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Purpose The system of equations for fractional thermo-viscoelasticity is used to investigate two-dimensional bioheat transfer and heat-induced mechanical response in human skin tissue with rheological properties. Design/methodology/approach Laplace and Fourier’s transformations are used. The resulting formulation is applied to human skin tissue subjected to regional hyperthermia therapy for cancer treatment. The inversion process for Fourier and Laplace transforms is carried out using a numerical method based on Fourier series expansions. Findings Comparisons are made with the results anticipated through the coupled and generalized theories. The influences of volume materials properties and fractional order parameters for all the regarded fields are examined. The results indicate that volume relaxation parameters, as well as fractional order parameters, play a major role in all considered distributions. Originality/value Bio-thermo-mechanics includes bioheat transfer, biomechanics, burn injury and physiology. In clinical applications, knowledge of bio-thermo-mechanics in living tissues is very important. One can infer from the numerical results that, with a finite distance, the thermo-mechanical waves spread to skin tissue, removing the unrealistic predictions of the Pennes’ model.

Viscoelasticity is a natural property of biological tissue that is often used as a diagnostic parameter of cancer diagnosis. Recently, it was discovered that the fractional calculus accurately describes the experimental effects of viscoelastic materials. In this work, the model of the thermo-viscoelasticity theory of fractional dual-phase-lag heat conduction law with rheological properties of the volume is constructed to investigate one-dimensional bioheat transfer and heat-induced mechanical response in human skin tissue. The model is applied to the one-dimensional problem of the skin tissue layer with an arbitrary thickness. The bounding plane of the skin tissue is subjected to three different types of thermal loading. The method of transforming Laplace was used and the reversal was measured using the Tzuo process. The influences of fractional dual-phase-lag relaxation parameters and rheological properties of the volume on the distributions of temperature, displacement and stress are obtained and illustrated graphically.

The generalized thermo-viscoelastic theory without energy dissipation is used in this work to investigate the transfer of bioheat and the mechanical heat-induced response in a human skin plane tiss...

In the present paper, a new fractional model of fractional order thermo-viscoelasticity theory with and without energy dissipation is constructed. Some basic theorems can easily obtain in both theo...

SEIR model is a widely used and acceptable model to distinguish the outbreak of the COVID-19 epidemic in many countries. In the current work, a new proposed SEIR model as a mathematical model for the outbreak of novel coronaviruses COVID-19 will be constructed. The new proposed SEIR pandemic model provides a new vision for evaluations and management of the epidemic of COVID-19 infection. For mathematical modeling and dynamic analyses, this paper uses the real data of spreading COVID-19 in Saudi Arabia. The dynamics of the proposed SEIR model are presented with the reproduction number and the extensive stability analysis. We discussed the domain of the solution and equilibrium situation based on the proposed SEIR model by using Jacobian's method of linearization. The condition of equilibrium and its uniqueness has been proved, and the stability analysis of disease-free equilibrium has been introduced. A sensitivity analysis of the reproduction number against its internal parameters has been done. The global stability of the equilibrium of this model has been proved by using Lyapunov's Stability theorem. A numerical verification and predictions of the proposed SEIR model have been made with comparing the results based on the SEIR model and the real data due to the spreading of the COVID-19 in Saudi Arabia. The proposed SEIR model is a successful model to analyze the spreading of epidemics like COVID-19. This work introduces the ideal protocol, which can help the Saudi population to breakdown spreading COVID-19 in a fast way.

PurposeIn the present paper, the new concept of “memory dependent derivative” in the Pennes’ bioheat transfer and heat-induced mechanical response in human living tissue with variable thermal conductivity and rheological properties of the volume is considered.Design/methodology/approachA problem of cancerous layered with arbitrary thickness is considered and solved analytically by Kirchhoff and Laplace transformation. The analytical expressions for temperature, displacement and stress are obtained in the Laplace transform domain. The inversion technique for Laplace transforms is carried out using a numerical technique based on Fourier series expansions.FindingsComparisons are made with the results anticipated through the coupled and generalized theories. The influence of variable thermal, volume materials properties and time-delay parameters for all the regarded fields for different forms of kernel functions is examined.Originality/valueThe results indicate that the thermal conductivity and volume relaxation parameters and MDD parameter play a major role in all considered distributions. This dissertation is an attempt to provide a theoretical thermo-viscoelastic structure to help researchers understand the complex thermo-mechanical processes present in thermal therapies.

