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TutorialDeIntegracionNumerica1.html
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<!DOCTYPE html>
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<head><meta charset="utf-8" />
<title>TutorialDeIntegracionNumerica1</title>
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<h1 id="Tutorial-de-solución-de-ecuaciones-diferenciales-ordinarias-con-parámetros-constantes">Tutorial de solución de ecuaciones diferenciales ordinarias con parámetros constantes<a class="anchor-link" href="#Tutorial-de-solución-de-ecuaciones-diferenciales-ordinarias-con-parámetros-constantes">¶</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>Resolvamos un problema con valor inicial para una variable $x$ que depende de $t$</p>
\begin{equation}
\partial_t x = \frac{a - x}{\tau}, \quad x(t_0) = 0, \quad (1)
\end{equation}<p>Utilizando las rutinas de integración numérica del módulo <em>scipy</em> de <em>Python</em></p>
<h2 id="Solución-analítica">Solución analítica<a class="anchor-link" href="#Solución-analítica">¶</a></h2><p>La solución de la ecuación (1) se puede obtener analíticamente separando variables e integrando en el intervalo [0,t]:</p>
<p>La solución analítica es de la forma</p>
\begin{equation}
x(t) = a - \left( a - x_0\right) \exp \left(-\frac{t}{\tau} \right)
\end{equation}<p>Compararemos $x(t)$ con la solución numérica $u(t)$ obtenida con <em>python</em></p>
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<p>Primero importemos 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>El módulo scipy tiene herramientas para hacer cálculos numéricos, el módulo pylab de matplotlib contiene herramientas para graficar arreglos, listas y otros objetos de Python. El comando matplotlib inline instruye al kernel de Jupyter (o ipython notebook) para que muestre las figuras dentro del documento.</p>
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<p>Definamos una función 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">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">p</span><span class="p">):</span>
<span class="n">du</span> <span class="o">=</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="n">u</span><span class="p">)</span><span class="o">/</span><span class="n">p</span><span class="p">[</span><span class="s2">"tau"</span><span class="p">]</span>
<span class="k">return</span> <span class="n">du</span>
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<p>¿Por qué así y no de otra forma? Por ejemplo:</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">linearEq2</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">a</span><span class="p">,</span><span class="n">b</span><span class="p">):</span>
<span class="n">du</span> <span class="o">=</span> <span class="p">(</span><span class="n">a</span><span class="o">-</span><span class="n">u</span><span class="p">)</span><span class="o">/</span><span class="n">b</span>
<span class="k">return</span> <span class="n">du</span>
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<p>Este modo de escribir la ecuación no nos permite variar los valores de los parámetros si así lo quisiéramos.</p>
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<p>Generemos un diccionario con los parámetros y arreglos necesarios para integrar la ecuación numéricamente.</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">"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="s2">"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="s2">"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="s2">"timeMax"</span><span class="p">]</span><span class="o">=</span><span class="mi">200</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="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>
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<pre>dict_keys(['timeStep', 'sampTimes', 'ic', 'timeMax', 'a', 'tau'])
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<p>Ahora iteremos la función usando la función <em>integrate.odeint</em> de <em>scipy</em></p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="c1"># Sintaxis:</span>
<span class="c1"># solucion = odeint(func, y0, t, args)</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="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>Explicar la sintaxis de la función odeint, y la forma de la salida de la función. De ahí, explicar la transposición.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="n">uSol</span><span class="p">[</span><span class="mi">0</span><span class="p">][</span><span class="mi">0</span><span class="p">:</span><span class="mi">10</span><span class="p">])</span>
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<pre>[ 200. 199.75582679 199.51219561 199.2691053 199.02655453
198.7845426 198.54306816 198.30212993 198.06172665 197.82185704]
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<p>Definamos otra función para comparar la solución analítica $x(t)$ con la solución numérica $u(t)$.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">x</span><span class="p">(</span><span class="n">t</span><span class="p">,</span><span class="n">p</span><span class="p">):</span>
<span class="k">return</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="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="n">p</span><span class="p">[</span><span class="s2">"ic"</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="o">-</span><span class="n">t</span><span class="o">/</span><span class="n">p</span><span class="p">[</span><span class="s2">"tau"</span><span class="p">])</span>
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<p>Otra manera de definir la función $x$ sería</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">x</span><span class="o">=</span> <span class="k">lambda</span> <span class="n">t</span><span class="p">,</span><span class="n">p</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="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="n">p</span><span class="p">[</span><span class="s2">"ic"</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="o">-</span><span class="n">t</span><span class="o">/</span><span class="n">p</span><span class="p">[</span><span class="s2">"tau"</span><span class="p">])</span>
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<p>Como el paso de integración es 0.1, el décimo valor de la solución numérica se puede comparar con el valor de la función $x$ en el tiempo $t=1$.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="n">x</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">p</span><span class="p">))</span>
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<pre>197.582515973
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<p>Graficamos la solución analítica $x(t)$ y comparamos contra la solución numérica $u(t)$ y un acercamiento</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">tt</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="o">*</span><span class="mi">100</span><span class="p">)</span>
<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">11</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="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">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">uSol</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span><span class="s1">'k'</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">0.35</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">r'($t,x(t)$)'</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">tt</span><span class="p">,</span><span class="n">x</span><span class="p">(</span><span class="n">tt</span><span class="p">,</span><span class="n">p</span><span class="p">),</span><span class="s1">'bo'</span><span class="p">,</span><span class="n">label</span><span class="o">=</span><span class="s1">r'($t,u(t)$)'</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">uSol</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span><span class="s1">'k'</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">0.35</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s1">r'($t,x(t)$)'</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">tt</span><span class="p">,</span><span class="n">x</span><span class="p">(</span><span class="n">tt</span><span class="p">,</span><span class="n">p</span><span class="p">),</span><span class="s1">'b'</span><span class="p">,</span><span class="n">label</span><span class="o">=</span><span class="s1">r'($t,u(t)$)'</span><span class="p">)</span>
<span class="n">ax2</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">(</span><span class="mf">0.008</span><span class="p">,</span><span class="mf">0.01</span><span class="p">);</span> <span class="n">ax2</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">(</span><span class="mf">199.95</span><span class="p">,</span><span class="mf">200.05</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">"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="s2">"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="s2">"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="s2">"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">ax2</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
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<pre><matplotlib.legend.Legend at 0x10b02cda0></pre>
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