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| parent_permlink | stemsocial |
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| author | ycam |
|---|
| permlink | calculo-fraccionario-un-problema-de-orden-arbitrario |
|---|
| title | "Cálculo fraccionario, un problema de orden arbitrario" |
|---|
| body | "<p></p><h1>Cálculo Fraccionario</h1><sup>Derivadas de orden aleatorio</sup><p></p><p></p>
<p></p>
<div class="text-justify">
<p>Ya les he presentado algunos ejemplos introductorios sobre la <a href="https://hive.blog/stem-espanol/@ycam/derivadas-un-criterio-para-analizar-funciones-matematicas">derivada de una función</a> para encontrar los máximos y los mínimos descritos por la curvatura de una función cuadrática, por ejemplo, vista como la pendiente en cada punto tangente de la curva estudiada. Yo les nombro la función cuadrática porque es fácil ver los cambios en la pendiente de una parábola para varios valores de la variable X en comparación con el estudio que podamos realizar sobre una función lineal. </p>
<p><center>https://images.hive.blog/DQmaRxPmw6jvubgWd2DrKy2SkitvQekpmgrLd1NPeE5vPX6/image.png</center></p>
<blockquote>
En ocasiones es conveniente analizar las tendencias de una función en intervalos intermedios antes de llegar a los extremos de la función, se los comenté en el artículo de <a href="https://hive.blog/stem-espanol/@ycam/el-limite-de-una-funcion-como-analisis-de-tendencias">"El LÍMITE de una función como análisis de tendencias,"</a> ya que nos puede interesar la información contenida en cierto rango de la variable X, lo que normalmente sucede en la vida real es que se trate de la variable "tiempo", con lo cual se pueden establecer estrategias basadas en las proyecciones del evento a futuro. Igual sucede con las derivadas e integrales fraccionarias que aparecieron a finales del siglo XXVII con L’Hôpital y Leibnitz y que les presentaré en esta publicación.
</blockquote>
<p><center>https://images.hive.blog/DQmNwpfi7RjqDjSLa1KYveq6Lr7P2m7ytF2DngSxwojfJyh/image.png</center></p>
<br />
En las gráficas de las funciones no lineales se espera una variación en la curvatura, lo que implica que la función tangente asociada a diferentes puntos de la curva sean distintos (en A = C, pero diferentes en B y D), es decir que la <strong>pendiente</strong> en cada punto sea variable, positiva o negativa, hasta pendiente 0. Si observamos la gráfica de arriba vemos que existen máximos (A) o mínimos (C) cuando se establece que la primera derivada de la función sea 0 (f´(X) = 0).<br />
<center>https://upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Derivada.gif/611px-Derivada.gif</center>
<br />
Note que para cada punto (o valor de X) se puede derivar la función original e igualarla a 0 para obtener el valor de la pendiente para cada X. Este cálculo lo muestro a continuación en forma de ecuaciones con su respectiva gráfica.<br />
<p><center>https://images.hive.blog/DQmTsM9oZr8zNP6T2V812Pia7BZnWSK9BVC7R3YmB9os3YJ/image.png</center></p>
<center>https://images.hive.blog/DQmWehL3adTRtb1NR7QeuELPvSApMk5ye3qmegsULLvyoEv/image.png</center>
<center>https://images.hive.blog/DQmaypfxTd6p3bLNBogMAQ8Fz5ZmZSKuZZwzSZ6HdRzQUJq/image.png</center>
<p>Observamos que la primera derivada corresponde a una línea recta de pendiente igual a 2 y corte en -2, mientras que la segunda derivada es un valor constante (2) para todo el rango de X. Sin embargo, a veces conviene analizar lo que sucede entre la primera y segunda derivada, es en este punto donde aparecen los cálculos fraccionarios para analizar la diferenciación e integración de orden n-ésimo, con n fraccionario.</p>
