A Numerical Approach for Solving Fuzzy Differential Equation Using Enhanced Euler’s Methods Based on Contra Harmonic Mean and Centroidal Mean

Abstract

Objectives: Fuzzy differential equations (FDEs) are crucial for modeling dynamic systems in science, economics, and engineering due to their unpredictable nature and the need for numerical methods for precise solutions. The objective of this work is to develop improved Euler's techniques for numerically solving fuzzy differential equations (FDEs) in order to produce approximate solutions for problems that are too complicated to be stated exactly. 

Methods: This study proposes improved Euler's approaches for solving differential equations with fuzzy initial conditions, including Euler's Method, Modified Euler's Method, and Improved Euler's Method  employing Contra Harmonic Mean and Centroidal Mean. Based on Zadeh's extension concept for fuzzy sets, we extend Euler's classical methods to address this dependence problem in a fuzzy setting.

Findings: A numerical example that contrasts the suggested approaches with the traditional Euler's Method is provided. The efficiency of the suggested methods is shown by numerical solutions and their geometrical representations. Furthermore, the numerical example demonstrates the acceptable accuracy of the improved Euler's techniques. 

Novelty: In this research proposal, we sought to solve first-order differential equations utilizing two creative approaches: the Contra Harmonic Mean and the Centroidal Mean of enhanced Euler's approaches for fuzzy primary value. The suggested methods yield great results and are quick, accurate, and easy to use.