmRR+7dsn//yvfgVbb72OC+TmwvjxMHYsn44cSZPevTfv+y9DMlYIhhACsApYDGzwfyKEcBgwCNg5xrgihHA2MBToXthkBPBIjPH2EEJVYE4I4eMY45RM5ZUkSZIqssaNkxmgJ5yQvP7yy2Tdk9deg0mvbsttD8PSU2HrrYbQ4u1ltJgLLeaOYs/X/kHT799k66XzCPk1oE0bGDIk2fl+772TEblBg6BdO2jdOpl6+fDDSdH344//CVCjRlLUNWqUFIKNGiXHAI44Alq1+k/hl5+/7uJur72SByRLqlaqBCSDmPPnw+zZMGtWsr3GrFnJPZRLliS3/+27L/ToAddeC/vss57Cb11yc6F7dz7Lz6dJ+/Y/6/++LMhYIRhjjMBHIYRNWYv2EuC5GOOKwtePA7eFEJoDWwMHAycUXndZCGEccAHQN1N5JUmSJP1Hw4bQu3fygGRNlQ8+gJkzKxc+YOSLB/J5QTJ2U4vF/KJgDr+YOIdddn2E7fmChn/bnoZtd6Dh5E+p33R38loD+++fXLz4dM21Rd9aAwb85+sddkge67F0abLWzfz5yfOXXyb3Q378MXz0UfK8dGlSs+22W1Ir9ugBl1ySjPzttFM6s13LklJfLCaEUAk4ABi39liM8YsQwhLgMCAfWBZj/KTIae8Dl5dqUEmSJKkCy8tL1ovZYw84+ujCg9cO5scr/sxH7MwcfvHvxyvb9+fLajsz7+IcliwB+D/CyGRx0tq1u1C7Nv9+5Ocni9YUfeTlJTNOV61KpmmufV6yBH74IZnx+cMPyePbb5PntWrXThY+bdIEdtkFOnRInnfeORlorFIlhf+8LJDGqqF1gcokU0iLWgw0Bqqu473vgK1CCLVjjP/1XgjhFOAUgPr16zNx4sSSyLxFCgoKymQulQ/2L5Uk+5dKkv1LJcn+VTLq5uWxe7VVtFz6Fi15C4BVVavy7hlX8O3++wPw00+5LFxYmW+/rcIPP+RRUJDHkiV5/PhjJQoK8pg/P5dVqwKrVwdWrsxh1arAqlU55ObGfz/y8taQmxupWnUN1auvol69VVSvvprq1Vex1VarqFt3BXXrrqBOnRVUqbJmnVkXLEgeJSXb+1gaheDa6aArix3PAULh++t6j8L3/0uMcQgwBKBNmzaxfRmcpztx4kTKYi6VD/YvlST7l0qS/Uslyf5VQg4+ONmQfsqUZLiuRg3y2ral+YUXZuWCKVsi2/vYRgvBEELcSJMBMcYHN/UfjDEuCiEsBWoVe6sO8DnJQjPrem9JjPG7Tf13JEmSJGVYkVUzmTEj2T8vS1fNrOg2WgjGGEviNspngKZrX4QQdiSZEjqBpBAcGELYNsb4TWGTZoXvSZIkSUpT4aqZdO++8bYqs3I23mSz/c+fA0Li1hDCHoWH7gQOL1w4BqAHMDnGODPGOB2YAvQuPLcy0KnwHEmSJEnSFsr0hvKdgV7AjiGE44ExhZvEVwWOAF4EZscYx4UQrgFuDSHMAdoARxW5VG/g5sI9BHcDrowxPpPJrJIkSZJUUWW0EIwxjgfGr+P4UqBJsWP/2MB1vgIGrO99SZIkSdLPVxJTQyVJkiRJZZiFoCRJkiRVMBaCkiRJklTBWAhKkiRJUgVjIShJkiRJFYyFoCRJkiRVMBaCkiRJklTBWAhKkiRJUgVjIShJkiRJFYyFoCRJkiRVMCHGmHaGjAkhLAA+SzvHOtQDFqYdQuWW/Uslyf6lkmT/Ukmyf6mkldU+tmOMcZuNNSpXhWBZFUJ4M8bYJu0cKp/sXypJ9i+VJPuXSpL9SyUt2/uYU0MlSZIkqYKxEJQkSZKkCsZCsHQMSTuAyjX7l0qS/Uslyf6lkmT/UknL6j7mPYKSJEmSVME4IihJkiRJFYyFoCRJkiRVMHlpB5AkSQIIIeQDJwKVgakxxhfTTSRJiRDCdjHGr9LOkUkWghkQQqgHXA/MBFoAE2OMw9bRrhowEPgE2BGYG2O8tTSzKvtsRv/aCxgMtAa+Ai6PMf6rNLMq+2xq/yp2zmnAfjHGE0s+obLZ5vSvEMJ+wP3AZTHGx0otpLLWZvx83Ab4EzAHqA9UAf4UXShDmyCEcChwDUk/O209bXKAq4ElQD6QC1waY1