Fractional modeling with optimized variable order for the thermal impacts of mobile phones on the auricular region

Fractional derivatives with variable order offer a powerful extension to the traditional fractional and integer derivatives, allowing adaptability well-suited for modeling complex, real-world systems with evolving memory and non-local properties. This study presents a variable-order fractional model to simulate the thermal effects of the mobile phone on the auricular region in different operating modes. Laplace transform is used to solve the proposed model to simulate the heat transfer dynamics between the ear and the smartphone. Numerical simulations are presented, demonstrating the effectiveness of the variable fractional order in predicting thermal performance over conventional approaches. The results demonstrate that the fractional model provides a more accurate representation of thermal memory effects and nonlocal heat conduction in biological tissues compared with traditional models. Furthermore, a detailed parametric analysis reveals the influence of the fractional order on heat dissipation rates, aligning well with thermographic observations. This study highlights the effectiveness of variable-order fractional modeling in capturing complex heat transfer dynamics, offering a more precise framework for thermal analysis in biomedical and electronic applications.