New Series Solution of the Caputo Fractional Ambartsumian Delay Differential Equationation by Mittag-Leffler Functions

The fractional generalization of the Ambartsumian delay equation with Caputo's fractional derivative is considered. The Ambartsumian delay equation is very difficult to be solved neither in the case of ordinary derivatives nor in the case of fractional derivatives. In this paper we combine the Laplace transform with the Adomian decomposition method to solve the studied equation. The exact solution is obtained as a series which terms are expressed by the Mittag-Leffler functions. The advantage of the present approach over the known in the literature ones is discussed.

Automated summary

This paper considers the fractional generalization of the Ambartsumian delay equation with Caputo's fractional derivative, which is difficult to solve using ordinary or fractional derivatives. By combining Laplace transform with Adomian decomposition method, the exact solution is obtained as a series expressed by the Mittag-Leffler functions. The advantage of this approach over existing methods in the literature is discussed.