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TutorialDeIntegracionNumerica2.html
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TutorialDeIntegracionNumerica2.html
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<!DOCTYPE html>
<html>
<head><meta charset="utf-8" />
<title>TutorialDeIntegracionNumerica2</title>
<script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.1.10/require.min.js"></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/jquery/2.0.3/jquery.min.js"></script>
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<h1 id="Tutorial-de-solución-de-ecuaciones-diferenciales-ordinarias-sujetas-a-una-variación-sistemática-de-parámetros">Tutorial de solución de ecuaciones diferenciales ordinarias sujetas a una variación sistemática de parámetros<a class="anchor-link" href="#Tutorial-de-solución-de-ecuaciones-diferenciales-ordinarias-sujetas-a-una-variación-sistemática-de-parámetros">¶</a></h1><h3 id="Lourdes-Martín-Aguilar,-César-Flores-López-&-Marco-Herrera-Valdez">Lourdes Martín Aguilar, César Flores López & Marco Herrera Valdez<a class="anchor-link" href="#Lourdes-Martín-Aguilar,-César-Flores-López-&-Marco-Herrera-Valdez">¶</a></h3><h4 id="Facultad-de-Ciencias,-Universidad-Nacional-Autónoma-de-México">Facultad de Ciencias, Universidad Nacional Autónoma de México<a class="anchor-link" href="#Facultad-de-Ciencias,-Universidad-Nacional-Autónoma-de-México">¶</a></h4><p>Resolveremos la ecuación</p>
\begin{equation}
\partial_t x = \frac{a - x}{\tau}, \quad
\end{equation}<p>para diferentes valores de $\tau$ (velocidad de cambio) y de las condiciones iniciales</p>
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<p>Primero importemos los módulos y funciones que necesitamos</p>
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<div class=" highlight hl-ipython3"><pre><span class="kn">import</span> <span class="nn">scipy</span> <span class="k">as</span> <span class="nn">sc</span>
<span class="kn">import</span> <span class="nn">matplotlib.pylab</span> <span class="k">as</span> <span class="nn">gr</span>
<span class="kn">from</span> <span class="nn">scipy.integrate</span> <span class="k">import</span> <span class="n">odeint</span>
<span class="o">%</span><span class="k">matplotlib</span> inline
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<p>El módulo scipy (scipy.org) tiene librerías para cálculos numéricos. El módulo matplotlib (matplotlib.org) tiene liberías para hacer gráficas en 2D, 3D, y más, con calidad de publicación.</p>
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<p>Definamos una función que calcule el cambio en $x$</p>
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<div class=" highlight hl-ipython3"><pre><span class="k">def</span> <span class="nf">linearEq</span><span class="p">(</span><span class="n">u</span><span class="p">,</span><span class="n">t</span><span class="p">,</span><span class="n">pa</span><span class="p">):</span>
<span class="n">du</span> <span class="o">=</span> <span class="p">(</span><span class="n">pa</span><span class="p">[</span><span class="s">"a"</span><span class="p">]</span><span class="o">-</span><span class="n">u</span><span class="p">)</span><span class="o">/</span><span class="n">pa</span><span class="p">[</span><span class="s">"tau"</span><span class="p">]</span>
<span class="k">return</span> <span class="n">du</span>
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<p>Note que la función <em>linearEq</em> toma como argumentos la variable de estado, el tiempo, y un diccionario de parámetros <em>p</em>. Definimos entonces un diccionario que contenga parámetros</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">p</span><span class="o">=</span><span class="nb">dict</span><span class="p">()</span>
<span class="n">p</span><span class="p">[</span><span class="s">"a"</span><span class="p">]</span><span class="o">=</span><span class="mf">90.0</span><span class="p">;</span> <span class="n">p</span><span class="p">[</span><span class="s">"tau"</span><span class="p">]</span><span class="o">=</span><span class="mf">45.0</span><span class="p">;</span> <span class="n">p</span><span class="p">[</span><span class="s">"ic"</span><span class="p">]</span><span class="o">=</span><span class="mf">200.0</span>
<span class="n">p</span><span class="p">[</span><span class="s">"timeMax"</span><span class="p">]</span><span class="o">=</span><span class="mi">300</span><span class="p">;</span> <span class="n">p</span><span class="p">[</span><span class="s">"timeStep"</span><span class="p">]</span><span class="o">=</span><span class="mf">0.1</span>
<span class="n">p</span><span class="p">[</span><span class="s">"sampTimes"</span><span class="p">]</span><span class="o">=</span> <span class="n">sc</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="n">p</span><span class="p">[</span><span class="s">"timeMax"</span><span class="p">],</span><span class="n">p</span><span class="p">[</span><span class="s">"timeStep"</span><span class="p">])</span>
<span class="n">uSol</span><span class="o">=</span> <span class="n">odeint</span><span class="p">(</span><span class="n">func</span><span class="o">=</span><span class="n">linearEq</span><span class="p">,</span> <span class="n">y0</span><span class="o">=</span><span class="n">p</span><span class="p">[</span><span class="s">"ic"</span><span class="p">],</span> <span class="n">t</span><span class="o">=</span><span class="n">p</span><span class="p">[</span><span class="s">"sampTimes"</span><span class="p">],</span> <span class="n">args</span><span class="o">=</span><span class="p">(</span><span class="n">p</span><span class="p">,))</span><span class="o">.</span><span class="n">transpose</span><span class="p">()</span>
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<p>Definimos de qué modo cambiará el valor en los parámetros $\tau$ y condiciones iniciales fijas</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">taus</span><span class="o">=</span> <span class="n">sc</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">20</span><span class="p">,</span><span class="mf">120.01</span><span class="p">,</span><span class="mi">20</span><span class="p">)</span>
<span class="n">tauSims</span><span class="o">=</span><span class="nb">list</span><span class="p">()</span>
<span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">taus</span><span class="p">:</span>
<span class="n">p</span><span class="p">[</span><span class="s">"tau"</span><span class="p">]</span><span class="o">=</span><span class="n">t</span>
