Thermal Diffusion Responses in an Infinite Medium with a ‎Spherical Cavity using the Atangana-Baleanu Fractional ‎Operator

Document Type : Research Paper


1 Department of Mathematics, College of Science, Qassim University, P.O. Box 6644, Buraydah 51482, Saudi Arabia

2 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt ‎


The main purpose of this study is to deal with a thermoelastic medium containing a spherical cavity within the framework of partial elastic thermal diffusion theory based on the Atangana-Baleanu operator which is characterized by the presence of a non-local single kernel. The chemical potential of the adjacent cavity is taken as a time-dependent function. The governing equations are represented and solved in the Laplace transform domain and the numerical solutions to the Laplace inversion are obtained to address the problem in the physical domain. In the physical field, the expansion of the Fourier approach is also used to obtain the numerical solutions and the stress-strain behavior of the studied medium is graphically illustrated.


Main Subjects

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‎[1]‎ Tripathi, J.J., Warbhe, S.D., Deshmukh, K.C., Verma J., Fractional Order Thermoelastic Deflection in a Thin Circular Plate, Applications and Applied ‎Mathematics Journal, 12(2), 2017, Article 17.‎
‎[2]‎ Abouelregal, A., Ahmad, H., A Modified Thermoelastic Fractional Heat Conduction Model ‎with a Single-Lag and Two Different Fractional-‎Orders, Journal of Applied and Computational Mechanics, 7(3), 2021, 1676-1686.‎
‎[3]‎ Jajarmi, A., Baleanu, D., On the fractional optimal control problems with a general derivative operator, Asian Journal of Control, 32(2), 2021, 1062-‎‎1071.‎
‎[4]‎ Piccolo, V., Alaimo, G., Chiappini, A., Ferrari, M., Zonta, D., Zingales, M., Deseri, L., Fractional-Order Theory of Thermoelasticity II: Quasi-Static ‎Behavior of Bars, Journal of Engineering Mechanics, 144(2), 2018, 346-354.‎
‎[5]‎ Caputo, M., Fabrizio, M., A new definition of fractional derivative without singular kernel, Progress in Fractional Differentiation and Applications, 2, ‎‎2015, 73–85.‎
‎[6]‎ Atangana, A., Baleanu, D., New fractional derivative without singular kernel: Theory and application to heat transfer modal, Journal of Thermal ‎Science, 20(2), 2016, 763–769.‎
‎[7]‎ Caputo, M., Mainardi, F., A new dissipation model based on memory mechanism, Pure and Applied Geophysics, 91(1), 1971, 134–147.‎
‎[8]‎ Alkahtani, B., Atangana, A., Analysis of non-homogeneous heat model with new trend of derivative with fractional order, Chaos, Solitons & ‎Fractals, 89, 2016, 566–571.‎
‎[9]‎ Atangana, A., Alqahtani, R., New numerical method and application to Keller-Segel model with fractional order derivative, Chaos, Solitons & ‎Fractals, 116, 2018, 14–21.‎
‎[10]‎ Owolabi, K., Atangana, A., Computational study of multi-species fractional reaction-diffusion system with ABC operator, Chaos, Solitons & ‎Fractals, 128, 2019, 280–289.‎
‎[11]‎ Owolabi, K., Hammouch, Z., Spatiotemporal patterns in the Belousov-Zhabotinskii reaction systems with Atangana-Baleanu fractional order ‎derivative, Physica A: Statistical Mechanics and its Applications, 523, 2019, 1072–1090.‎
‎[12]‎ Parveen, L., Fractional order thermoelastic thick circular plate with two temperatures in frequency domain, Journal of Applications and Applied ‎Mathematics, 13(2), 2018, 1216 – 1229.‎
