A New Model for Drug Diffusion Using Memory-Dependent Derivative with a Time Delay

This article introduces two high-order numerical approximations based on the quartic interpolation formula to estimate both the first- and second-order memory-dependent derivatives. The convergence orders of these approximations are determined to be the third and second orders of convergence, respectively. Subsequently, these approximations are employed to solve the diffusion equation with a time delay numerically. This equation represents a generalized form of the classical diffusion equation with hyperbolicity, as it incorporates memory-dependent derivatives in the two terms of time-derivatives. Also, the stability and convergence of the numerical scheme are thoroughly discussed. To demonstrate the consistency between the numerical and theoretical results, the developed scheme is tested on various numerical problems. Furthermore, a novel memory-dependent model is described, specifically designed to investigate the release and diffusion of drugs from multi-layer capsules/tablets. This model is designed to address the infinite propagation speed in Fickian-based models. The accuracy and compatibility of the proposed model are demonstrated through comparisons with the corresponding in vivo data, the classical diffusion model, and the fractional-order model, while the accompanying error analysis confirms its improved predictive accuracy and agreement with experimental observations.