<p></p><h2>¿Derivadas Enteras o Fraccionarias?</h2><p></p>
<div class="text-justify">
Ya sabemos el procedimiento normal o común para determinar la <strong>derivada</strong> de una función, pero para los fines de mostrar la metodología en el caso del cálculo fraccionario les presentaré el análisis de una función simple como la X<sup>3</sup>, que se reproduce en el caso de monomios.<p></p>
<center>https://images.hive.blog/DQmSpVX1YVLZMEFzsu9ypR5zpPRqkwd6EwxEZmnmsFh42H2/image.png</center>
<hr />
<p></p>
<p>Como ya les comenté arriba, el cálculo de las derivadas enteras es un procedimiento que cumple las reglas de la derivación y el resultado es otra función que puede graficarse, así que la tendencia en la primera derivada se relaciona, en este caso, con la curva de una parábola (3x<sup>2</sup>). Se supone que el resultado de la primera derivada de la función cúbica está relacionada con la pendiente de la curva descrita por esta función cuadrática, veamos esto en la siguiente gráfica:</p>
<center>https://images.hive.blog/DQmU6afnGqKjuJHpmmL4zvsHotW5E9DnXfDen5DrWdJ9K6g/image.png</center>
<hr />
<p>Es interesante observar que a partir de la función x<sup>3</sup>, podamos derivar 1 y 2 veces (números enteros) y obtener las gráficas con dependencia cuadrática y lineal, respectivamente!
</p>
<p>Este tratamiento matemático fue ampliado a comienzos del siglo XIX, cuando Lacroix propone el desarrollo de la función Gamma más el uso del Factorial de un número para resolver la derivada de una función. La ecuación matemática es muy sencilla y nos conduce al mismo resultado que cuando aplicamos las reglas de las derivadas, veamos el ejemplo aplicado a la función x<sup>3</sup>:
</p>
<center>https://images.hive.blog/DQmc4zbkKhTgXswXjiU1CBpnCSckDsiRyE32aXW5U91yFYN/image.png</center>
<hr />
<strong><div class="phishy">Función Gamma</div></strong>, es una función introducida por Legendre para ampliar el alcance de la notación factorial a los números complejos con n entero, positivo y considerando sólo la parte real. Sin embargo, con la ampliación de la función Gamma se introducen todos los números reales (ℝ) y se generaliza la operación de los factoriales, quedando la notación:
<p><center>https://images.hive.blog/DQmV15JB7T9iMhq9FatQcNBCttGKDHVxcFWeeAEuGwtTezb/image.png</center></p>
<p></p>
<div class="text-justify">
<p>Esta última representación es la que vamos a utilizar en el cálculo fraccionario para determinar las derivadas fraccionarias entre los intervalos de las derivadas normales (primera derivada, segunda derivada, etc). Note que aparece la notación <strong>n!! doble factorial</strong>, que debe ser calculado de la siguiente manera, dependiendo si n es par o impar:</p>
<p><center>
https://images.hive.blog/DQmSfLEr5dHpPz8ViHK24Dz2rrRNJAxz5qFieBmNs24GvgY/image.png</center></p>
<div class="text-justify">
<p>Ya tenemos las bases teóricas del cálculo fraccionario y las expresiones matemáticas necesarias para analizar lo que sucede en el intermedio de la función inicial y la primera derivada de esa función, en otras palabras, podemos decir que vamos a ver el comportamiento a la media derivada de la función, siempre y cuando combinemos la función Gamma con la operación factorial y la derivada de la función, veamos:</p>
<p><center>https://images.hive.blog/DQmTz3qV27x7APa3PVjFE75nM25DzPszAZRNtbBoXrQnkvH/image.png</center></p>
