Una introducción al cálculo fraccional con aplicaciones en la física
DOI:
https://doi.org/10.31349/RevMexFis.23.010211Keywords:
Fractional Calculus, numerical methodsAbstract
Este trabajo ofrece una introducción a los conceptos fundamentales del cálculo fraccional, destacando las definiciones más utilizadas de la derivada fraccional y poniendo énfasis en sus propiedades y aplicaciones. En particular, se analiza la derivada fraccional de Fourier, subrayando su relativa simplicidad para la implementación numérica, y se presentan códigos en Matlab y Python basados en métodos espectrales para su cálculo. Además, se exploran aplicaciones clave de las derivadas fraccionales en física, como su uso en haces ópticos, estructuras viscoelásticas, mecánica cuántica y el oscilador armónico fraccional. Como complemento, se introducen los fundamentos de la función de Mittag-Leffler, ampliamente utilizada en el cálculo fraccional. Finalmente, se discuten posibles aplicaciones futuras del cálculo fraccional, destacando su relevancia en diversas áreas de la física.
This work provides an introduction to the fundamental concepts of fractional calculus, highlighting the most commonly used definitions of fractional derivatives and emphasizing their properties and applications. In particular, the fractional Fourier derivative is analyzed, underscoring its relative simplicity for numerical implementation, and Matlab and Python codes based on spectral methods for its calculation are presented. Additionally, key applications of fractional derivatives in physics are explored, such as their use in optical beams, viscoelastic structures, quantum mechanics, and the fractional harmonic oscillator. As a complement, the fundamentals of the Mittag-Leffler function, widely used in fractional calculus, are introduced. Finally, potential future applications of fractional calculus are discussed, highlighting its relevance in various areas of physics.
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