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TutorialDeIntegracionNumerica3.html
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TutorialDeIntegracionNumerica3.html
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<!DOCTYPE html>
<html>
<head><meta charset="utf-8" />
<title>TutorialDeIntegracionNumerica3</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-con-forzamiento-temporal-determinista">Tutorial de solución de ecuaciones diferenciales ordinarias con forzamiento temporal determinista<a class="anchor-link" href="#Tutorial-de-solución-de-ecuaciones-diferenciales-ordinarias-con-forzamiento-temporal-determinista">¶</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>Supongamos que la concentración de un metabolito ($x$, mM) en el torrente sanguíneo es influenciada por dos hormonas de manera distinta. Si la concentración del metabolito tiende a un valor $a$ en la ausencia de influencias externas, los cambios en $x$ con respecto al tiempo se pueden determinar resolviendo una ecuación de la forma<br>
\begin{equation}
\partial_t x = \frac{a - x}{\tau} + f(t),
\end{equation}
donde $f$ es una función de forzamiento que representa una influencia externa que comienza en un tiempo $t_0$, a partir de una condición inicial $x(0) = x_0$. Si las influencias de las dos hormonas mencionadas anteriormente se pueden representar con funciones $f_1(t)$ y $f_2(t)$, entonces nos interesa comparar el comportamiento de $x$ con respecto a $f_1$ y $f_2$ en el tiempo.</p>
<p>Las soluciones son obtenidas
utilizando las rutinas de integración numérica del módulo <em>scipy</em> de <em>python</em>.</p>
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<p>Primero debemos importar los módulos necesarios.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><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>Después hacemos un diccionario con los parámetros y arreglos necesarios para la integración numérica.</p>
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<p>Las funciones de forzamiento que utilizaremos como ejemplos son:
\begin{eqnarray}
f_1(t)=\alpha\left(t,t<em>0,\tau</em>{\alpha}, A_1\right)&=& A_1 H(t>t_0)\frac{t-t<em>0}{\tau</em>{\alpha}}\left(1- \frac{t-t<em>0}{\tau</em>{\alpha}} \right)
\
f_2(t)=\sin\left(t,t_0,\omega, A_2\right)&=& A_2 H(t>t_0) \sin \left[\left(t-t_0\right)2\pi \right]
\end{eqnarray}
La función $f_1$ representa un pulso con comienzo súbito en un tiempo $t_0$ y decaimiento exponencial con una constante de tiempo $\tau_{\alpha}$. La función $f_2$ representa una fluctuación periódica de forma sinusoidal que comienza también en $t=t_0$.</p>
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<p>Escribimos y graficamos las funciones alfa y seno:</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">alpha</span><span class="o">=</span> <span class="k">lambda</span> <span class="n">t</span><span class="p">,</span><span class="n">q</span><span class="p">:</span> <span class="n">q</span><span class="p">[</span><span class="s2">"A1"</span><span class="p">]</span><span class="o">*</span><span class="n">sc</span><span class="o">.</span><span class="n">int32</span><span class="p">(</span><span class="n">t</span><span class="o">></span><span class="n">q</span><span class="p">[</span><span class="s2">"t0"</span><span class="p">])</span><span class="o">*</span><span class="p">((</span><span class="n">t</span><span class="o">-</span><span class="n">q</span><span class="p">[</span><span class="s2">"t0"</span><span class="p">])</span><span class="o">/</span><span class="n">q</span><span class="p">[</span><span class="s2">"tauAlpha"</span><span class="p">])</span><span class="o">*</span><span class="n">sc</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="p">(</span><span class="n">t</span><span class="o">-</span><span class="n">q</span><span class="p">[</span><span class="s2">"t0"</span><span class="p">])</span><span class="o">/</span><span class="n">q</span><span class="p">[</span><span class="s2">"tauAlpha"</span><span class="p">])</span>
<span class="n">sine</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">t</span><span class="p">,</span><span class="n">q</span><span class="p">:</span> <span class="n">sc</span><span class="o">.</span><span class="n">int32</span><span class="p">(</span><span class="n">t</span><span class="o">></span><span class="n">q</span><span class="p">[</span><span class="s2">"t0"</span><span class="p">])</span><span class="o">*</span> <span class="n">q</span><span class="p">[</span><span class="s2">"A2"</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">sc</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">t</span><span class="o">*</span><span class="n">q</span><span class="p">[</span><span class="s2">"freq"</span><span class="p">]</span><span class="o">*</span><span class="mi">2</span><span class="o">*</span><span class="n">sc</span><span class="o">.</span><span class="n">pi</span><span class="p">))</span>
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<p>y las graficamos para verificar que tienen el curso temporal correcto</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="c1"># Parámetros para usar con las funciones alpha y sine</span>
