In this article, we study two proposed fractional models. Initially, we introduce a new generalized Marshall–Hoare model using the Mittag-Leffler type function to simulate the cooling of the internal solid organs during the time immediately after death. Furthermore, we provide an explicit solution for this new model. Secondly, by using \(\Psi \) -Caputo derivative of order \(\alpha \in (0,1)\) , we study Newton’s law of cooling under a general variable ambient temperature with two heat transfer coefficients. By transforming the main problem to a general delayed Volterra integral equation of the second kind and in light of the Banach fixed point theorem, we establish the existence and uniqueness results under a condition on both heat transfer coefficients. Additionally, via the delayed Henry–Gronwall integral inequality, we derive a criterion for finite-time stability. Moreover, we give a first-order numerical scheme to approximate our proposed model by approximating the integral form of the given solution. Finally, to show the efficacy and precision of our model, we look into a pharmacokinetics clinical trial. Our delayed fractional model anticipates values of drug concentration in plasma as close as possible to the ones observed in experimental data.
