The Approximate Method for Solving Second-Order Fuzzy Boundary Value Problems

Abstract

Nowadays, the topic of fuzzy differential equations (FDEs) has received a lot of attention among researchers. The FDE formed a mathematical modelling of the real-world problems, such as in medicine, hydraulic systems, population models and modelling of periodic phenomena. The FDE can be divided into two parts, which are fuzzy initial value problem (FIVP) and fuzzy boundary value problem (FBVP). Due to many real-world problems modelled using FBVP, there has been a lot of interest in investigating the solution of FBVP. The purpose of this study is to provide a method of solution for second-order FBVP. Based on the generalized fuzzy derivative, four systems of FBVP are formulated. For each system, the second-order FBVP is split into two parts, namely fuzzy non-homogeneous and fuzzy homogeneous equations. By appropriate substitution, these two equations are then reduced to first-order FDE. By proposing the Runge-Kutta Cash-Karp (RKCK) method in a fuzzy setting, the approximate solution is obtained. To make sure the result is acceptable, the approximate solution is then compared with Runge-Kutta of Order Four (RK4) method. From numerical solutions, the result showed that the approximate solution of the proposed method is better compared to the result obtained using RK4 method.