<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="s2">"t0"</span><span class="p">]</span><span class="o">=</span><span class="mi">5</span><span class="p">;</span> <span class="n">p</span><span class="p">[</span><span class="s2">"ic"</span><span class="p">]</span><span class="o">=</span><span class="mf">10.0</span><span class="p">;</span>
<span class="n">p</span><span class="p">[</span><span class="s2">"a"</span><span class="p">]</span><span class="o">=</span><span class="mf">5.0</span><span class="p">;</span> <span class="n">p</span><span class="p">[</span><span class="s2">"A1"</span><span class="p">]</span><span class="o">=</span><span class="mi">2</span><span class="p">;</span> <span class="n">p</span><span class="p">[</span><span class="s2">"A2"</span><span class="p">]</span><span class="o">=</span><span class="mf">0.3</span>
<span class="n">p</span><span class="p">[</span><span class="s2">"tau"</span><span class="p">]</span><span class="o">=</span><span class="mf">5.0</span><span class="p">;</span> <span class="n">p</span><span class="p">[</span><span class="s2">"tauAlpha"</span><span class="p">]</span><span class="o">=</span><span class="mf">0.5</span><span class="p">;</span> <span class="n">p</span><span class="p">[</span><span class="s2">"height"</span><span class="p">]</span><span class="o">=</span><span class="mf">0.3</span><span class="p">;</span>
<span class="n">p</span><span class="p">[</span><span class="s2">"freq"</span><span class="p">]</span><span class="o">=</span><span class="mf">0.3</span><span class="p">;</span>
<span class="n">p</span><span class="p">[</span><span class="s2">"timeMax"</span><span class="p">]</span><span class="o">=</span><span class="mf">30.0</span><span class="p">;</span> <span class="n">p</span><span class="p">[</span><span class="s2">"timeStep"</span><span class="p">]</span><span class="o">=</span><span class="mf">0.1</span><span class="p">;</span>
<span class="n">p</span><span class="p">[</span><span class="s2">"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="s2">"timeMax"</span><span class="p">],</span><span class="n">p</span><span class="p">[</span><span class="s2">"timeStep"</span><span class="p">])</span>
<span class="nb">print</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">keys</span><span class="p">())</span>
<span class="c1"># Graficas de las funciones alpha y sine</span>
<span class="n">fi</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">11</span><span class="p">,</span><span class="mi">4</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">fi</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</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">fi</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</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="s2">"sampTimes"</span><span class="p">],</span><span class="n">alpha</span><span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="s2">"sampTimes"</span><span class="p">],</span><span class="n">p</span><span class="p">),</span><span class="n">label</span><span class="o">=</span><span class="s2">r"$f_1(t)$ ((mg/dl)/seg), $t_0$=</span><span class="si">%g</span><span class="s2">"</span><span class="o">%</span><span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="s2">"t0"</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="s2">"sampTimes"</span><span class="p">],</span><span class="n">sine</span><span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="s2">"sampTimes"</span><span class="p">],</span><span class="n">p</span><span class="p">),</span><span class="n">label</span><span class="o">=</span><span class="s2">r"$f_2$ ((mg/dl)/seg), </span><span class="si">%g</span><span class="s2"> Hz, $t_0$=</span><span class="si">%g</span><span class="s2">"</span><span class="o">%</span><span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="s2">"freq"</span><span class="p">],</span><span class="n">p</span><span class="p">[</span><span class="s2">"t0"</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="s2">"seg"</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="s2">"seg"</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">ax2</span><span class="o">.</span><span class="n">legend</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>
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<pre>dict_keys(['A1', 'tauAlpha', 'ic', 'a', 'timeMax', 'A2', 't0', 'freq', 'sampTimes', 'tau', 'height', 'timeStep'])
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<p>Definimos una función que calcule el cambio en $x$ para que <em>Python</em> la integre numéricamente:</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">linearForcedEq</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">t</span><span class="p">,</span><span class="n">par</span><span class="p">):</span>
<span class="n">dx</span> <span class="o">=</span> <span class="p">(</span><span class="n">par</span><span class="p">[</span><span class="s2">"a"</span><span class="p">]</span><span class="o">-</span><span class="n">x</span><span class="p">)</span><span class="o">/</span><span class="n">par</span><span class="p">[</span><span class="s2">"tau"</span><span class="p">]</span> <span class="o">+</span> <span class="n">par</span><span class="p">[</span><span class="s1">'f'</span><span class="p">](</span><span class="n">t</span><span class="p">,</span><span class="n">par</span><span class="p">)</span>