<p><em>Lacroix formuló esta expresión en el año 1819 y comenzó su análisis planteando la media derivada de un monomio, yo también hubiese comenzado con lo más fácil! </em></p>
<p><center>https://images.hive.blog/DQmWvDFzPb2TP5DMMocLqTURpx85wXMbCCYv3kQMF5uwiVU/image.png</center></p>
<div class="text-justify">
<p>Nada difícil hasta ahora sólo hemos sustituido valores en la ecuación, ahora vamos a resolver la <strong>función Gamma</strong> para (4) y (7/2).</p>
<p><center>https://images.hive.blog/DQmTAcKiTUovPwpnsDJJSQ6t8kFbC9hDnTBhdmRwwKau61g/image.png</center></p>
<div class="text-justify">
<p>Luego procedemos a graficar esta nueva función, la media derivada, la cual se espera que se prolongue entre la función original y la primera derivada.</p>
<p><center>https://images.hive.blog/DQmNyNuKizddPvL2mz9YJ4fiZDThrpysGdtRRDDRRsiqRMu/image.png</center></p>
<p><center>https://images.hive.blog/DQmZwWnQJYZg4HGKZ1TvTDiUTYZfnJWgWMCq7dHAepebDaJ/image.png</center></p>
<p>Con agrado y satisfacción puedo exclamar ¡Voilà!</p>
<p><blockquote>
La media derivada me puede proporcionar información intermedia sobre la <strong>tendencia del evento que estemos estudiando</strong> en un intervalo de tiempo o dependencia de cualquier variable.</blockquote></p>
<p><strong><div class="phishy">¿Qué sucede entre la primera y segunda derivada?</div></strong>, pues en este caso tenemos que buscar un valor de "a" entre 1 y 2, por ejemplo <strong>1,5 = 3/2</strong>, para seguirlo viendo como cálculo fraccionario, aunque en realidad se trate de una derivada de orden "arbitrario".</p>
<p><center>https://images.hive.blog/DQmS8C5aaNhpkZomfCkDgWYQYoBhAxMorEeKm7TpsygF9ec/image.png</center></p>
<p></p><h3><strong>Apoyo bibliográfico y fuente de imágenes</strong></h3><p></p>
<p>Nuestras ideas y conocimientos que podamos tener sobre el tema tratado en este artículo pueden ampliarse de manera voluntaria al consultar el siguiente catálogo de referencias:</p>
<ul>
<li>Imagen de geralt: <a href="https://pixabay.com/es/illustrations/matem%C3%A1ticas-f%C3%B3rmula-f%C3%ADsica-escuela-3393240/">Portada con fórmulas</a></li>
<li>Gráfica de Moriel: <a href="https://commons.wikimedia.org/wiki/File:Derivada.png">Derivada de una función</a></li>
<li>Gif dy/dx de Ovichan: <a href="https://commons.wikimedia.org/wiki/File:Derivada.gif">Pendiente de una curva</a></li>
<li>Nota: <a href="https://www.chegg.com/homework-help/questions-and-answers/define-double-factorial-n-denoted-n-follows-n-1-middledot-3-middledot-5-four-middledot-n-2-q12993677">La doble factorial</a></li>
<li>Nota: <a href="http://www.guiasdeapoyo.net/guias/cuart_mat_e/Funci%C3%B3n%20gamma.pdf">La función Gamma</a></li>
<li>Nota: <a href="https://www.redalyc.org/pdf/104/10444319011.pdf">La disyuntiva: ¿derivadas enteras o fraccionarias?</a></li>
<li>Wikipedia: <a href="https://es.wikipedia.org/wiki/Factorial">Factorial</a></li>
<li>Wikipedia: <a href="https://labellephysique.wordpress.com/2019/08/16/las-derivadas-fraccionarias/">Las derivadas fraccionarias</a></li>
<li>Tesis: <a href="https://core.ac.uk/download/pdf/19711062.pdf">Estudio de generalizaciones fraccionarias</a></li>
<li>Wikipedia: <a href="https://en.wikipedia.org/wiki/Double_factorial#:~:text=In%20mathematics%2C%20the%20double%20factorial,odd%20or%20even)%20as%20n.">Doble factorial</a></li></ul>
<p></p>
<center><strong>Con los números enteros positivos o los naturales</strong><br />
<em>podemos hacer algunos cálculos básicos<br />
con la derivación e integración entera o fraccionaria</em></center>
<p></p></div>" |
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