Generalized thermoelasticity based on higher-order memory-dependent derivative with time delay

Abstract In this investigation, based on the theory of generalized thermoelasticity and memory-dependent derivative (MDD) with time delay, a new model of heat conduction has been constructed. The new model has been incorporated by introducing the higher-order Taylor’s series expansion of Fourier’s law involving MDD with a kernel function. The derived model is an extension and generalization of many models presented in this field which can be obtained as special cases. The thermoelastic vibrations in an infinite medium that is exposed to an instant heat source and a concentrated magnetic field have been discussed based on the formulation model. The solutions and the numerical results of the studied fields are obtained by using the Laplace transform technique. Some comparisons with figures and tables have been presented to examine the effects of various choices for kernel function, time delay and the higher order of derivatives in all fields studied.

[1]  A. Abouelregal,et al.  Analysis of a functionally graded thermopiezoelectric finite rod excited by a moving heat source , 2020 .

[2]  Wei Hu,et al.  A novel generalized thermoelasticity model based on memory-dependent derivative , 2014 .

[3]  S. Mondal.  Memory Response in a Magneto-Thermoelastic Rod with Moving Heat Source Based on Eringen’s Nonlocal Theory Under Dual-Phase Lag Heat Conduction , 2020 .

[4]  D. Tzou.  Experimental support for the lagging behavior in heat propagation , 1995 .

[5]  S. Mondal.  Interactions due to a moving heat source in a thin slim rod under memory-dependent dual-phase lag magneto-thermo-visco-elasticity , 2019, Mechanics of Time-Dependent Materials.

[6]  Hijaz Ahmad,et al.  Reproducing kernel method for Fangzhu's oscillator for water collection from air , 2020 .

[7]  D. Tzou.  A Unified Field Approach for Heat Conduction From Macro- to Micro-Scales , 1995 .

[8]  Francesco Mainardi,et al.  Linear models of dissipation in anelastic solids , 1971 .

[9]  A. Akgül.  A novel method for a fractional derivative with non-local and non-singular kernel , 2018, Chaos, Solitons & Fractals.

[10]  Analysis of New Trends of Fractional Differential Equations , 2020 .

[11]  Rakhi Tiwari,et al.  Analysis of wave propagation in the presence of a continuous line heat source under heat transfer with memory dependent derivatives , 2018 .

[12]  P. M. Naghdi,et al.  ON UNDAMPED HEAT WAVES IN AN ELASTIC SOLID , 1992 .

[13]  A. El-Bary,et al.  Electro-thermoelasticity theory with memory-dependent derivative heat transfer , 2016 .

[14]  Abdon Atangana,et al.  Models of fluid flowing in non-conventional media: New numerical analysis , 2020, Discrete & Continuous Dynamical Systems - S.

[15]  Jin-Liang Wang,et al.  Surpassing the fractional derivative: Concept of the memory-dependent derivative , 2011, Comput. Math. Appl..

[16]  A. Abouelregal.  Three-phase-lag thermoelastic heat conduction model with higher-order time-fractional derivatives , 2019, Indian Journal of Physics.

[17]  Phatiphat Thounthong,et al.  Analytic approximate solutions for some nonlinear Parabolic dynamical wave equations , 2020, Journal of Taibah University for Science.

[18]  Abdon Atangana,et al.  Can transfer function and Bode diagram be obtained from Sumudu transform , 2020 .

[19]  Multistage Optimal Homotopy Asymptotic Method for the Nonlinear Riccati Ordinary Differential Equation in Nonlinear Physics , 2020, Applied Mathematics & Information Sciences.

[20]  Yu-Ming Chu,et al.  Modified Variational Iteration Algorithm-II: Convergence and Applications to Diffusion Models , 2020, Complex..

[21]  A. Akgül,et al.  Solutions of fractional gas dynamics equation by a new technique , 2019, Mathematical Methods in the Applied Sciences.

[22]  Sonal Jain,et al.  Numerical analysis for the fractional diffusion and fractional Buckmaster equation by the two-step Laplace Adam-Bashforth method , 2018 .

[23]  Said Mesloub,et al.  Even Higher Order Fractional Initial Boundary Value Problem with Nonlocal Constraints of Purely Integral Type , 2019, Symmetry.

[24]  Sonal Jain,et al.  Mathematical model of Ebola and Covid-19 with fractional differential operators: Non-Markovian process and class for virus pathogen in the environment , 2020, Chaos, Solitons & Fractals.