xTWjl/DgvBzBgBPBJjvD2EUBWYE0L4OMY4pVi7IcDXMcbbQggBmBpC+NwfeNqIjfavEEIN4AqSH3TLgcuAh0II78UY30oltbLFpn5+ARBC2Ac4A5hWmiGVtTapf4UQ9gCeAU6IMY5II6iy0qZ+ft0BPBpjfAQghDAUOAZ4uHTjKtuEEGoDs0iKuw25Gtg9xti38LyRwHnALSWbcMt4j+AWCiG0Aw4GRgPEGJcB44ALirVrAhxXpF0k+QC7uDTzKrtsav8CDgMuiDG+GmOcBhwPrC48V1qnzehfa9vXBXqQfHZJG7SZ/etvwBsWgdpUm9m/dge2LvL6O6BuSWdU9osxLo4xfg0sWl+bwj/Gn0thXyz0OHBeCKFSCUfcIhaCW64dsCzG+EmRY+8DnYq1OxgIwOxi7VqHELZGWrdN6l8xxidijJ8Vef0T8D3waWmEVNba1M+vtdNeLgJuLKVsyn6b1L9CCDsAhwCrQgh3hhDeCiE8EUKoX4pZlX02+fML+BdwfQihXeHoczPgwVLIqIqhNVCD//0dfzugeSqJNpGF4JZrCCwuduw7YKvC4eSi7SjW9rvC58YllE3Zb1P7138JIewEfEvy11FpfTanf50LDIkxLi2VZCoPNrV/tSp8viHGeDqwD1CPLN+oWSVucz6/rgOGAi+QFIX9YozfIWVG1v6ObyG45VYAK4sdW/v/Goq1o1jbdbWTitrU/lXcecCvY4yrSiSVyotN6l8hhK7ABzHGOaUVTOXCpn5+VQfWAC8BxBhXAHcDh4cQKpd0SGWtzfn5WJ1k8Y4LgQbApMIFZKRMyNrf8S0Et9w8oFaxY3WAJcX+2jSv8LlWsXYAn5dQNmW/Te1f/xZCOAZ4Jsb4RkmHU9bb1P71W+DREMKyEMIyksWIBhS+3qOUsir7bGr/+pLk95FqRY59RvKLe/HzpbU25+fjP4FXYoy3kKwumkcZX8RDWSVrf8e3ENxyE4CaIYRtixxrVni8qOdJFu9oWqzd9BjjwpKNqCy2qf0L+PfIzU8xxqeLHCvTNyorVZvUv2KMfWOMVdc+SKZZPVD4uug9EVJRm/r59SZQQLKgx1rVgR9ijAtKNqKy2Ob8fOwKzAAo3AfuKpL7uqRMmEpyO07x3/EXUNjvyioLwS0UY5wOTAF6AxROY+kE3BlC2DOEcGthuwXAY2vbFepBsqSxtE6b2r8K3zuM5F6b90MIzUIIzUMIl5P8QiX9j83pX9Lm2oyfj0uAe0lWO16rI/581AZs5ufXVKBtkdfVAWfNaHPkFn0RQmhYuLhVtcLbcO7hf3/HH1LWb9EJ7qW55UII2wE3k/xVczdgWozxnhBCH+DPwJ4xxmUhhK2AQcAHJBuaLgJudENTbcim9C9gb+BZ/rfouy/G+JvSzKvssqmfX8XOuQrYyQ3ltTGb8fOxGsnPxx9IRgfrAhcW3i8ordNm9K8dSFY8/gD4EdgRuCrGWHyxGem/FG4NcQTwd+ALkvtMR5MsajUCaBtj/KJw9tVAkhXbc4CawEUxxuL3sZYpFoKSJEmSVME4NVSSJEmSKhgLQUmSJEmqYCwEJUmSJKmCsRCUJEmSpArGQlCSJEmSKhgLQUmSJEmqYCwEJUmSJKmCsRCUJEmSpArm/wEGkX6KJdJNcgAAAABJRU5ErkJggg==\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"Greska je E_n=2.1159e-02\n",
"Teo. greska je 5.9172e-03\n"
]
}
],
"source": [
"import math \n",
"import numpy as np\n",
"from matplotlib import pyplot \n",
"from numpy import linalg\n",
"from matplotlib import rc\n",
"noviparametri = {'figure.figsize': (15, 7), 'axes.grid': True,\n",
" 'lines.markersize': 5, 'lines.linewidth': 1.25,\n",
" 'font.size': 15, 'mathtext.fontset': 'stix',\n",
