Fractional Differential Equations with Variable-Order Operators: A Survey, Unified Framework, and Extended Stability Analysis

Variable-order fractional differential equations (VO-FDEs) generalise classical fractional calculus by allowing the differentiation order to depend on time, space, or the solution itself. Despite growing interest in their modelling potential—spanning viscoelastic mechanics, anomalous diffusion, and control systems—the theoretical foundations remain fragmented across the literature. This paper provides a self-contained survey of VO-FDE formulations (Riemann–Liouville, Caputo, and Grünwald–Letnikov), unifies the existence-and-uniqueness theory under a single Lipschitz-type framework, and extends the Lyapunov-based stability analysis to encompass a broad class of nonlinear variable-order systems. We introduce the notion of a ‘uniform Grönwall kernel,’ which bridges constant-order and variable-order regimes, and derive explicit conditions under which Mittag–Leffler stability is preserved. Illustrative numerical experiments using a generalised Adams–Bashforth–Moulton scheme corroborate the theoretical predictions. The results unify and extend those of Samko, Kilbas, and Marichev (1993), Lorenzo and Hartley (2002), and Coimbra (2003).

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