The New Runge-Kutta Fehlberg Method for the Numerical Solution of Second-Order Fuzzy Initial Value Problems

Abstract

This study presents a novel computational methodology for resolving second-order fuzzy initial value problems (FIVPs), encompassing ordinary differential equations. The proposed approach modifies the conventional crisp fifth-order Runge-Kutta Fehlberg method to suit the resolution of second-order FIVPs within the fuzzy domain, drawing on concepts from fuzzy set theory. It is demonstrated that by reducing them to a system of first-order FIVPs, all second-order FIVPs can be effectively solved. The novel method is subsequently applied to both linear and non-linear second-order FIVPs. The results attest to the high efficiency and accuracy of the approach, while also preserving the inherent properties of fuzzy solutions. Therefore, this study offers a promising new avenue for addressing second-order FIVPs, with potential applicability across a broad range of scenarios.