  This research deals with the numerical solution of non-linear fractional
differential equations with delay using the method of steps and shifted
Legendre (Chebyshev) collocation method. This article aims to present a new
formula for the fractional derivatives (in the Caputo sense) of shifted
Legendre polynomials. With the help of this tool and previous work of the
authors, efficient numerical schemes for solving nonlinear continuous
fractional delay differential equations are proposed. The proposed schemes
transform the nonlinear fractional delay differential equations to a non-delay
one by employing the method of steps. Then, the approximate solution is
expanded in terms of Legendre (Chebyshev) basis. Furthermore, the convergence
analysis of the proposed schemes is provided. Several practical model examples
are considered to illustrate the efficiency and accuracy of the proposed
schemes.
