Generalized Coupled System of ψ-Hilfer Fractional Differential Equations with Self and Cross-Dependence

Abstract

In this paper, we extend the study of coupled systems of ψ-Hilfer fractional differential equations with terminal conditions to a more general case where each equation depends on both unknown functions. This formulation incorporates both self- and cross-coupling, which is essential for many realistic models in fields such as biology, physics, and engineering. Furthermore, we establish Ulam–Hyers stability of our generalized coupled system. Finally, we provide two concrete numerical examples (a linear one with ψ(t) = t and a nonlinear one with ψ(t) = t^2) that illustrate the applicability of the theoretical results to both linear and genuinely nonlinear problems . The extension presented here opens the door to modeling more complex interactions in fractional dynamical systems while preserving the rigorous analysis of existence, uniqueness, and stability.