Modeling Stiff Differential Equations: A COVID-19 Case Study in Verification and Application

Abstract

In this research, we explore the application of the Differential Transform Method (DTM) and Local Differential Transform Method (LDTM) to solve diverse types of nonlinear differential equations encountered in stiff systems. Stiff differential equations represent a category of mathematical models arising in scientific and engineering contexts where the solution displays both rapid and gradual changes on distinct time scales. The numerical solution of these systems can be challenging due to the broad range of time scales involved, leading to potential issues of numerical instability and accuracy. The exploration of stiff differential equations remains an active research area, with continuous efforts focused on devising novel numerical methods to enhance the accuracy, efficiency, and robustness of solving these intricate systems. Various types of initial value problems are considered comparatively, examining the stability of the DTM and LDTM methods across different time intervals. To effectively characterize the nonlinear behavior of the studied problems, we employ the Duffing model, which represents a forced oscillator coupled with a spring exerting a rebalancing force. Within this investigation, a novel numerical estimation method is introduced based on the modified LDTM to address the nonlinear cubic Duffing equation, both dissipative and non-dissipative cases. Jagannath University Journal of Science, Volume 12, Number 1, Jun. 2025, pp. 49−64