Research Article Analytical Solution of the Fractional Linear Time-Delay Systems and their Ulam-Hyers Stability Nazim I. Mahmudov Department of Mathematics, Eastern Mediterranean University, Famagusta, 99628 Northern Cyprus, Mersin 10, Turkey Correspondence should be addressed to Nazim I. Mahmudov; nazim.mahmudov@emu.edu.tr Received 28 July 2022; Accepted 8 September 2022; Published 23 September 2022 Academic Editor: Yansheng Liu Copyright © 2022 Nazim I. Mahmudov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We introduce the delayed Mittag-Leffler type matrix functions, delayed fractional cosine, and delayed fractional sine and use the Laplace transform to obtain an analytical solution to the IVP for a Hilfer type fractional linear time-delay system D μ,ν 0,t zðt Þ + Az ðt Þ + Ωzðt − hÞ = f ðt Þ of order 1< μ <2 and type 0 ≤ ν ≤ 1, with nonpermutable matrices A and Ω. Moreover, we study Ulam- Hyers stability of the Hilfer type fractional linear time-delay system. Obtained results extend those for Caputo and Riemann- Liouville type fractional linear time-delay systems with permutable matrices and new even for these fractional delay systems. 1. Introduction Khusainov et al. [1] studied the following Cauchy problem for a second order linear differential equation with pure delay: x ′ ′ t ðÞ + Ω 2 xt − τ ð Þ = ft ðÞ, t ≥ 0, τ > 0, xt ðÞ = φ t ðÞ, x ′ t ðÞ = φ′ t ðÞ, −τ ≤ t ≤ 0, ( ð1Þ where f : ½0,∞Þ ⟶ ℝ n , Ω is a n × n nonsingular matrix, τ is the time delay and φ is an arbitrary twice continuously dif- ferentiable vector function. A solution of (1) has an explicit representation of the form ([1], Theorem 2): xt ðÞ = cos τ Ωt ð Þφ −τ ð Þ + Ω −1 sin τ Ωt ð Þφ ′ −τ ð Þ + Ω −1 ð 0 −τ sin τ Ω t − τ − s ð Þφ ′ ′ s ðÞds + Ω −1 ð t 0 sin τ Ω t − τ − s ð Þ fs ðÞds, ð2Þ where cos τ Ω : ℝ ⟶ ℝ n×n and sin τ Ω : ℝ ⟶ ℝ n×n denote the delayed matrix cosine of polynomial degree 2k on the intervals ðk − 1Þτ ≤ t < kτ and the delayed matrix sine of polynomial degree 2k +1 on the intervals ðk − 1Þτ ≤ t < kτ, respectively. It should be stressed out that pioneer works [1, 2] led to many new results in integer and noninteger order time-delay differential equations and discrete delayed system; see [3–18]. These models have applications in spatially extended fractional reaction-diffusion models [19], oscillating systems [20, 21], numerical solutions [22], and so on. Introducing the fractional analogue delayed matrices cosine/sine of a polynomial degree, see Formulas (5) and (6), Liang et al. [23] gave representation of a solution to the initial value problem (3): Theorem 1 (see [23]). Let h > 0, φ ∈ C 2 ð½−h, 0, ℝ n Þ, Ω be a nonsingular n × n matrix. The solution x : ½−h,∞Þ ⟶ ℝ n of the initial value problem C D α −h xt ðÞ + Ω 2 xt − h ð Þ = 0, t ≥ 0, h > 0, xt ðÞ = φ t ðÞ, x ′ t ðÞ = φ ′ t ðÞ, −h ≤ t ≤ 0, ( ð3Þ Hindawi Journal of Applied Mathematics Volume 2022, Article ID 2661343, 7 pages https://doi.org/10.1155/2022/2661343