Efficiency of Runge-Kutta-Fehlberg Method in Resolving Uncertainty: a Fuzzy Approach to Delay Differential Equations

Abstract

This research delves into the examination of the convergence behavior of the Runge-Kutta-Fehlberg Method (RKFM) when applied to address the challenges posed by Delay Differential Equations (DDE) within the framework of fuzzy concepts. The integration of fuzzy logic is explored as a strategic means to grapple with the inherent uncertainty in obtained results, acknowledging the limitations of conventional approaches in dealing with nuanced uncertainties in DDEs. The core of the study involves a meticulous analysis of numerical outcomes, with a specific emphasis on evaluating the effectiveness and efficiency of the Runge-Kutta-Fehlberg method. The assessment includes a thorough comparison of results obtained through the RKFM against those derived from the Runge-Kutta 4th-order method, as well as benchmarking against the exact solution. The findings of this research not only validate the suitability of the RKFM for solving Delay Differential Equations but also highlight its comparative advantages over the traditional Runge-Kutta 4th-order method. The results contribute to an enhanced understanding of the performance and applicability of the RKFM in the context of fuzzy DDEs, thereby advancing the landscape of numerical methods for addressing complex dynamic systems.