[25]  Guy Jumarie,et al.  Fractional Brownian motions via random walk in the complex plane and via fractional derivative. Comparison and further results on their Fokker–Planck equations , 2004 .

[26]  P. Thounthong,et al.  Generalized thermoviscoelastic model with memory dependent derivatives and multi-phase delay for an excited spherical cavity , 2020, Physica Scripta.

[27]  G. Jumarie,et al.  Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results , 2006, Comput. Math. Appl..

[28]  A. El-Bary,et al.  Memory-dependent derivatives theory of thermo-viscoelasticity involving two-temperature , 2015 .

[29]  S. Mondal,et al.  Transient response in a piezoelastic medium due to the influence of magnetic field with memory-dependent derivative , 2019, Acta Mechanica.

[30]  S. Choudhuri,et al.  On A Thermoelastic Three-Phase-Lag Model , 2007 .

[31]  A. Abouelregal,et al.  Thermodynamic modeling of viscoelastic thin rotating microbeam based on non-Fourier heat conduction , 2021 .

[32]  Abdon Atangana,et al.  Analysis of fractal fractional differential equations , 2020 .

[33]  K. Diethelm.  The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type , 2010 .

[34]  V. Zampoli.  Uniqueness theorems about high-order time differential thermoelastic models , 2018 .

[35]  H. Lord,et al.  A GENERALIZED DYNAMICAL THEORY OF THERMOELASTICITY , 1967 .

[36]  A. Abouelregal.  On Green and Naghdi Thermoelasticity Model without Energy Dissipation with Higher Order Time Differential and Phase-Lags , 2020 .

[37]  M. Caputo.  Linear Models of Dissipation whose Q is almost Frequency Independent-II , 1967 .

[38]  A. Abouelregal,et al.  Functionally Graded Piezoelectric Medium Exposed to a Movable Heat Flow Based on a Heat Equation with a Memory-Dependent Derivative , 2020, Materials.

[39]  Phatiphat Thounthong,et al.  Analysing time-fractional exotic options via efficient local meshless method , 2020 .

[40]  Phatiphat Thounthong,et al.  Numerical simulation of 3-D fractional-order convection-diffusion PDE by a local meshless method , 2020 .

[41]  P. M. Naghdi,et al.  A re-examination of the basic postulates of thermomechanics , 1991, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[42]  Phatiphat Thounthong,et al.  Solution of Multi-Term Time-Fractional PDE Models Arising in Mathematical Biology and Physics by Local Meshless Method , 2020, Symmetry.

[43]  A. Abouelregal,et al.  Response of thermoviscoelastic microbeams affected by the heating of laser pulse under thermal and magnetic fields , 2020, Physica Scripta.

[44]  P. M. Naghdi,et al.  Thermoelasticity without energy dissipation , 1993 .

[45]  Abdon Atangana,et al.  A new numerical approximation of the fractal ordinary differential equation , 2018 .

[46]  M. Sadd.  Thermoelasticity , 2021, Elasticity.

[47]  S. Chiriţă.  On high-order approximations for describing the lagging behavior of heat conduction , 2019 .

[48]  Thermoelastic interaction in a two-dimensional infinite space due to memory-dependent heat transfer , 2017 .

[49]  M. Biot.  Thermoelasticity and Irreversible Thermodynamics , 1956 .

[50]  A. Abouelregal.  Modified fractional thermoelasticity model with multi-relaxation times of higher order: application to spherical cavity exposed to a harmonic varying heat , 2019, Waves in Random and Complex Media.

[51]  A. Abouelregal.  Rotating magneto-thermoelastic rod with finite length due to moving heat sources via Eringen’s nonlocal model , 2019 .

[52]  M. Othman,et al.  Memory-dependent derivative effect on 2D problem of generalized thermoelastic rotating medium with Lord–Shulman model , 2020, Indian Journal of Physics.

[53]  Hijaz Ahmad,et al.  New Oscillation Criteria for Advanced Differential Equations of Fourth Order , 2020, Mathematics.

[54]  H. Sherief.  State space approach to thermoelasticity with two relaxation times , 1993 .

[55]  S. Mondal,et al.  Thermoelastic Interactions in a Slim Strip Due to a Moving Heat Source Under Dual-Phase-Lag Heat Transfer , 2019, Journal of Heat Transfer.

[56]  Y. J. Yan,et al.  Influences of longitudinal magnetic field on wave propagation in carbon nanotubes embedded in elastic matrix , 2010 .