Delayed analogue of three-parameter Mittag-Leffler functions and their applications to Caputo-type fractional time delay differential equations - Huseynov, Ismail T.; Mahmudov, Nazim I. (Wiley, 2024)
In this paper, we consider a Cauchy problem for a Caputo-type time delay linear system of fractional differential equations with permutable matrices. First, we provide a new representation of solutions to linear homogeneous fractional differential equations using the Laplace integral transform and variation of constants formula via a newly defined delayed Mittag-Leffler type matrix function introduced through a three-parameter Mittag-Leffler function. Second, with the help of a delayed perturbation of a Mittag-Leffler type matrix function, we attain an explicit formula for solutions to a linear nonhomogeneous time delay fractional order system using the superposition principle. Furthermore, we prove the existence and uniqueness of solutions to nonlinear fractional delay differential equations using the contraction mapping principle. Finally, we present an example to illustrate the applicability of our results.
Keywords: Caputo fractional derivative; delayed Mittag-Leffler type matrix function; existence and uniqueness; fractional linear time delay system; permutable matrices; three-parameter Mittag-Leffler function