In this work, the new concept of “memory-dependent derivative” in heat transfer process in a perfectly conducting viscoelastic metal films was used to investigate the action of a thin metal film, irradiated by a femtosecond laser pulse. The temporal profile of the ultrafast laser was regarded as being non-Gaussian. Laplace transform technique is utilized to solve the problem. Analytical results have been obtained for the fields of temperature, displacement and stress fields, as well as induced magnetic and electrical in a metal film. We have attempted to show the significance of a kernel function and a time delay parameter characteristic of memory-dependent derivative heat transfer in field variables actions with the aid of numerical tests. In the absence of memory-dependent derivative parameters, associations are made with the results obtained. The effects of the time delay on all distributions in a metal film for various forms of kernel functions are examined.

In this study, the fractional model of thermo-viscoelasticity theory with variable thermal conductivity and rheological properties of the volume is considered to investigate bio-thermo-mechanics be...

Although the thermo-viscoelastic properties of solids have been reported in many articles, no effort has been made to investigate the coupled thermo-viscoelastic and plasma wave behavior of semicon...

In this work, we construct a mathematical model of two-temperature Green–Naghdi (III) magneto-thermo viscoelasticty theory based on the fractional order of heat transfer. Some essential theorems of coupled and generalized thermo-viscoelasticity can be easily obtained. The Laplace transform and state-space techniques are used to obtain the general solution for any set of boundary conditions. The resulting formulation is applied to the specific problem of a perfect electrically conducting viscoelastic half-space subjected to a moving heat source with constant velocity and ramp-type heating. The inverse transforms are obtained by using a numerical method based on Fourier expansion techniques. The effects of the heat source speed, the two-temperature parameter, Alfven velocity and the ramping time parameter on a polymethyl methacrylate (Plexiglas) material are discussed.

The model of equations of thermo-viscoelasticity with fractional order heat transfer is constructed. Some fundamental theorems on the linear coupled and generalized theories of thermo-visco...

We develop a model of generalized thermoelasticity with memory-dependent derivative (MDD) heat conduction law for a thermoelectric half-space. Some urgent theories take after as most remote point cases. The Laplace transform and state-space procedures are utilized to urge the overall account for any arrangement of limit conditions. The general solution acquired is connected to the particular issue of a half-space exposed to a uniform magnetic field, a moving heat source with consistent speed and ramp-type heating. The inverse Laplace transforms are registered numerically. The impacts of various estimations of the figure-of-merit quantity, heat source speed, MDD parameters, the magnetic number and the ramping time parameter are thought about.

A mathematical model of electro-thermoelasticity has been constructed in the context of a new consideration of heat conduction with phase-lag Green-Naghdi theory without energy dissipation (GN-II)....

A new model of critical epidemic dynamics for the emergence of the new coronavirus COVID-19 is being established in this paper. A new approach to the assessment and control of the COVID 19 epidemic is given with the SEIRQ pandemic model. This paper uses real knowledge on the distribution of COVID-19 in Saudi Arabia for mathematical modeling and dynamic analyses. The reproductive number and detailed stability analysis are provided in the SEIRQ model dynamics. In a Jacobian method of linearization, we will address the domain of the solution and the equilibrium situation based on the SEIRQ model. The equilibrium and its importance have been proven, and a study of the stability of the equilibrium free from diseases has been implemented. The reproduction number was evaluated in accordance with its internal parameters. The Lyapunov theorem of stability has proven the global stability of the current model's equilibrium. The SEIRQ model was contrasted by comparing the results based on the SEIRQ model with the real COVID-19 spread data in Saudi Arabia. Numerical evaluation and predictions were given. The results indicate that the SEIRQ model is a strong model for the study of the spread of epidemics, such as COVID-19. At the end of this work, we implemented an optimum protocol that can quickly stop the spread of COVID-19 among the Saudi populations.

A new mathematical model of generalized magneto-thermo-viscoelasticity theories with memory-dependent derivatives (MDD) of dual-phase-lag heat conduction law is developed. The equations for one-dimensional problems including heat sources are cast into matrix form using the state space and Laplace transform techniques. The resulting formulation is applied to a problem for the whole space with a plane distribution of heat sources. It is also applied to a perfect conducting semi-space problem with a traction-free surface and plane distribution of heat sources located inside the medium. The inversion of the Laplace transforms is carried out using a numerical approach. Numerical results for the temperature, displacement, stress and heat flux distributions as well as the induced magnetic and electric fields are given and illustrated graphically. A comparison is made with the results obtained in the coupled theory. The impacts of the MDD heat transfer parameter and Alfven velocity on a viscoelastic material, for example, poly (methyl methacrylate) (Perspex) are discussed.

