# Photo-Thermoelasticity Heat Transfer Modeling with Fractional Differential Actuators for Stimulated Nano-Semiconductor Media

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulation

## 3. Statement of the Problem

## 4. Problem Solution

## 5. Numerical Results

#### 5.1. Implications of the Nonlocal Coefficient

#### 5.2. The Influence of the Fractional Operators

## 6. Conclusions

- Nonlocal factors have a significant role in changing the behavior of thermomechanical interactions in small-sized semiconductor materials. As a result, when modeling nonlocal microstructures, the value of the nonlocal coefficient must be chosen very carefully.
- The new nonlocal photothermal model predicts smaller amounts than those in the case of the traditional (local) photothermal model. For this reason, nanoscale factors must be included in reducing the mechanical wave behavior of (nonlocal) nanostructures.
- The fractional-order index can be used to reclassify semiconductor materials in terms of photoelectric thermal conductivity. The fractional coefficient of the derivative operator of Atangana and Baleanu slightly affects the rate of temperature fluctuation. Thermoplastic models with fractional derivatives have much larger standard deviations than thermoplastic models. As a result, the fractional coefficient is gaining ground as an excellent thermal indicator.
- When using fractional actuators, the values of the thermo-photophysical fields were found to be lower compared to what would be expected by conventional thermophotometric models. Therefore, by varying the fractional parameter, we may be able to estimate the function that the Atangana and Baleanu derivative operators play in heat transfer regimes and perform more detailed examinations of elastic thermal deformation in rigid mechanics. The method and results from this work can also be used to solve similar problems in thermoelasticity and thermodynamics.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Gärtner, W.W. Photothermal Effect in Semiconductors. Phys. Rev.
**1961**, 122, 419. [Google Scholar] [CrossRef] - Jin, H.; Lin, G.; Bai, L.; Zeiny, A.; Wen, D. Steam generation in a nanoparticle-based solar receiver. Nano Energy
**2016**, 28, 397–406. [Google Scholar] [CrossRef] [Green Version] - Liu, F.; Lai, Y.; Zhao, B.; Bradley, R.; Wu, W. Photothermal materials for efficient solar powered steam generation. Front. Chem. Sci. Eng.
**2019**, 13, 636–653. [Google Scholar] [CrossRef] [Green Version] - Abbas, I.; Hobiny, A.D. Analytical-numerical solutions of photothermal interactions in semiconductor materials. Inf. Sci. Lett.
**2021**, 10, 189. [Google Scholar] - Todorovic, D.M.; Galović, S.; Popovic, M. Optically excited plasmaelastic waves in semiconductor plate-coupled plasma and elastic phenomena. J. Phys. Conf. Ser.
**2010**, 214, 012106. [Google Scholar] [CrossRef] [Green Version] - Vasil’ev, A.N.; Sablikov, V.A.; Sandomirskii, V.B. Photothermal and photoacoustic effects in semiconductors and semi-conductor structures. Soviet Phys. J.
**1987**, 30, 544–554. [Google Scholar] [CrossRef] - Abouelregal, A.E.; Sedighi, H.M.; Sofiyev, A.H. Modeling photoexcited carrier interactions in a solid sphere of a sem-iconductor material based on the photothermal Moore–Gibson–Thompson model. Appl. Phys. A
**2021**, 127, 845. [Google Scholar] [CrossRef] - Pradhan, S.C.; Murmu, T. Small scale effect on the buckling of single-layered graphene sheets under biaxial com-pression via nonlocal continuum mechanics. Comput. Mater. Sci.
**2009**, 47, 268–274. [Google Scholar] [CrossRef] - Wang, C.Y.; Murmu, T.; Adhikari, S. Mechanisms of nonlocal effect on the vibration of nanoplates. Appl. Phys. Lett.
**2011**, 98, 153101. [Google Scholar] [CrossRef] - Eringen, A. Theory of nonlocal thermoelasticity. Int. J. Eng. Sci.
**1974**, 12, 1063–1077. [Google Scholar] [CrossRef] - Eringen, A.C. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys.
**1983**, 54, 4703–4710. [Google Scholar] [CrossRef] - Eringen, A.C.; Edelen, D.G.B. On nonlocal elasticity. Int. J. Eng. Sci.
**1972**, 10, 233–248. [Google Scholar] [CrossRef] - Eringen, A.; Wegner, J. Nonlocal Continuum Field Theories. Appl. Mech. Rev.
**2003**, 56, B20–B22. [Google Scholar] [CrossRef] - Biot, M.A. Thermoelasticity and Irreversible Thermodynamics. J. Appl. Phys.
