Introdução ao cálculo fracionário com aplicações

Abstract

This paper presents a historical review of fractional calculus, describing the main definitions and applications of the theme since its inception in 1695 to the present day, briefly commenting on publications on the subject in Brazil. Prerequisites for fractional calculus are presented, such as the gamma function, the beta function, and the Mittag-Leffler function The Riemann-Liouville fractional integral and the Caputo and Riemann-Liouville fractional derivatives are defined and explored with examples compared to classical calculus. Several applications of fractional calculus are covered as: (i) the fractional generalization of the radioactive decay and population growth equations, which are applied to the radioactive decay of cesium-137 and population growth in Brazil; (ii) the generalization of Newton’s second law to a fractional differential equation and solved for some fractional orders; (iii) the generalization of the logistic equation to an arbitrary order. With the fractional solution of the logistic equation, a curve fitting is performed involving those infected by Covid-19 in Chapecó-SC.