Background: Fractional differential equations (FDEs) have emerged as essential tools for modeling complex systems exhibiting anomalous diffusion, memory effects, and hereditary properties that classical integer-order models fail to capture adequately.
Objective: This study systematically investigates stability frameworks applicable to FDEs, benchmarks analytical and numerical methods, and evaluates their efficacy for real-world mathematical modeling scenarios.
Methods: Lyapunov stability theory adapted for fractional operators, Laplace transform analysis, and several numerical schemes — including Adams-Bashforth-Moulton and spectral collocation — were applied to representative FDE models of order α ∈ (0, 1] ∪ (1, 2).
Results: The Mittag-Leffler stability criterion was validated across test cases; spectral collocation achieved the lowest maximum error (1.08×10⁻⁶) with a stability index of 0.9999, while the Grünwald-Letnikov finite-difference scheme offered the fastest runtime at 1.2 s but with higher truncation error (2.90×10⁻⁴).
Conclusion: Selecting an appropriate stability framework and numerical scheme is critical; no universal method dominates across all problem classes. Hybrid strategies combining Lyapunov analysis with high-order numerical solvers offer promising avenues for future research.