Exact solutions and finite time stability for higher fractional-order differential equations with pure delay

We obtain the exact solutions to the higher fractional-order nonhomogeneous delayed differential equations with Caputo-type fractional derivative by using a set of newly defined generalized delayed Mittag-Leffler matrix functions. The Laplace transform and inductive construction are taken up as the major solving approaches. Thereafter, we consider some special cases and prove that the new exact solutions are suitable for the delayed differential equations with arbitrary order alpha>0$$ \alpha >0 $$. Additionally, we propose some criteria on the finite time stability of the higher fractional-order delay differential equations. Finally, an illustrative example is presented to test the correctness of the theoretical results.