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# Model of Fractional Heat Conduction in a Thermoelastic Thin Slim Strip under Thermal Shock and Temperature-Dependent Thermal Conductivity

F. S. Bayones1, S. M. Abo-Dahab2,*, Ahmed E. Abouelregal3, A. Al-Mullise1, S. Abdel-Khalek1,4, E. M. Khalil1,5

1 Department of Mathematics and Statistics, College of Science, Taif University, Taif, 21944, Saudi Arabia
2 Department of Mathematics, Faculty of Science, South Valley University, Qena, 83523, Egypt
3 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt
4 Department of Mathematics, Faculty of Science, Sohag University, Sohag, Egypt
5 Department of Mathematics, Faculty of Science, Al-Azhar University, Giza, Egypt

* Corresponding Author: S. M. Abo-Dahab. Email:

Computers, Materials & Continua 2021, 67(3), 2899-2913. https://doi.org/10.32604/cmc.2021.012583

## Abstract

The present paper paper, we estimate the theory of thermoelasticity a thin slim strip under the variable thermal conductivity in the fractional-order form is solved. Thermal stress theory considering the equation of heat conduction based on the time-fractional derivative of Caputo of order α is applied to obtain a solution. We assumed that the strip surface is to be free from traction and impacted by a thermal shock. The transform of Laplace (LT) and numerical inversion techniques of Laplace were considered for solving the governing basic equations. The inverse of the LT was applied in a numerical manner considering the Fourier expansion technique. The numerical results for the physical variables were calculated numerically and displayed via graphs. The parameter of fractional order effect and variation of thermal conductivity on the displacement, stress, and temperature were investigated and compared with the results of previous studies. The results indicated the strong effect of the external parameters, especially the time-fractional derivative parameter on a thermoelastic thin slim strip phenomenon.

## Keywords

F. S. Bayones, S. M. Abo-Dahab, A. E. Abouelregal, A. Al-Mullise, S. Abdel-Khalek et al., "Model of fractional heat conduction in a thermoelastic thin slim strip under thermal shock and temperature-dependent thermal conductivity," Computers, Materials & Continua, vol. 67, no.3, pp. 2899–2913, 2021.

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