The cauchy problem for the delay differential equation with Dzhrbashyan – Nersesyan fractional derivative

In recent, the number of works devoted to the study of problems for fractional order differential equations is growing noticeably. The interest of researchers is due to the fact that the number of areas of science in which equations containing fractional derivatives are used varies from biology and medicine to control theory, engineering, finance, as well as optics, physics, and so on. The inclusion of delay in the fractional order equation significantly affects the course of the process described by this equation, since the unknown function is given for different values of the argument, which includes a history effect into the equation. Therefore, mathematical models containing a fractional operator and a delay argument are more accurate than models containing integer derivatives. In this paper, we study the Cauchy problem for a linear ordinary delay differential equation with the Dzhrbashyan – Nersesyan fractional differentiation operator, which is generalizing the Riemann – Liouville and Gerasimov – Caputo fractional operators. The results of the work are obtained using the methods of the theory of integer and fractional calculus, methods of the theory of delay differential equations, the method of special functions. In this paper proves a theorem on the validity of an analogue of the Lagrange formula. It is also proved that the special function $W_{\gamma_m}(t)$, which is defined in terms of the generalized Mittag-Leffler function (or the Prabhakar function), satisfies the equation and conditions associated with the one under study, and is the fundamental solution of the considered equation. The main result is that the existence and uniqueness theorem to the initial value problem is proved. The solution to the problem is written out in terms of the special function $W_\nu(t)$.