**1956**, 27, 240–253. [Google Scholar] [CrossRef] - Lord, H.; Shulman, Y. A generalized dynamical theory of thermoelasticity. J. Mech. Phys. Solids
**1967**, 15, 299–309. [Google Scholar] [CrossRef] - Green, A.E.; Naghdi, P.M. A re-examination of the basic postulates of thermomechanics. Proc. R. Soc. London Ser. A Math. Phys. Sci.
**1991**, 432, 171–194. [Google Scholar] [CrossRef] - Green, A.E.; Naghdi, P.M. ON UNDAMPED HEAT WAVES IN AN ELASTIC SOLID. J. Therm. Stress.
**1992**, 15, 253–264. [Google Scholar] [CrossRef] - Green, A.E.; Naghdi, P.M. Thermoelasticity without energy dissipation. J. Elast.
**1993**, 31, 189–208. [Google Scholar] [CrossRef] - Lasiecka, I.; Wang, X. Moore–Gibson–Thompson equation with memory, part II: General decay of energy. J. Diff. Eqns.
**2015**, 259, 7610–7635. [Google Scholar] [CrossRef] - Quintanilla, R. Moore-Gibson-Thompson thermoelasticity. Math. Mech. Solids
**2019**, 24, 4020–4031. [Google Scholar] [CrossRef] - Quintanilla, R. Moore-Gibson-Thompson thermoelasticity with two temperatures. Appl. Eng. Sci.
**2020**, 1, 100006. [Google Scholar] [CrossRef] - Abouelregal, A.; Ahmed, I.-E.; Nasr, M.; Khalil, K.; Zakria, A.; Mohammed, F. Thermoelastic Processes by a Continuous Heat Source Line in an Infinite Solid via Moore–Gibson–Thompson Thermoelasticity. Materials
**2020**, 13, 4463. [Google Scholar] [CrossRef] [PubMed] - Abouelregal, A.E.; Marin, M. The response of nanobeams with temperature-dependent properties using state-space method via modified couple stress theory. Symmetry
**2020**, 12, 1276. [Google Scholar] [CrossRef] - Marin, M.; Öchsner, A.; Bhatti, M.M. Some results in Moore-Gibson-Thompson thermoelasticity of dipolar bodies. J. Appl. Math. Mech.
**2020**, 100, e202000090. [Google Scholar] [CrossRef] - Abouelregal, A.E.; Marin, M.; Askar, S. Thermo-Optical Mechanical Waves in a Rotating Solid Semiconductor Sphere Using the Improved Green–Naghdi III Model. Mathematics
**2021**, 9, 2902. [Google Scholar] [CrossRef] - Abouelregal, A.; Ersoy, H.; Civalek, Ö. Solution of Moore–Gibson–Thompson Equation of an Unbounded Medium with a Cylindrical Hole. Mathematics
**2021**, 9, 1536. [Google Scholar] [CrossRef] - Podlubny, I.; Chechkin, A.; Skovranek, T.; Chen, Y.; Jara, B.M.V. Matrix approach to discrete fractional calculus II: Partial fractional differential equations. J. Comput. Phys.
**2009**, 228, 3137–3153. [Google Scholar] [CrossRef] [Green Version] - Hobiny, A.; Abbas, I. The Effect of a Nonlocal Thermoelastic Model on a Thermoelastic Material under Fractional Time Derivatives. Fractal Fract.
**2022**, 6, 639. [Google Scholar] [CrossRef] - Podlubny, I. Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equa-tions, to Methods of Their Solution and Some of Their Applications; Elsevier: Amsterdam, The Netherlands, 1998; Volume 198. [Google Scholar]
- Wu, X.; Chen, G.Y.; Owens, G.; Chu, D.; Xu, H. Photothermal materials: A key platform enabling highly efficient water evaporation driven by solar energy. Mater. Today Energy
**2019**, 12, 277–296. [Google Scholar] [CrossRef] - Atangana, A.; Baleanu, D. New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model. Therm. Sci.
**2016**, 20, 763–769. [Google Scholar] [CrossRef] [Green Version] - Caputo, M.; Fabrizio, M. A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl.
**2015**, 1, 73–85. [Google Scholar] - Bachher, M.; Sarkar, N. Nonlocal theory of thermoelastic materials with voids and fractional derivative heat transfer. Waves Random Complex Media
**2019**, 29, 595–613. [Google Scholar] [CrossRef] - Ebrahimi, F.; Haghi, P. Elastic wave dispersion modelling within rotating functionally graded nanobeams in thermal environment. Adv. Nano Res.
**2018**, 6, 201–217. [Google Scholar] - Kaur, I.; Singh, K. Plane wave in non-local semiconducting rotating media with Hall effect and three-phase lag fractional order heat transfer. Int. J. Mech. Mater. Eng.
**2021**, 16, 14. [Google Scholar] [CrossRef] - Zhou, H.; Shao, D.; Song, X.; Li, P. Three-dimensional thermoelastic damping models for rectangular micro/nanoplate resonators with nonlocal-single-phase-lagging effect of heat conduction. Int. J. Heat Mass Transf.
**2022**, 196, 123271. [Google Scholar] [CrossRef] - Todorović, D.M. Plasma, thermal, and elastic waves in semiconductors. Rev. Sci. Instrum.
**2003**, 74, 582–585. [Google Scholar] [CrossRef] - Song, Y.Q.; Bai, J.T.; Ren, Z.Y. Study on the reflection of photothermal waves in a semiconducting medium under gen-eralized thermoelastic theory. Acta Mech.