A unified mathematical model of electro-thermoelasticity has been constructed in the context of a new consideration of heat conduction law with fractional order derivative. Some essential theories follow as limit cases. The governing coupled equations are applied to several concrete problems: (a) time-dependent thermal shock problem; (b) a problem for a half-space subjected to an arbitrary heating and (c) a layer media problem. Laplace transforms are used to derive the solution in the Laplace transform domain. A numerical method is employed for the inversion of the Laplace transforms. According to the numerical results and its graphs, conclusion about the new theory has been constructed. The predictions of the theory are discussed and compared with dynamic classical coupled theory, Lord–Shulman, Green–Naghdi (GN) and fractional coupled thermoelasticity theories. The result provides a motivation to investigate conducting fractional thermoelectric materials as a new class of applicable materials.

A mathematical model of magneto electro-thermoelasticity has been constructed in the context of a new consideration of fractional Green-Naghdi heat conduction law without energy dissipation. The governing coupled equations are applied to a one-dimensional problem for a perfect conducting spherical cavity subjected to an arbitrary thermal shock in the presence of a uniform magnetic field. By means of the Laplace transform and numerical Laplace inversion, the problems are solved. The distributions of the considered temperature, stress and displacement are represented graphically. Some comparisons are shown in the figures to estimate the effects of the fractional order and relaxation time.

The constitutive equations are given with a fractional Maxwell–Cattaneo heat conduction law using the Caputo fractional derivative and the fractional order heat transport equation is given. The uniqueness theorem is proved, the reciprocity relation is deduced and the variational characterization of solution is given. Seven thermoelasticity theories result from the given problem as special cases (Coupled, Lord–Shulman, phase-lag Green–Naghdi theories, Green–Naghdi theory without energy dissipation and the Green–Naghdi theory of type III).

A unified mathematical model of Green–Naghdi’s thermoelasticty theories (GN), based on fractional time-derivative of heat transfer is constructed. The model is applied to solve a one-dimensional problem of a perfect conducting unbounded body with a cylindrical cavity subjected to sinusoidal pulse heating in the presence of an axial uniform magnetic field. Laplace transform techniques are used to get the general analytical solutions in Laplace domain, and the inverse Laplace transforms based on Fourier expansion techniques are numerically implemented to obtain the numerical solutions in time domain. Comparisons are made with the results predicted by the two theories. The effects of the fractional derivative parameter on thermoelastic fields for different theories are discussed.

A mathematical model of electro-thermoelasticity has been constructed in the context of a new consideration of heat conduction with memory-dependent derivative. The governing coupled equations with time-delay and kernel function, which can be chosen freely according to the necessity of applications, are applied to several concrete problems. The exact solutions for all fields are obtained in the Laplace transform domain for each problem. According to the numerical results and its graphs, conclusion about the proposed model has been constructed. The predictions of the theory are discussed and compared with dynamic classical coupled theory. The result provides a motivation to investigate conducting thermoelectric viscoelastic materials as a new class of applicable materials.

A mathematical model of two-temperature phase-lag Green–Naghdi thermoelasticty theories based on fractional derivative heat transfer is given. The GN theories as well as the theories of coupled and of generalized thermoelasticity with thermal relaxation follow as limit cases. The resulting non dimensional coupled equations together with the Laplace transforms techniques are applied to a specific problem of a half space subjected to arbitrary heating which is taken as a function of time and is traction free. The inverse transforms are obtained by using a numerical method based on Fourier expansion techniques. The predictions of the theory are discussed and compared with those for the generalized theory of thermoelasticity with one relaxation time. The effects of temperature discrepancy and fractional order parameters on copper-like material are discussed in different types of GN theories.