‎[13]‎ Povstenko, Y., Kyrylych, T., Fractional thermoelasticity problem for an infinite solid with a penny-shaped crack under prescribed heat flux ‎across its surfaces, Journal of Philosophical Transactions of the Royal Society A, 378(2172), 2020, 1471-2962.‎
‎[14]‎ Abouelregal, A.E., Fractional derivative Moore-Gibson-Thompson heat equation without singular kernel for a thermoelastic medium with a ‎cylindrical hole and variable properties, ZAMM‐Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, ‎‎102(1), 2021, e202000327.‎
‎[15]‎ Owolabi, K., Hammouch, Z., Spatiotemporal patterns in the Belousov-Zhabotinskii reaction systems with Atangana-Baleanu fractional order ‎derivative, Physica A: Statistical Mechanics and its Applications, 523, 2019, 1072–1090.‎
‎[16]‎ Al-Refai, M., Hajji, M.A., Analysis of a fractional eigenvalue problem involving Atangana-Baleanu fractional derivative a maximum principle and ‎applications, Chaos: An Interdisciplinary Journal of Nonlinear Science, 29, 2019, 13135.‎
‎[17]‎ Rakhi, T., Ravi K., Analysis of Plane wave propagation under the purview of three phase lags theory of thermoelasticity with non-local effect, ‎European Journal of Mechanics - A/Solids, 88, 2021, 104235.‎
‎[18]‎ Tiwari, R., Kumar, R., Abouelregal, A.E., Analysis of a magneto-thermoelastic problem in a piezoelastic medium using the non-local memory-‎dependent heat conduction theory involving three phase lags, Mechanics of Time-Dependent Materials, 2021, 1-17.‎
‎[19]‎ Rakhi, T., Santwana, M., Boundary Integral Equations Formulation for Fractional Order Thermoelasticity, Computational Methods in Science and ‎Technology, 20(2), 2014, 49-58‎
‎[20]‎ Lord, H., Shulman, Y., A generalized dynamical theory of thermoelasticity, Journal of the Mechanics and Physics of Solids, 15(5), 1967, 299–309.‎
‎[21]‎ Green, A.E., Lindsay, K.E., Thermoelasticity, J. Elasticity, 2, 1972, 1–7.‎
‎[22]‎ Green, A.E., Naghdi, P.M., A re-examination of the basic postulates of thermomechanics, Proceedings of the Royal Society A, 432, 1991, 171–94.‎
‎[23]‎ Green, A.E., Naghdi, P.M., On undamped heat waves in an elastic solid, Journal of Thermal Stresses, 15, 1992, 253–64.‎
‎[24]‎ Green, A.E., Naghdi, P.M., Thermoelasticity without energy dissipation, Journal of Elasticity, 31, 1993, 189–208.‎
‎[25]‎ Dhaliwal, R., Sherief, H., Generalized thermoelasticity for anisotropic media, The Quarterly of Applied Mathematics, 38(1), 1980, 1–8.‎
‎[26]‎ Sherief, H., Hamza, F., El-Sayed, A., Theory of generalized micropolar thermoelasticity and an axisymmetric half-space problem, Journal of ‎Thermal Stresses, 28, 2005, 409–437.‎
‎[27]‎ Ghosh, D., Lahiri, A., A Study on the Generalized Thermoelastic Problem for an Anisotropic Medium, Journal of Heat Transfer, 140(9), 2018, 094501.‎
‎[28]‎ Abouelregal, A.E., Thermoelastic fractional derivative model for exciting viscoelastic microbeam resting on Winkler foundation, Journal of ‎Vibration and Control, 27(17), 2020, 2123-2135.‎
‎[29]‎ Sadeghi, M., Kiani, Y., Generalized magneto-thermoelasticity of a layer based on the Lord–Shulman and Green–Lindsay theories, Journal of ‎Thermal Stresses, 45(2), 2022, 101-171.‎
‎[30]‎ Marin, M., An uniqueness result for body with voids in linear thermoelasticity, Rendiconti di Matematica, 17(7), 1997, 103-113.‎
