Abstract
In this study, we investigate numerical methods for non-smooth differential equations (NSDEs), which are pivotal in simulating abrupt phenomena in natural and engineering systems. We introduce the fractional interpolation method (FIM), a novel technique that utilizes fractional power functions to approximate solutions at points where derivatives are infinite. This method’s principal innovation is its adept handling of NSDEs' inherent discontinuities, offering a stable and convergent solution framework. Our findings confirm that FIM is both theoretically sound and practically reliable. Through rigorous numerical experiments, we have demonstrated its superior performance compared to conventional high-order numerical methods and MATLAB’s built-in functions. To further affirm FIM’s practicality, we applied it to two distinct non-smooth system types: systems with dry friction and binary wing systems with clearances. These applications substantiate the effectiveness of FIM and highlight its potential to tackle real-world challenges. Furthermore, this research equips scientists and engineers with a robust new tool for addressing NSDEs, setting the stage for further exploration and practical uses, especially in scenarios requiring accurate simulation of abrupt system behaviors. We anticipate the broader application of FIM in analyzing and designing non-smooth systems and are enthusiastic about its role in enhancing our understanding and prediction of complex dynamics across various natural and technical systems.