On initial conditions for fractional delay differential equations

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.