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 explores the application of generalized conformable derivatives in modeling hotel demand dynamics in Mexico, using the Gompertz-type model. The research focuses on customizing conformable functions to fit the unique characteristics of the Mexican hotel industry, considering the Tourist Area Life Cycle (TALC) model and aiming to enhance forecasting accuracy. The parameter adjustment in all cases was made by designing a convex function, which represents the difference between the theoretical model and real data. Results demonstrate the effectiveness of the generalized conformable derivative approach in predicting hotel demand trends, showcasing its potential for improving decision-making processes in the Mexican hospitality sector. The comparison between the logistic and Gompertz models, in both integer and fractional versions, provides insights into the suitability of these modeling techniques for capturing the dynamics of hotel demand in the studied regions.

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.

This study introduces a novel approach to chess analysis based on center-of-mass dynamics and discrete fractal derivatives, offering an alternative framework for evaluating gameplay strategies. Unlike conventional methods that rely on exhaustive search and statistical simulations, our model provides a macroscopic perspective by analyzing the collective motion of pieces over time. By representing chess positions as a dynamic system in ${\mathbb{R}}^2$, we identify key movement patterns—such as oblique, parallel, and orthogonal trends—revealing strategic tendencies throughout the game. Additionally, fractal derivatives enable the detection of subtle momentum shifts and long-term imbalances, enhancing the understanding of decision-making processes. This approach is computationally efficient and adaptable, extending beyond chess to applications in sports analytics and real-time strategy games. These findings highlight the potential of interdisciplinary techniques in capturing complex strategic behavior within dynamic environments.

This work investigates the reduction of Generalized Conformable Partial Differential Equations (GCPDEs) to classical Partial Differential Equations (PDEs) through a tailored change of variables based on integral conformable operators. We concentrate on second-order linear PDEs due to their central role in modeling physical and engineering systems. The proposed transformation yields an equivalent equation expressed in terms of conformable derivatives, preserving the original order and degree while introducing local kernel-dependent modulation. This equivalence establishes that solving a GCPDE can be systematically reduced to solving a standard PDE in transformed coordinates. We demonstrate the utility of this framework in modeling anomalous diffusion and anisotropic transport phenomena, where traditional formulations may fail to capture essential spatial or temporal variability.