The current research aims to investigate thermodynamic responses to thermal media based on a modified mathematical model in the field of thermoelasticity. In this context, it was considered to present a new model with a fractional time derivative that includes Caputo-Fabrizio and Atangana-Baleanu fractional differential operators within the framework of the two-phase delay model. The proposed mathematical model is employed to examine the problem of an unbounded material with a spherical hole experiencing a reduced moving heat flow on its inner surface. The problem is solved analytically within the modified space utilizing the Laplace transform as the solution mechanism. An arithmetic inversion of the Laplace transform was performed and presented visually and tabularly for the studied distributions. In the tables, specific comparisons are introduced to evaluate the influences of different fractional operators and thermal properties on the response of all the fields examined.

In this paper, a new fractional model of non-Fourier heat conduction is presented that includes phase delays and two fractional orders. To derive the proposed model, the fractional integral Atangana-Baleanu (AB) operator with non-singular and non-local kernels was used. The proposed model has been applied to solve a one-dimensional thermoelasticity problem that includes an annular cylinder of a flexible material whose inner and outer surfaces are subjected to a variable heat flux that depends on time and temperature and is free from traction. The Laplace transform approach was applied to find the general solution to the problem and to obtain the expressions for the different physical fields. To estimate the effects of the fractional-order parameters and instantaneous time on the responses of all thermophysical field variables, comparisons are presented in figures and tables.

This study investigated the impact of the fractional Caputo-tempered two-phase-lag (FCT-TPL) heat conduction model on thermoelastic vibrations within a medium containing spherical cavities and voids. In the proposed model, the nonlocality of time and space is integrated to unify classical and generalized thermoelastic theories, enabling a thorough investigation of size-dependent phenomena and the scattering characteristics of thermo-mechanical waves. Also, by integrating fractional calculus with tempered derivatives, the proposed model adeptly captures the complex interaction between localized thermal effects and nonlocal mechanical responses, particularly in materials with pronounced microstructural features. The fractional order and tempering parameter are shown to play a crucial role in controlling thermal relaxation times and the amplitude of thermoelastic vibrations. The findings reveal that the integration of the fractional Caputo-tempered derivative, along with temporal and spatial nonlocal effects, into the two-phase-lag model significantly improves the accuracy of predicting transient thermoelastic responses in materials with cavities and voids.

In this work, we present the nonlocal Moore-Gibson-Thompson photothermal (NMGTPT) theory, a novel framework that integrates spatial and temporal nonlocality to address limitations in both traditional and advanced thermoelastic models. Specifically tailored for semiconductor materials with microstructural features, memory effects, and photo-excited phenomena, the NMGTPT theory unifies nonlocal elasticity, MGT thermal relaxation, and photothermal effects to model the complex interplay between heat, deformation, and photo-induced processes. Unlike prior models, the NMGTPT framework incorporates spatial and temporal nonlocalities, enabling the accurate representation of long-range interactions and memory effects. Additionally, the Atangana-Baleanu (AB) fractional operator is integrated into the NMGTPT model to further enhance its ability to describe nonlocal and memory-dependent behavior, making it particularly suitable for advanced material systems. By incorporating a thermal relaxation coefficient, the framework ensures finite-speed thermal wave propagation, effectively addressing the unrealistic prediction of infinite heat speed found in classical models. The theory also integrates photo-excited free carriers, thermal waves, and acoustic waves, proving highly effective in photothermal and photoacoustic studies involving semiconductors. With the inclusion of an internal length scale, the NMGTPT theory successfully captures size-dependent behaviors, which are essential for accurately modeling nanostructured materials, thin films, and composites. This innovation provides a robust platform for investigating the complex dynamics of photothermal and thermoelastic phenomena in advanced material systems.

The objective of this study was to investigate the thermodynamic reactions of thermoelastic materials by utilizing a modified mathematical fractional thermoelastic model. This model combines a fractional derivative with Rabotnov's exponential kernel and the idea of a two-phase delay, which makes it possible to show thermoelastic behavior more accurately. The model was utilized to investigate an unbounded material with a spherical cavity subjected to a decreasing and shifting heat flux on its inner surface. The problem was solved using analytical approaches, with a strong focus on the Laplace transform. The transform was numerically inverted to provide time-domain results. The study presented graphs that compared the outcomes of utilizing a single kernel fractional derivative with the results obtained using the Rabotnov kernel and fractional order. These graphs showed how the Rabotnov kernel and fractional order affected the physical fields under investigation. This novel theoretical framework has the potential to be advantageous in diverse domains, including engineering, solid mechanics, and materials science.

In this study, we presented a novel fractional nonlocal thermoelastic heat conduction model that extends the Guyer–Krumhansl framework by incorporating size-dependent nonlocal thermal effects and non-Fourier heat conduction characteristics. The model extends the traditional approach using the single-phase-lag (SPL) method derived from Moore–Gibson–Thompson (MGT) heat theory. By employing the Atangana–Baleanu (AB) fractional derivative with a non-singular kernel, we integrated nonlocal features through fractional derivatives, enhancing its applicability to complex thermal behaviors in materials exhibiting combined nonlocal and fractional dynamics. To validate the model, thermoelastic interactions were examined in a long, hollow cylinder subjected to a uniform electromagnetic field. The outer surface was thermally insulated and traction-free, while the inner surface, also traction-free, experienced thermal shock. Governing equations were solved using the Laplace transform method, and numerical solutions were obtained via the Dubner–Abate algorithm. The results were compared with conventional and generalized thermoelastic models to assess accuracy and effectiveness. Additional analysis explored material properties through graphical data, considering various fractional orders and operators, thereby enriching the understanding of system behavior under different conditions. The findings demonstrated the advantages of the fractional nonlocal thermoelastic model in capturing complex thermal interactions within advanced materials, contributing significantly to heat conduction theory.