Publication: Cálculo fraccionario, geometría fractal y modelos de crecimiento tumoral
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2020-07-10
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Abstract
El cálculo fraccionario estudia la posilidad de extender los operadores de derivación e integración clásicos a operadores de órdenes no enteros. El carácter no local de estos nuevos operadores fraccionarios ofrece nuevas perspectivas para formular modelos matemáticos en ramas muy diversas.
En este trabajo presentamos una introducción teórica a las definiciones y propiedades más importantes utilizadas en cálculo fraccionario y en el estudio de las ecuaciones diferenciales fraccionarias.
Introducimos el método numérico que surge de manera natural de la definición del operador fraccionario de Grünwald-Letnikov y lo aplicamos al estudio numérico de ecuaciones diferenciales fraccionarias que resultan de extender modelos clásicos de crecimiento tumoral.
Finalmente, se definen algunos conceptos de cálculo fraccionario discreto y se propone un modelo de crecimiento fractal en el plano complejo, basado en el conjunto de Mandelbrot, que presenta aspectos que pueden compararse con el crecimiento de tumores reales.
Fractional calculus studies the possibility of extending classical derivation and integration operators to non-integer order operators. The non-local nature of these new fractional operators offers new perspectives for formulating mathematical models in very diverse branches. In this work we present a theoretical introduction to the most important definitions and properties used in fractional calculus and in the study of fractional differential equations. We introduce the numerical method that arises naturally from the definition of the Grünwald-Letnikov fractional operator and apply it to the numerical study of fractional differential equations that result from extending classical models of tumor growth. Finally, we define some concepts of discrete fractional calculus and propose a fractal growth model in the complex plane, based on the Mandelbrot set, that presents aspects that can be compared with the growth of real tumors.
Fractional calculus studies the possibility of extending classical derivation and integration operators to non-integer order operators. The non-local nature of these new fractional operators offers new perspectives for formulating mathematical models in very diverse branches. In this work we present a theoretical introduction to the most important definitions and properties used in fractional calculus and in the study of fractional differential equations. We introduce the numerical method that arises naturally from the definition of the Grünwald-Letnikov fractional operator and apply it to the numerical study of fractional differential equations that result from extending classical models of tumor growth. Finally, we define some concepts of discrete fractional calculus and propose a fractal growth model in the complex plane, based on the Mandelbrot set, that presents aspects that can be compared with the growth of real tumors.