Numerical Methods for Solving Impulsive Differential Equations: Challenges, Methods, and Innovations

Impulsive differential equations (IDEs) are a class of differential equations that model dynamic systems subject to sudden changes at distinct moments in time, called impulses. These equations arise in a variety of fields such as physics, biology, economics, and engineering, often-representing phenomena where processes experience discontinuities, such as shock waves, sudden force applications, or rapid switching. The study of numerical methods for solving IDEs is crucial for their practical applications, as analytical solutions are rarely available due to the non-smooth nature of impulses. This paper discusses the challenges in solving IDEs numerically, the methods developed to tackle these challenges, and recent innovations that have improved the accuracy and efficiency of numerical simulations.