Analytical Solution and Stability of Ψ-Prabhakar Delayed Systems: Application to Antibiotic Production

Abstract

The analytical solution of the linear ψ-Prabhakar delay differential equations is explored using the method of variation of parameters, in which a ψ-general delayed exponential matrix function is introduced. The ψ-Prabhakar calculus is enhanced in terms of the semi-group property, inversion results, and several simplified calculations and relations. The system’s stability is analyzed within the framework of Ulam-Hyers stability. Finally, the theoretical findings are validated through an application to antibiotic production. © 2025 Elsevier B.V., All rights reserved.