‎[31]‎ Marin, M., On the domain of influence in thermoelasticity of bodies with voids, Archivum Mathematicum, 33(4), 1997, 301-308.‎
‎[32]‎ Marin, M., Marinescu, C., Thermoelasticity of initially stressed bodies. Asymptotic equipartition of energies, International Journal of Engineering ‎Science, 36(1), 1998, 73- 86.‎
‎[33]‎ Rakhi, T., Santwana, M., On electromagneto-thermoelastic plane waves under Green–Naghdi theory of thermoelasticity-II, Journal of Thermal ‎Stresses, 40(8), 2017, 1040-1062.‎
‎[34]‎ Rakhi, T., Misra, J.C., Rashmi, P., Magneto-thermoelastic wave propagation in a finitely conducting medium: A comparative study for three types ‎of thermoelasticity I, II, and III, Journal of Thermal Stresses, 44(7), 2021, 785-806.‎
‎[35]‎ Rakhi, T., Magneto-thermoelastic interactions in generalized thermoelastic half-space for varying thermal and electrical conductivity, Waves in ‎Random and Complex Media, 2021, 1-7. ‎
‎[36]‎ Abouelregal, A.E., A Modified Law of Heat Conduction of Thermoelasticity with Fractional Derivative and Relaxation Time, Journal of Molecular and ‎Engineering Materials, 8(1), 2020, 2050003.‎
‎[37]‎ Abouelregal, A.E., A Modified fractional thermoelasticity model with multi-relaxation times of higher order: application to spherical cavity ‎exposed to a harmonic varying heat, Waves in Random and Complex Media, 31(5), 2021, 812-832.‎
‎[38]‎ Abouelregal, A.E., Mohammad-Sedighi, H., Faghidian, S.A., Shirazi, A.H., Temperature-dependent physical characteristics of the rotating ‎nonlocal nanobeams subject to a varying heat source and a dynamic load, Facta Universitatis, Series: Mechanical Engineering, 19(4), 2021, 633-56.‎
‎[39]‎ Sedighi, H.M., Ouakad, H.M., Dimitri, R., Tornabene, F., Stress-driven nonlocal elasticity for the instability analysis of fluid-conveying C-BN ‎hybrid-nanotube in a magneto-thermal environment, Physica Scripta, 95(6), 2020, 065204.‎
‎[40]‎ Sedighi, H.M., Divergence and flutter instability of magneto-thermo-elastic C-BN hetero-nanotubes conveying fluid, Acta Mechanica Sinica, 36(2), ‎‎2020, 381-96.‎
‎[41]‎ Abouelregal, A.E., Sedighi, H.M., The effect of variable properties and rotation in a visco-thermoelastic orthotropic annular cylinder under the ‎Moore–Gibson–Thompson heat conduction model, Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials: Design and ‎Applications, 235(5), 2021, ‪1004-20‬.‬‬‎
‎[42]‎ Nowacki, W., Dynamical Problems of Thermoelastic Diffusion in Solids I, Bull. Acad. Pol. Sci., Ser. Sci. Tech., 22, 1974, 55–64.‎
‎[43]‎ Nowacki, W., Dynamical Problems of Thermoelastic Diffusion in Solids II, Bull. Acad. Pol. Sci., Ser. Sci. Tech., 22, 1974, 129–135.‎
‎[44]‎ Nowacki, W., Dynamical Problems of Thermoelastic Diffusion in Solids I, Bull. Acad. Pol. Sci., Ser. Sci. Tech., 22, 1974, 257–266.‎
‎[45]‎ Nowacki, W., Dynamical Problems of Thermoelastic Diffusion in Elastic Solids, Proc. Vib. Prob., 15, 1974, 105–128.‎
‎[46]‎ Sherief, H., Saleh, H., A Half-Space Problem in the Theory of Generalized Thermoelastic Diffusion, International Journal of Solids and Structure, 42, ‎‎2005, 4484–4493.‎
‎[47]‎ Alzahrani, F.S., Abbas, I.A., Generalized thermoelastic diffusion in a nanoscale beam using eigenvalue approach, Acta Mechanica, 227, 2016, 955–‎‎968.‎
‎[48]‎ Abouelregal, A.E., Hakan, E., Ömer, C., Solution of Moore–Gibson–Thompson Equation of an Unbounded Medium with a Cylindrical Hole, ‎Mathematics, 9(13), 2021, 1536.‎