<span class="n">uSol</span><span class="o">=</span> <span class="n">odeint</span><span class="p">(</span><span class="n">func</span><span class="o">=</span><span class="n">linearEq</span><span class="p">,</span> <span class="n">y0</span><span class="o">=</span><span class="n">p</span><span class="p">[</span><span class="s">"ic"</span><span class="p">],</span> <span class="n">t</span><span class="o">=</span><span class="n">p</span><span class="p">[</span><span class="s">"sampTimes"</span><span class="p">],</span> <span class="n">args</span><span class="o">=</span><span class="p">(</span><span class="n">p</span><span class="p">,))</span><span class="o">.</span><span class="n">transpose</span><span class="p">()</span>
<span class="n">tauSims</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">uSol</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
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<p>Soluciones iteradas para distintas condiciones iniciales y $\tau$ fija</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">p</span><span class="p">[</span><span class="s">"tau"</span><span class="p">]</span><span class="o">=</span><span class="mf">45.0</span><span class="p">;</span>
<span class="n">ics</span><span class="o">=</span> <span class="n">sc</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">70</span><span class="p">,</span><span class="mf">120.01</span><span class="p">,</span><span class="mi">10</span><span class="p">)</span>
<span class="n">icsSims</span><span class="o">=</span><span class="nb">list</span><span class="p">()</span>
<span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">ics</span><span class="p">:</span>
<span class="n">p</span><span class="p">[</span><span class="s">"ic"</span><span class="p">]</span><span class="o">=</span><span class="n">t</span>
<span class="n">uSol</span><span class="o">=</span> <span class="n">odeint</span><span class="p">(</span><span class="n">func</span><span class="o">=</span><span class="n">linearEq</span><span class="p">,</span> <span class="n">y0</span><span class="o">=</span><span class="n">p</span><span class="p">[</span><span class="s">"ic"</span><span class="p">],</span> <span class="n">t</span><span class="o">=</span><span class="n">p</span><span class="p">[</span><span class="s">"sampTimes"</span><span class="p">],</span> <span class="n">args</span><span class="o">=</span><span class="p">(</span><span class="n">p</span><span class="p">,))</span><span class="o">.</span><span class="n">transpose</span><span class="p">()</span>
<span class="n">icsSims</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">uSol</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
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<p>Graficamos la solución numérica para los diferentes valores de $\tau$ (gráfica izquierda) y de condiciones iniciales (gráfica derecha)</p>
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<div class=" highlight hl-ipython3"><pre><span class="n">fig</span><span class="o">=</span><span class="n">gr</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">15</span><span class="p">,</span><span class="mi">5</span><span class="p">))</span>
<span class="n">rows</span><span class="o">=</span><span class="mi">1</span><span class="p">;</span> <span class="n">cols</span><span class="o">=</span><span class="mi">2</span><span class="p">;</span> <span class="n">gr</span><span class="o">.</span><span class="n">ioff</span><span class="p">()</span>
<span class="n">ax1</span><span class="o">=</span><span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="n">rows</span><span class="p">,</span><span class="n">cols</span><span class="p">,</span><span class="mi">1</span><span class="p">)</span>
<span class="n">ax2</span><span class="o">=</span><span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="n">rows</span><span class="p">,</span><span class="n">cols</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
<span class="n">nTaus</span><span class="o">=</span><span class="nb">len</span><span class="p">(</span><span class="n">taus</span><span class="p">)</span>
<span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">sc</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">nTaus</span><span class="p">):</span>
<span class="n">s0</span><span class="o">=</span><span class="s">r'$\tau$=%g mins'</span><span class="o">%</span><span class="p">(</span><span class="n">taus</span><span class="p">[</span><span class="n">n</span><span class="p">])</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="s">"sampTimes"</span><span class="p">],</span><span class="n">tauSims</span><span class="p">[</span><span class="n">n</span><span class="p">],</span><span class="n">lw</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span><span class="n">alpha</span><span class="o">=</span><span class="mf">0.9</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="n">s0</span><span class="p">)</span>
<span class="n">nIcs</span><span class="o">=</span><span class="nb">len</span><span class="p">(</span><span class="n">ics</span><span class="p">)</span>
<span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">sc</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">nIcs</span><span class="p">):</span>
<span class="n">s0</span><span class="o">=</span><span class="s">r'$x_0$=%g mg/dl'</span><span class="o">%</span><span class="p">(</span><span class="n">ics</span><span class="p">[</span><span class="n">n</span><span class="p">])</span>
<span class="n">ax2</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="s">"sampTimes"</span><span class="p">],</span> <span class="n">icsSims</span><span class="p">[</span><span class="n">n</span><span class="p">],</span><span class="n">lw</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.9</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="n">s0</span><span class="p">)</span>
<span class="n">ax2</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s">"mins"</span><span class="p">)</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s">"mins"</span><span class="p">)</span>
<span class="n">ax2</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s">"mg/dl"</span><span class="p">)</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s">"mg/dl"</span><span class="p">)</span>
<span class="n">ax1</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">ncol</span><span class="o">=</span><span class="mi">2</span><span class="p">);</span> <span class="n">ax2</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">ncol</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<span class="n">gr</span><span class="o">.</span><span class="n">ion</span><span class="p">();</span><span class="n">gr</span><span class="o">.</span><span class="n">draw</span><span class="p">()</span>
</pre></div>
</div>
</div>
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