" 'font.family': 'STIXGeneral'}\n",
"pyplot.rcParams.update(noviparametri)\n",
"\n",
"ep=1.0\n",
"N=12\n",
"h=1/N\n",
"alpha=0 # rubni uslovi\n",
"beta=np.sin(5) # rubni uslovi\n",
"\n",
"B=np.zeros(N-1)\n",
"A=np.zeros((N-1,N-1))\n",
"\n",
"def b(x):\n",
" return -x\n",
"def c(x):\n",
" return np.sin(x)\n",
"def f(x):\n",
" return -5*x*np.cos(5*x)+25*ep**2*np.sin(5*x)+np.sin(x)*np.sin(5*x)\n",
"def tacno(x):\n",
" return np.sin(5*x)\n",
"\n",
"# funkcija f i tačno rješenje y(x)=x^2, treba promijeniti rubni uslov y(1)=1\n",
"#def f(x):\n",
"# return -2*ep**2-2*x**2+x**2*np.sin(x)\n",
"#def tacno(x):\n",
"# return x**2\n",
"\n",
"B[0]=f(h)-alpha*(-ep**2/h**2-b(h)/(2*h))\n",
"A[0][0]=c(h)+2*ep**2/h**2; A[0][1]=b(h)/(2*h)-ep**2/h**2;\n",
"\n",
"B[N-2]=f((N-1)*h)-beta*(b((N-1)*h)/(2*h)-ep**2/h**2)\n",
"A[N-2][N-3]=-ep**2/(h**2)-b((N-1)*h)/(2*h)\n",
"A[N-2][N-2]=c((N-1)*h)+2*ep**2/h**2\n",
"\n",
"for i in range(1,(N-2)):\n",
" B[i]=f((i+1)*h)\n",
" A[i][i-1]=-ep**2/h**2-b((i+1)*h)/(2*h)\n",
" A[i][i]=c((i+1)*h)+2*ep**2/h**2\n",
" A[i][i+1]=b((i+1)*h)/(2*h)-ep**2/h**2\n",
" \n",
"rjesenje=linalg.solve(A,B)\n",
" \n",
"rjesenje=np.append(rjesenje,[beta])\n",
"rjesenje=np.append([alpha],rjesenje) \n",
"\n",
"x2=np.linspace(0,1,N+1)\n",
"x3=np.linspace(0,1,1000)\n",
"\n",
"pyplot.plot(x2,rjesenje,'ro')\n",
"pyplot.plot(x2,rjesenje,'r-.')\n",
"pyplot.plot(x3,tacno(x3),'b-')\n",
"pyplot.legend(('numeričko','linearni interpolat','tačno'))\n",
"#pyplot.savefig('/home/samir/Dropbox/Python/slika1.pdf', dpi=1000)\n",
"pyplot.show()\n",
"pyplot.close()\n",
"\n",
"print('Greska je E_n=%2.4e'%max(abs(rjesenje-tacno(x2))))\n",
"print('Teo. greska je %2.4e'%(1/(N+1)**2))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
" \n",
"## Rješavanje nelinearnih rubnih problema \n",
"Idi na: \n",
"[početak](#uvod)\n",
"[linearni](#linearni)\n",
"\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Rješavanje opšteg nelinearnog rubnog problema \n",
"$$y''=F(x,y,y'),\\:x\\in[a,b],\\:y(a)=\\alpha,\\:y(b)=\\beta, $$\n",
"slično je rješavanju linearnog problema. Sistem jednačina, čijim rješavanjem ćemo dobiti aproksimativna rješenje u tačkama mreže, više neće biti linearan, što je najveća teškoća. \n",
