  Implementing and executing numerical algorithms to solve fractional
differential equations has been less straightforward than using their
integer-order counterparts, posing challenges for practitioners who wish to
incorporate fractional calculus in applied case studies. Hence, we created an
open-source Julia package, FdeSolver, that provides numerical solutions for
fractional-order differential equations based on product-integration rules,
predictor-corrector algorithms, and the Newton-Raphson method. The package
covers solutions for one-dimensional equations with orders of positive real
numbers. For high-dimensional systems, the orders of positive real numbers are
limited to less than (and equal to) one. Incommensurate derivatives are allowed
and defined in the Caputo sense. Here, we summarize the implementation for a
representative class of problems, provide comparisons with available
alternatives in Julia and Matlab, describe our adherence to good practices in
open research software development, and demonstrate the practical performance
of the methods in two applications; we show how to simulate microbial community
dynamics and model the spread of Covid-19 by fitting the order of derivatives
based on epidemiological observations. Overall, these results highlight the
efficiency, reliability, and practicality of the FdeSolver Julia package.