**2012**, 223, 1545–1557. [Google Scholar] [CrossRef] - Nonnenmacher, T.F.; Metzler, R. On the Riemann-Liouville Fractional Calculus and Some Recent Applications. Fractals
**1995**, 3, 557–566. [Google Scholar] [CrossRef] - Wang, J.; Ye, Y.; Pan, X.; Gao, X.; Zhuang, C. Fractional zero-phase filtering based on the Riemann–Liouville integral. Signal Process.
**2014**, 98, 150–157. [Google Scholar] [CrossRef] - Honig, G.; Hirdes, U. A method for the numerical inversion of Laplace transforms. J. Comput. Appl. Math.
**1984**, 10, 113–132. [Google Scholar] [CrossRef] [Green Version] - Abouelregal, A.E.; Sedighi, H.M.; Shirazi, A.H. The Effect of Excess Carrier on a Semiconducting Semi-Infinite Medium Subject to a Normal Force by Means of Green and Naghdi Approach. Silicon
**2021**, 14, 4955–4967. [Google Scholar] [CrossRef] - Sumelka, W.; Zaera, R.; Fernández-Sáez, J. A theoretical analysis of the free axial vibration of nonlocal rods with frac-tional continuum mechanics. Meccanica
**2015**, 50, 2309–2323. [Google Scholar] [CrossRef] [Green Version] - Karličić, D.; Kozić, P.; Pavlović, R. Free transverse vibration of nonlocal viscoelastic orthotropic multi-nanoplate system (MNPS) embedded in a viscoelastic medium. Compos. Struct.
**2014**, 115, 89–99. [Google Scholar] [CrossRef] - Aydogdu, M. Axial vibration of the nanorods with the nonlocal continuum rod model. Phys. E Low-Dimens. Syst. Nanostructures
**2009**, 41, 861–864. [Google Scholar] [CrossRef] - Kaminski, W. Hyperbolic heat conduction for materials with a non-homogeneous inner structure. ASME J. Heat Transf.
**1990**, 112, 555–560. [Google Scholar] [CrossRef] - Wall, D.J.N.; Olsson, P. Invariant imbedding and hyperbolic heat waves. J. Math. Phys.
**1997**, 38, 1723–1749. [Google Scholar] [CrossRef] [Green Version] - Mitra, K.; Kumar, S.; Vedevarz, A.; Moallemi, M.K. Experimental Evidence of Hyperbolic Heat Conduction in Processed Meat. J. Heat Transf.
**1995**, 117, 568–573. [Google Scholar] [CrossRef] - Abdel-Hamid, B. Modelling non-Fourier heat conduction with periodic thermal oscillation using the finite integral transform. Appl. Math. Model.
**1999**, 23, 899–914. [Google Scholar] [CrossRef] [Green Version] - Abouelregal, A.E. A novel model of nonlocal thermoelasticity with time derivatives of higher order. Math. Methods Appl. Sci.
**2020**, 43, 6746–6760. [Google Scholar] [CrossRef] - Cong, P.H.; Duc, N.D. Effect of nonlocal parameters and Kerr foundation on nonlinear static and dynamic stability of micro/nano plate with graphene platelet reinforcement. Thin-Walled Struct.
**2023**, 182, 110146. [Google Scholar] [CrossRef] - Numanoğlu, H.M.; Ersoy, H.; Akgöz, B.; Civalek, Ö. A new eigenvalue problem solver for thermo-mechanical vibration of Timoshenko nanobeams by an innovative nonlocal finite element method. Math. Methods Appl. Sci.
**2021**, 45, 2592–2614. [Google Scholar] [CrossRef] - Herwig, H.; Beckert, K. Fourier Versus Non-Fourier Heat Conduction in Materials with a Nonhomogeneous Inner Structure. J. Heat Transf.
**1999**, 122, 363–365. [Google Scholar] [CrossRef] - Sheikh, N.A.; Ali, F.; Saqib, M.; Khan, I.; Alam Jan, S.A.; Alshomrani, A.S.; Alghamdi, M.S. Comparison and analysis of the Atangana–Baleanu and Caputo–Fabrizio fractional derivatives for generalized Casson fluid model with heat generation and chemical reaction. Results Phys.
**2017**, 7, 789–800. [Google Scholar] [CrossRef] - Ali, F.; Alam Jan, S.A.; Khan, I.; Gohar, M.; Sheikh, N.A. Solutions with special functions for time fractional free convection flow of Brinkman-type fluid. Eur. Phys. J. Plus
**2016**, 131, 310. [Google Scholar] [CrossRef] - Khan, A.; Abro, K.A.; Tassaddiq, A.; Khan, I. Atangana–Baleanu and Caputo Fabrizio Analysis of Fractional Derivatives for Heat and Mass Transfer of Second Grade Fluids over a Vertical Plate: A Comparative Study. Entropy
**2017**, 19, 279. [Google Scholar] [CrossRef] [Green Version] - Xu, H.; Zhang, L.; Wang, G. Some New Inequalities and Extremal Solutions of a Caputo–Fabrizio Fractional Bagley–Torvik Differential Equation. Fractal Fract.
**2022**, 6, 488. [Google Scholar] [CrossRef] - Alqahtani, R.T. Fixed-point theorem for Caputo--Fabrizio fractional Nagumo equation with nonlinear diffusion and convection. J. Nonlinear Sci. Appl.
**2016**, 9, 1991–1999. [Google Scholar] [CrossRef] [Green Version] - Goufo, E.F.D. Application of the Caputo-Fabrizio fractional derivative without singular kernel to Korteweg-de Vries-Bergers equation. Math. Model. Anal.
**2016**, 21, 188–198. [Google Scholar] [CrossRef]