" \n",
"\n",
" Napomena 2: I u ovom slučaju koristeći funkciju $x\\mapsto\\frac{x-a}{b-a}$ segment $[a,b]$ možemo zamijeniti sa segmentom $[0,1].$ \n",
"
\n",
"\n",
"Podijelimo ponovo segment $[a,b]$ na $N$ jednakih podsegmenata tačkama $x_i=a+ih,\\:i=0,1,\\ldots,N,\\:h=\\frac{b-a}{N}.$ \n",
"Prvi i drugi izvod aproksimiraćemo na isti način kao i u linearnom slučaju\n",
"$$y'(x_i)\\approx\\frac{y_{i+1}-y_{i-1}}{2h},\\quad y''(x_i)\\approx\\frac{y_{i+1}-2y_i+y_{i-1}}{h^2},$$\n",
"gdje je $y_i$ aproksimacija tačne vrijednosti $y(x_i)$ u tačkama mreže $x_i,\\,i=0,1,\\ldots,N.$\n",
"\n",
"\n",
"Koristeći aproksimacije za prvi i drugi izvod dobijamo diferencnu šemu\n",
"$$\\frac{y_{i+1}-2y_i+y_{i-1}}{h^2}=F\\left(x_i,y_i,\\frac{y_{i+1}-y_{i-1}}{2h} \\right),\\,i=1,2,\\ldots,N-1.$$\n",
"\n",
"Ponovo kao i linearnom slučaju uvrštavajući vrijednosti indeksa $i$ u diferencnu šemu i uvrštavajući $y_0=\\alpha,\\,y_N=\\beta,$ dobijamo \n",
"\n",
"\\begin{align}\n",
" \\frac{\\alpha-2y_1+y_{2}}{h^2}&=F\\left(x_1,y_1,\\frac{y_2-\\alpha}{2h} \\right)\\nonumber\\\\\n",
" \\frac{y_1-2y_2+y_{3}}{h^2}&=F\\left(x_2,y_2,\\frac{y_3-y_1}{2h} \\right)\\nonumber\\\\\n",
" \\quad\\vdots\\nonumber\\\\\n",
" \\frac{y_{N-2}-2y_{N-1}+\\beta}{h^2}&=F\\left(x_{N-1},y_{N-1},\\frac{\\beta -y_{N-2}}{2h} \\right). \\nonumber\n",
"\\end{align}\n",
"Rješavanjem prethodnog sistema dobijamo aproksimativne vrijednosti $y_i,\\,i=1,2,\\ldots,N-1,$\n",
"\n",
"\n",
"\n",
"\n",
"\n",
"\n",
"\n",
"\n",
"\n",
"\n",
"\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
" Primjer. Izračunati numeričko rješenje rubnog problema \n",
"$y''=y+y^2-6e^{-x}\\cos 3x-9e^{-x}\\sin 3x -e^{-2x}\\sin^2 3x,\\quad x\\in[0,10],$$\\:y(0)=0,\\,y(10)=e^{-10}\\sin 30.$\n",
"
"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {},
"outputs": [
{
"data": {
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\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"Greska je En=9.39907e-03\n",
"Teorijska vrijednost greske je Et=2.44141e-02\n"
]
}
],
"source": [
