On the Quaternion-Valued Fractional Differential Equation with Hyers-Ulam Stability

Abstract

This paper investigates the Hyers-Ulam stability of linear Quaternion-Valued Fractional Differential Equations (QVFDEs) in both homogeneous and non-homogeneous forms. By exploiting the correspondence between quaternion modules and vector 2-norms, we transform QVFDEs into equivalent real fractional differential systems. Within this framework, new theoretical results on Hyers-Ulam and generalized Hyers-Ulam stability are established, supported by rigorous proofs and illustrative examples. These results not only reinforce the theoretical foundations of fractional stability analysis but also extend its applicability to systems characterized by quaternionic structures. The study offers valuable insights for modeling and analyzing phenomena in fields such as robotics, control theory, signal processing, and quantum mechanics, where quaternion-based representations naturally arise. Overall, this work contributes to the broader understanding of stability in fractional systems and opens avenues for future research on nonlinear and delayed quaternionic fractional models.