In this study, we introduced the $\psi$-Laplace transform Adomian decomposition method, which is a combination of the efficient Adomian decomposition method with the generalization of the classical Laplace transform to treat fractional differential equations with respect to another function, $\psi$, in the Caputo sense. To validate the effectiveness of this method, we applied the derived recurrent scheme of the $\psi$-Laplace Adomian decomposition on several test numerical problems, including a real-life scenario in pharmacokinetics that models the movement of drug concentration in human blood. The solutions obtained closely matched the known solutions for the test problems. Additionally, in the pharmacokinetics case, the results were consistent with the available physical data. Consequently, this method simplifies the verification of numerous related aspects and proves advantageous in solving various $\psi$-fractional differential equations.

This paper combines the Sumudu transform with the Adomian decomposition method to address Caputo-type fractional-order pantograph delay differential equations. It features numerical evaluations that confirm the effectiveness of the proposed methods. The study introduces a powerful computational technique for solving these equations, providing results that establish its efficiency and relevance through comparisons with existing methods. The findings underscore both the efficiency and accuracy of the proposed algorithm.