‎[49]‎ Chenlin, L., Huili, G., Xiaogeng, T., Tianhu, H., Generalized thermoelastic diffusion problems with fractional order strain, European Journal of ‎Mechanics - A/Solids, 78, 2019, 103827.‎
‎[50]‎ Aseem, M., Rajneesh, K., Rekha, R., Two Dimensional Axisymmetric Thermoelastic Diffusion Problem of Micropolar Porous Circular Plate with ‎Dual Phase Lag Model, Mechanics and Mechanical Engineering, 22(4), 2018, 1389–1406.‎
‎[51]‎ Davydov, S.A., Zemskov, A.V., Thermoelastic diffusion phase-lag model for a layer with internal heat and mass sources, International Journal of ‎Heat and Mass Transfer, 183, 2022, 122213.‎
‎[52]‎ Masood, K., Zahoor, I., Awais, A., Stagnation point flow of magnetized Burgers’ nanofluid subject to thermal radiation, Applied Nanoscience, ‎‎10, 2020, 5233–5246. ‎
‎[53]‎ ‎ Zahoor, I., Masood, K., Awais, A., Jawad, A., Abdul, H., Thermal energy transport in Burgers nanofluid flow featuring the Cattaneo–Christov ‎double diffusion theory, Applied Nanoscience, 10, 2020, 5331–5342. ‎
‎[54]‎ Masood, K., Zahoor, I., Awais, A., A mathematical model to examine the heat transport features in Burgers fluid flow due to stretching cylinder, ‎Journal of Thermal Analysis and Calorimetry, 147, 2022, 827–841.‎
‎[55]‎ Zahoor, I., Masood, K., Awais, A.,On modified Fourier heat flux in stagnation point flow of magnetized Burgers' fluid subject to homogeneous–‎heterogeneous reactions, Journal of Thermal Analysis and Calorimetry, 147, 2022, 815–826.‎
‎[56]‎ Zahoor, I., Masood, K., Awais, A., Sohail N., Features of thermophoretic and Brownian forces in Burgers fluid flow subject to Joule heating and ‎convective conditions, Physica Scripta, 96(1), 2021, 015211.‎
‎[57]‎ Zahoor, I., Masood, K., Awais, A., Burgers fluid flow in perspective of Buongiorno’s model with improved heat and mass flux theory for ‎stretching cylinder, The European Physical Journal of Applied Physics, 92(3), 2020, 14.‎
‎[58]‎ Abouelregal, A.E., Generalized mathematical novel model of thermoelastic diffusion with four phase lags and higher-order time derivative, The ‎European Physical Journal Plus, 135(2), 2020, 263.‎
‎[59]‎ Aouadi, M., A problem for an infinite elastic body with a spherical cavity in the theory of generalized thermoelastic diffusion, International Journal ‎of Solids and Structures, 44, 2007, 5711–5722.‎
‎[60]‎ Elhagary, M.A., Fractional thermoelastic diffusion problem for an infinite medium with a spherical cavity using Modified Caputo-Fabrizio’s ‎definition, Waves in Random and Complex Media, 44, 2021, 281-94.‎
‎[61]‎ Sherief, H., Hamza, F., Saleh, H., The theory of generalized thermoelastic diffusion, International Journal of Engineering Science, 42, 2004, 591–608.‎
‎[62]‎ Watson, G.N., A treatise on the theory of Bessel functions, London: Cambridge University Press, 1996.‎
‎[63]‎ Durbin, F., Numerical Inversion of Laplace Transforms: An Efficient Improvement to Dubner and Abate's Method, The Computer Journal, 17(4), ‎‎1973, 371.‎
‎[64]‎ Sherief, H., Raslan, W., 2D problem for a long cylinder in the fractional theory of thermoelasticity, Latin American Journal of Solids and Structures, ‎‎13, 2016, 1596–1613.‎
‎[65]‎ Aouadi, M., A problem for an infinite elastic body with a spherical cavity in the theory of generalized thermoelastic diffusion, International Journal ‎of Solids and Structures, 44, 2007, 5711–5722.‎