Analytical solution of the fractional linear time-delay systems and their Ulam-Hyers stability

The candidate solution reproduces the paper’s Laplace-transform method for the Hilfer-type fractional delay IVP with 1<μ<2, 0≤ν≤1, arriving at the same explicit representation in terms of the delayed two-parameter matrix kernel Y^h_{μ,γ} and the same Ulam–Hyers stability bound. Key identities used by the model match the paper’s Lemma 4 (Laplace transform of the Hilfer derivative) and Lemma 8 (transform of Y^h_{μ,γ}) and yield Theorem 9’s solution formula and Theorem 10’s stability statement. Notational differences (e.g., writing initial Hilfer data as w(0+), w′(0+) with w=I^{(1−ν)(2−μ)}z) are consistent with the paper’s data in (6), and the model adds a small-time asymptotics check of the initial data. Overall, the proofs are the same in structure and substance: Laplace-domain algebra followed by inverse transform using Y^h_{μ,γ}, plus a convolution estimate for stability. See the statement of the problem and solution formula in (6) and Theorem 9, the Laplace identities in Lemma 4, the kernel transform in Lemma 8, and the Ulam–Hyers stability definition and theorem in Section 2 of the paper .