Fractional variable order systems with unusual dynamics in the order are a little-studied topic. In this study, we present three examples of very simple fractional systems with unusual dynamics in the derivative order. These cases involve different approaches to define the variable-order dynamics: (1) an integer-order differential equation that includes the state variable, (2) a differential equation that incorporates the state variable and features both integer- and fractional-order derivatives, and (3) fractional variable-order differential equations nested in the derivative orders. We prove a result that shows how the extended recursion of the last case is generalized. These examples illustrate the richness that simple dynamical systems can reveal through the order of their derivatives.

This article addresses critical challenges in the transition to clean energy sources by highlighting the importance of advanced mathematical modeling and computational techniques in turbine design and operation. Specifically, we extend and generalize the work of Camporeale to advance the modeling of hydrogen turbine systems. By utilizing conformable calculus, we develop dynamic equations that analyze key aspects of turbine performance, including temperature variations in turbine blades, angular velocities of rotating shafts, and mass–energy balances within the plenum and combustion chamber. Furthermore, we incorporate Kirchhoff’s equation in its generalized conformable integral form, enhancing the precision of energy balance calculations and improving the representation of heat transfer processes in the combustion chamber. This methodology introduces novel perspectives in hydrogen turbine research, contributing to the advancement of sustainable and efficient technologies. Our comprehensive approach aims to provide more accurate and efficient predictions of turbine behavior, thereby impacting the design and optimization of hydrogen-based clean energy systems.

This study focuses on optimizing the thermal performance of hydrogen turbine blades through a sensitivity analysis using generalized fractional calculus. The approach is designed to capture the transient temperature dynamics and optimize thermal profiles by analyzing the influence of a fractional-order parameter on the system’s behavior. The model was implemented in Python, using Monte Carlo simulations to evaluate the impact of the parameter on the temperature evolution in different thermal regimes. Three distinct regions were identified: the Quasi-Uniform Region (where fractional effects are negligible), the Sub-Classical Region (characterized by delayed thermal behavior), and the Super-Classical Region (exhibiting enhanced heat accumulation). Regression analyses reveal quadratic and cubic dependencies of blade temperature on the fractional-order parameter, confirming the robustness of the model with R² values greater than 0.96. The study highlights the potential of using fractional calculus to optimize the thermal response of turbine blades, helping to identify the most suitable parameters for faster stabilization and efficient heat management in hydrogen turbines. Furthermore, it was found that by adjusting the fractional-order parameter, the system can be optimized to reach equilibrium more rapidly while achieving higher temperatures. Importantly, the equilibrium is not altered but rather accelerated based on the chosen parameter, ensuring a more efficient thermal stabilization process.