**Figure 5.**The dimensionless carrier intensity $N$ under the influence of small-scale coefficient $\xi $.

**Figure 6.**Effect of the AB and RL fractional derivative operators on the dimensionless temperature $\theta $.

**Figure 7.**Effect of the AB and RL fractional derivative operators on the nonlocal thermal stress ${\sigma}_{zz}$.

**Figure 8.**Effect of the CF and RL fractional derivative operators on the dimensionless displacement $w$.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Askar, S.; Abouelregal, A.E.; Marin, M.; Foul, A.
Photo-Thermoelasticity Heat Transfer Modeling with Fractional Differential Actuators for Stimulated Nano-Semiconductor Media. *Symmetry* **2023**, *15*, 656.
https://doi.org/10.3390/sym15030656

**AMA Style**

Askar S, Abouelregal AE, Marin M, Foul A.
Photo-Thermoelasticity Heat Transfer Modeling with Fractional Differential Actuators for Stimulated Nano-Semiconductor Media. *Symmetry*. 2023; 15(3):656.
https://doi.org/10.3390/sym15030656

**Chicago/Turabian Style**

Askar, Sameh, Ahmed E. Abouelregal, Marin Marin, and Abdelaziz Foul.
2023. "Photo-Thermoelasticity Heat Transfer Modeling with Fractional Differential Actuators for Stimulated Nano-Semiconductor Media" *Symmetry* 15, no. 3: 656.
https://doi.org/10.3390/sym15030656