Extension of the Fuzzy Laplace Transform Method for Solving Higher-Order Nonlinear Fuzzy Ordinary Differential Equations under Generalized Hukuhara Differentiability

Abstract

Higher-order nonlinear fuzzy ordinary differential equations (FODEs) arise in many real-world systems where uncertainty and imprecision are inherent, such as engineering processes, biological models, and economic dynamics.  Traditional analytical and numerical techniques often encounter difficulties when dealing with such equations, particularly under generalized differentiability conditions.  This study presents an extension of the Fuzzy Laplace Transform Method (FLTM) for solving higher-order nonlinear FODEs under generalized Hukuhara differentiability.  The proposed approach combines the Laplace transform technique with fuzzy calculus to transform complex fuzzy differential equations into more manageable algebraic forms in the Laplace domain.  The method accommodates different cases of generalized Hukuhara differentiability and provides systematic procedures for handling nonlinear terms in higher-order fuzzy systems.  By applying the inverse fuzzy Laplace transform, explicit fuzzy solutions can be obtained in parametric form.  The effectiveness and reliability of the extended method are demonstrated through illustrative examples of higher-order nonlinear FODEs, where the obtained solutions show strong agreement with known analytical results and highlight the method’s capability in dealing with uncertainty in dynamic systems.  The results indicate that the extended FLTM offers a powerful, efficient, and flexible framework for solving a wide class of higher-order nonlinear fuzzy differential equations under generalized Hukuhara differentiability, thereby contributing to the advancement of analytical techniques in fuzzy differential analysis.  The study concludes with recommendations for future research, focusing on establishing solutions for both odd- and even-order FODEs.