Spectral Vieta–Lucas Projection Method for Neutral Fuzzy Fractional Functional Differential Equations: Theory and Well-Posedness

This work investigates a sophisticated class of neutral fuzzy fractional functional differential equations (N3FDEs), where the fractional order α satisfies 0<α≤1. We present a comprehensive analysis of the existence, uniqueness, and well-posedness of solutions under the generalized Hukuhara framework. First, we examine the existence and uniqueness of solutions under the generalized Hukuhara framework, providing an refined iterative formula for linear systems. We further verify the system’s well-posedness, proving that solutions remain stable and respond continuously to changes in initial data and parameters. Second, we introduce a novel spectral Vieta–Lucas projection method to approximate the solution. By leveraging the unique properties of Vieta–Lucas polynomials, we transform complex memory-dependent fuzzy equations into a streamlined algebraic system. Finally, numerical examples and error analysis show the method is accurate and efficient.