Delayed analogue of three‐parameter Mittag‐Leffler functions and their applications to Caputo‐type fractional time delay differential equations

Abstract

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.

Construction of solutions for delay evolution equations in a Banach space: A delayed Dyson-Phillips series

“…It should be note that this particular case is a natural extension of the results are attained in [6] in terms of commutative matrix coefficients by Khusainov and Shuklin. Corresponding results are derived for fractional-order time-delay systems in [7] by Huseynov and Mahmudov. Proof.…”

“…In [6], Khusainov et al have proposed an exact analytical representation of solutions for a linear differential system with permutable matrix coefficients and a constant delay. In [7,8], the classical results are extended to time-delay systems of fractional order with permutable [7] and nonpermutable [8] matrices with the help of delayed Mittag-Leffler matrix functions. For more general results, using the Laplace transform technique, Mahmudov has proposed in [9], a closed-form representation of solutions for linear delay differential equations using a multiple delayed perturbation of the Mittag-Leffler matrix functions.…”

A new technique for solving Sobolev type fractional multi-order evolution equations

“…Several results have been investigated on solving multi-dimensional time-delay deterministic and stochastic systems with permutable matrices [3,16] in classical and fractional senses. In [8], Diblik et al have considered inhomogeneous system of linear differential equations of second order with multiple different delays & pairwise permutable matrices and represented a solution of corresponding initial value problem by using matrix polynomials.…”

Asymptotic stability analysis of Riemann-Liouville fractional stochastic neutral differential equations

“…Several outcomes have been investigated on the qualitative theory and applications of fractional stochastic functional differential equations (FSDEs) [2,13,38,40]. For instance, Mahmudov and McKibben [21] studied the approximate controllability of fractional evolution equations involving generalized Riemann-Liouville fractional derivative in Hilbert spaces.…”