Examining the One-Step Implicit Scheme with Fuzzy Derivatives to Investigate the Uncertainty Behavior of Several First-Order Real-Life Models

Abstract

Differential Equations (DE) are useful for representing a variety of concepts and circumstances. However, when considering the initial or boundary conditions for these DEs models, the usage of fuzzy numbers is more realistic and flexible since the parameters can fluctuate within a certain range. Such scenarios are referred to be unexpected conditions, and they introduce the idea of uncertainty. These issues are dealt with using fuzzy derivatives and fuzzy differential equations (FDEs). When there is no precise solution to FDEs, numerical methods are utilized to obtain an approximation solution. In this study, the One-step Implicit Scheme (OIS) with a higher fuzzy derivative is extensively used to discover optimum solutions to first-order FDEs with improved accuracy in terms of absolute accuracy. We evaluate the method competency by investigating first-order real-life models with fuzzy initial value problems (FIVPs) in the Hukuhaa derivative category. The principles of fuzzy sets theory and fuzzy calculus were utilized to give a new generic fuzzification formulation of the OIS approach with the Taylor series, followed by a detailed fuzzy analysis of the existing problems. OIS is acknowledged as a practical, convergent, and zero-stable with absolute stability region approach for solving linear and nonlinear fuzzy models, as well as a useful methodology for properly managing the convergence of approximate solutions. The developed scheme capabilities is proved by providing approximate solutions to real-life problems. The numerical findings demonstrate that OIS is a viable and transformative approach for solving linear and nonlinear first-order FIVPs. The results provide a concise, efficient, and user-friendly approach to dealing with larger FDEs.