<span class="k">return</span> <span class="n">dx</span> <span class="c1">#la función integrate.odeint de _scipy_</span>
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<p>Ahora integramos la función usando la función integrate.odeint de <em>scipy</em></p>
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<div class=" highlight hl-ipython3"><pre><span></span><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="s2">"t0"</span><span class="p">]</span><span class="o">=</span><span class="mi">5</span><span class="p">;</span> <span class="n">p</span><span class="p">[</span><span class="s2">"ic"</span><span class="p">]</span><span class="o">=</span><span class="mf">10.0</span><span class="p">;</span> <span class="n">p</span><span class="p">[</span><span class="s2">"a"</span><span class="p">]</span><span class="o">=</span><span class="mf">3.0</span><span class="p">;</span> <span class="n">p</span><span class="p">[</span><span class="s2">"A1"</span><span class="p">]</span><span class="o">=</span><span class="mi">2</span><span class="p">;</span> <span class="n">p</span><span class="p">[</span><span class="s2">"A2"</span><span class="p">]</span><span class="o">=</span><span class="mf">0.3</span>
<span class="n">p</span><span class="p">[</span><span class="s2">"tau"</span><span class="p">]</span><span class="o">=</span><span class="mf">5.0</span><span class="p">;</span> <span class="n">p</span><span class="p">[</span><span class="s2">"tauAlpha"</span><span class="p">]</span><span class="o">=</span><span class="mf">0.5</span><span class="p">;</span> <span class="n">p</span><span class="p">[</span><span class="s2">"freq"</span><span class="p">]</span><span class="o">=</span><span class="mf">0.3</span><span class="p">;</span>
<span class="n">p</span><span class="p">[</span><span class="s2">"timeMax"</span><span class="p">]</span><span class="o">=</span><span class="mf">30.0</span><span class="p">;</span> <span class="n">p</span><span class="p">[</span><span class="s2">"timeStep"</span><span class="p">]</span><span class="o">=</span><span class="mf">0.1</span><span class="p">;</span>
<span class="n">p</span><span class="p">[</span><span class="s2">"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="s2">"timeMax"</span><span class="p">],</span><span class="n">p</span><span class="p">[</span><span class="s2">"timeStep"</span><span class="p">])</span>
<span class="c1"># Integración numérica</span>
<span class="n">p</span><span class="p">[</span><span class="s2">"f"</span><span class="p">]</span><span class="o">=</span><span class="n">sine</span>
<span class="n">uSine</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">linearForcedEq</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="s2">"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="s2">"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">p</span><span class="p">[</span><span class="s2">"f"</span><span class="p">]</span><span class="o">=</span><span class="n">alpha</span>
<span class="n">uAlpha</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">linearForcedEq</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="s2">"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="s2">"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>Graficamos la solución.</p>
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<div class=" highlight hl-ipython3"><pre><span></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">11</span><span class="p">,</span><span class="mi">4</span><span class="p">))</span>
<span class="n">gr</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="s2">"sampTimes"</span><span class="p">],</span><span class="n">uAlpha</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span><span class="n">label</span><span class="o">=</span><span class="s2">r"$f(t)=\alpha(t$,</span><span class="si">%g</span><span class="s2">,</span><span class="si">%g</span><span class="s2">$)$"</span><span class="o">%</span><span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="s2">"tauAlpha"</span><span class="p">],</span><span class="n">p</span><span class="p">[</span><span class="s2">"t0"</span><span class="p">]))</span>
<span class="n">gr</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="s2">"sampTimes"</span><span class="p">],</span><span class="n">uSine</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span><span class="n">label</span><span class="o">=</span><span class="s2">r"$f(t)=\sin(t$,</span><span class="si">%g</span><span class="s2">,</span><span class="si">%g</span><span class="s2">$)$"</span><span class="o">%</span><span class="p">(</span><span class="n">p</span><span class="p">[</span><span class="s2">"freq"</span><span class="p">],</span><span class="n">p</span><span class="p">[</span><span class="s2">"t0"</span><span class="p">]))</span>
<span class="n">gr</span><span class="o">.</span><span class="n">ylim</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="s2">"ic"</span><span class="p">])</span>
<span class="n">gr</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s2">"lower left"</span><span class="p">,</span><span class="n">ncol</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
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<pre><matplotlib.legend.Legend at 0x10b8a2ba8></pre>
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