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This study proposes a comprehensive heat conduction model that incorporates fractional time derivatives and two-phase lags to describe the behavior of non-simple thermoelastic materials accurately. Generalized fractional differential operators with non-singular kernels are introduced. This type of fractional derivative includes the Caputo–Fabrizio and the Atangana–Baleanu fractional derivatives. The model also consists of the two-temperature idea, which considers the effect of microstructure through a two-stage delay approach. Interactions of a thermoelastic nature caused by the rapid heating of an isotropic substance under the influence of an external body force were studied as a practical application of the new concept. There has been some discussion about the effect of the discrepancy index and fractional differential operators. Finally, the graphical representations obtained from the numerical simulations were used to explain the behavior of the studied physical fields. The generalized fractional heat transfer model is demonstrated to be capable of producing a temperature forecast that is in close agreement with experimental data. As a result, the proposed model may be useful for solving difficulties in heat transfer, anomalous transport, and other branches of engineering analysis.

The current work is devoted to introduce a novel thermoelastic heat conduction model where the Moore-Gibson-Thompson (MGT) equation describes the heat equation. The constructed model is characterized by allowing limited velocities of heat wave propagation within the material, consistent with physical phenomena. The Green–Naghdi Type III model is improved by introducing the delay factor into the modified Fourier law. Also, from the presented model, some other models of thermoelasticity can be derived at specific states. Based on the suggested model, an infinite orthotropic material with a cylindrical hole exposed to time-dependent temperature variation was studied. It has also been considered that the coefficient of thermal conductivity varies with temperature, unlike in many other cases where this value is considered constant. The viscoelastic material of the investigated medium was assumed to be of the Kelvin–Voigt type. The Laplace transform method provides general solutions to the studied field variables equations. The effects of viscosity and thermal variability parameters on these fields are discussed and graphically presented. In addition, the numerical results were presented in tables, and a comparison with previous models was made to ensure the accuracy of the results of the proposed model.

This article deals with a new modified heat conduction model with fractional order that includes the Caputo–Fabrizio differential operator (CF) and the thermal relaxation time. This new approach to the CF fractional derivative has attracted many researchers because it includes a nonsingular kernel. The nonlocal theory proposed by Eringen has also been applied to demonstrate the effect of scale-dependent thermoelastic materials. The problem of thermal isotropic semi-infinite space is addressed as an application of the presented model. The medium is exposed to regularly changing heat sources and is initially placed in a continuous external magnetic field. The system of governing equations was expressed in the field of the Laplace transform, and the problem in this field was solved by the state-space operation. The inverse of the transformed expressions of physical quantities is found numerically using Zakian’s algorithm. The effects of the nonlocal parameter, the fractal order parameter, and the magnetic field were graphically presented and analyzed in detail. Some of the previous investigations were extracted in some special cases.