  Derivatives of fractional order are introduced in different ways: as
left-inverse of the fractional integral or by generalizing the limit of the
difference quotient defining integer-order derivatives. Although the two
approaches lead (under standard smoothness assumptions) to equivalent
operators, the first one does not involve the function at the left of the
initial point where, instead, the latter forces the function to assume selected
values. With fractional delay differential equations new problems arise: the
presence of the delay imposes to assign the solution not just at the initial
point but on an entire interval. Due to the freedom in the choice of the
initial function, some inconsistencies with the values forced by the fractional
derivative are possible and the operators may no longer be equivalent. In this
paper we discuss the initialization of fractional delay differential equations,
we investigate the effects of the initial condition not only on the solution
but also on the fractional operator as well and we study the difference between
solutions obtained by incorporating or not the initial function in the memory
of the fractional derivative. The exact solution of a family of linear
equations is obtained by the Laplace transform whilst numerical methods are
used to solve nonlinear problems; the different results are therefore shown and
commented.

Classical Analysis and ODEs;Numerical Analysis;Numerical Analysis