  In this paper we consider a Caputo type fractional derivative with respect to
another function. Some properties, like the semigroup law, a relationship
between the fractional derivative and the fractional integral, Taylor's
Theorem, Fermat's Theorem, etc, are studied. Also, a numerical method to deal
with such operators, consisting in approximating the fractional derivative by a
sum that depends on the first-order derivative, is presented. Relying on
examples, we show the efficiency and applicability of the method. Finally, an
application of the fractional derivative, by considering a Population Growth
Model, and showing that we can model more accurately the process using
different kernels for the fractional operator is provided.

Classical Analysis and ODEs