Analysis of Second-Order Linear Fuzzy Differential Equation Under an Innovative Fuzzy Derivative Approach and Its Application

Abstract

Fuzzy derivatives are a concept based on fuzzy calculus, which extends classical calculus to handle uncertainty and vagueness, habitually represented using fuzzy sets or fuzzy numbers. In fuzzy calculus, the derivative incorporates fuzziness, meaning both the input and output can have imprecise or uncertain values. In this paper, the step size is also considered as a fuzzy number. The impact of fuzziness in step size is studied here for first- and second-order fuzzy derivatives. Under this proposed derivative, differentiation of fuzzy exponential function is estimated. The concept is applied for solving second-order linear fuzzy differential equation. The proposed approaches are applied to spring mass system dynamics in fuzzy environment. Lastly, numerical illustrations on second-order fuzzy differential equations are shown using analytical technique and graphically the results are displayed. © 2025 Elsevier B.V., All rights reserved.