A modified Green–Naghdi fractional-order model for analyzing thermoelectric semispace heated by a moving heat source

The classical Green–Naghdi (GN-II) model encounters challenges in accurately describing the thermo-mechanical behavior of electro-thermoelastic materials; in particular, the model does not consider the memory effect. To address this, a novel mathematical model of the Green–Naghdi (GN-II) theory is developed, incorporating a fractional order of heat transfer. This enhanced model offers a more comprehensive understanding by including several theories as limiting examples. Central to this approach is the use of the matrix exponential method, foundational to the state-space approach in modern theory. Additionally, the Laplace transform is employed to facilitate the model formulation. This formulation is applied to a specific half-space problem, which involves exposure to a uniform magnetic field and heating by a moving heat source at a constant speed. For the practical application of this model, a numerical method is utilized for the inverse Laplace transform. The roles of various factors on the solution are examined, including the figure-of-merit quantity, speed of the heat source, fractional parameter, magnetic number, and thermal shock parameter. By exploring these variables the model provides a thorough understanding of the interaction between heat transfer and magnetic fields in electro-thermoelastic materials. This research represents a significant advancement in the modeling of electro-thermoelastic materials, offering a more accurate and comprehensive tool for predicting their behavior under varying thermal and magnetic conditions.