"from scipy.optimize import fsolve\n",
"import numpy as np\n",
"import matplotlib \n",
"from matplotlib import pyplot\n",
"\n",
"from matplotlib import rc\n",
"noviparametri = {'figure.figsize': (12, 8), 'axes.grid': True,\n",
" 'lines.markersize': 5, 'lines.linewidth': 1.25,\n",
" 'font.size': 15, 'mathtext.fontset': 'stix',\n",
" 'font.family': 'STIXGeneral'}\n",
"pyplot.rcParams.update(noviparametri)\n",
"\n",
"n=64; #broj podsegmenata \n",
"alpha=0; beta=np.exp(-10)*np.sin(30);\n",
"xa=0;xb=10\n",
"pocetni=.5\n",
"h=(xb-xa)/n\n",
" \n",
"#def F(x,y):\n",
"# F=(-y+y*y-2+x**2-x**4) \n",
"# return F\n",
"#def tacno(x):\n",
"# return x**2\n",
"\n",
"def F(x,y):\n",
" return y+y**2-6*np.exp(-x)*np.cos(3*x)-9*np.exp(-x)*np.sin(3*x)-np.exp(-2*x)*(np.sin(3*x))**2\n",
"def tacno(x):\n",
" return np.exp(-x)*np.sin(3*x)\n",
"\n",
"poc=np.zeros(n-1)\n",
"for i in range(n-1):\n",
" poc[i]=pocetni\n",
" \n",
"x=np.zeros(n+1)\n",
"for i in range(n+1):\n",
" x[i]=xa+i*h\n",
"\n",
"def nelin(y):\n",
" shema=np.zeros(n-1)\n",
" shema[0]=alpha-2*y[0]+y[1]-h*h*F(x[1],y[0])\n",
" shema[n-2]=y[n-3]-2*y[n-2]+beta-h*h*F(x[n-1],y[n-2])\n",
" for i in range(1,n-2):\n",
" shema[i]=y[i-1]-2*y[i]+y[i+1]-h*h*F(x[i+1],y[i])\n",
" return shema\n",
"\n",
"rjesenje=fsolve(nelin,poc)\n",
"rjesenje=np.append([alpha],rjesenje)\n",
"rjesenje=np.append(rjesenje,[beta])\n",
"\n",
"x1=np.linspace(xa,xb,1000)\n",
"\n",
"pyplot.plot(x1,tacno(x1),'b-.-.',x,rjesenje,'ro',x,rjesenje,'ro-.')\n",
"pyplot.legend(('Tačno ','Numeričko ','Linearni interpolat'))\n",
"pyplot.show()\n",
"pyplot.close()\n",
"\n",
"greska=max(abs(rjesenje-tacno(x)))\n",
"print('Greska je En=%2.5e'%greska)\n",
"print('Teorijska vrijednost greske je Et=%2.5e'%h**2)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Literatura \n",
"\n",
" R.L.Burden, J.D. Faires, Numerical analisys, BROOKS/COLE, Pacific Grove, CA, USA, 2001. \n",
" H. B. Keller, Numerical methods for two-points boundary value problems, Dover publications, Inc., Mineola, New York, USA, 1992. \n",
" \n",
"[Vrati se na početak](#uvod)"
]
}
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