In this work, the model of fractional magneto-thermoelasticity is applied to a one-dimensional thermal shock problem for a functionally graded half-space whose surface is assumed to be traction free and subjected to an arbitrary thermal loading. The Lame’s modulii is taken as functions of the vertical distance from the surface of thermoelastic perfect conducting medium in the presence of a uniform magnetic field. Laplace transform and the perturbation techniques are used to derive the solution in the Laplace transform domain. Numerical inversion of the Laplace transform is carried out to obtain the temperature, displacement, stress and induced magnetic and electric field distributions. Numerical results are represented graphically and discussed.

The model of fractional magneto-thermoelasticity is applied to one-dimensional problems of a thermoelectric spherical shell subjected to an arbitrary thermal loading of its external boundary in the presence of a uniform magnetic field. By means of the Laplace transform and numerical Laplace inversion, the problems are solved. The distributions of the considered temperature, stress and displacement, are represented graphically. The theories of coupled and generalized magneto-thermoelasticity follow as limited cases. Some comparisons are shown in the figures to estimate the effects of the fractional order and ramping parameters.

The model of fractional time-derivative of generalized magneto–thermo–viscoelasticity equations for isotropic media in the presence of a constant magnetic field is considered. Some essential theorems on the linear coupled and generalized theories of thermo-viscoelasticity can be easily obtained. This model is applied to solve a problem of an infinite body with a cylindrical cavity in the presence of an axial uniform magnetic field. The boundary of the cavity is subjected to a combination of thermal and mechanical shock acting for a finite period of time. Laplace transform techniques are used to derive the solution in the Laplace transform domain. The inversion process is carried out using a numerical method based on Fourier series expansions. Numerical computations for the temperature, the displacement and the stress distributions as well as for the induced magnetic and electric fields are carried out and represented graphically. Comparisons are made with the results predicted by the generalizatio...

The present work is concerned with the investigation of disturbances in a homogeneous, isotropic elastic medium with memory-dependent derivatives (MDDs). A one-dimensional problem is considered for a half-space whose surface is traction free and subjected to the effects of thermodiffusion. For treatment of time variations, the Laplace-transform technique is utilized. The theories of coupled and of generalized thermoelastic diffusion with one relaxation time follow as limit cases. A direct approach is introduced to obtain the solutions in the Laplace transform domain for different forms of kernel functions and time delay of MDDs, which can be arbitrarily chosen. Numerical inversion is carried out to obtain the distributions of the considered variables in the physical domain and illustrated graphically. Some comparisons are made and shown in figures to estimate the effects of MDD parameters on all studied fields.

A new mathematical model of generalized thermoelasticity with memory-dependent derivatives for the dual-phase-lag heat conduction law is constructed. The governing equations of the new model are applied to a half-space subjected to ramp-type heating. Laplace transforms technique is used. The solution is obtained for different types of functions representing the thermal shock and for different values of the parameter of the time fraction derivative of the model. The effects of time-delay and arbitrary kernel function on elastic material are studied and represented graphically. The predictions of the theory are discussed and compared with dynamic classical coupled theory.

The University of the corresponding author should be Department of Mathematics, College of Science and Arts, Qassim University, Al Bukairiyah, Saudi Arabia.

A fractional model of the equations of generalized magneto-thermoelasticity for a perfect conducting isotropic thermoelastic media which is assumed to have variable thermal conductivity depending on the temperature is given. This model is applied to solve a problem of an infinite long hollow cylinder in the presence of an axial uniform magnetic field. The solution is obtained by a direct approach. Laplace transform techniques are used to derive the solution in the Laplace transform domain. The inversion process is carried out using a numerical method based on Fourier series expansions. Numerical computations for the temperature, the displacement and the stress distributions as well as the induced magnetic and electric fields are carried out and represented graphically. The results indicate that the thermal conductivity and time-fractional order play a major role in all considered distributions.

Present work deals with the problem of an infinite long hollow cylinder with variable thermal conductivity in the context of generalized thermo-viscoelasticity theory with thermal relaxation. A mapping of Kirchhoff's transformation was used to solve a problem with variable thermal conductivity. The Laplace transform is used. The inversion process is carried out using a numerical method based on Fourier series expansions. Numerical computations for the temperature, the displacement and the stress distributions are carried out and represented graphically. The results indicate that the thermal conductivity play a major role in all considered distributions.

A three-dimensional model of thermoelasticity with fractional order heat transfer is established. The resulting nondimensional coupled equations together with the Laplace and double Fourier transforms techniques are applied to a half space, which is assumed to be traction free and subjected to a thermal shock that is a function of time. The inverses of Fourier transforms and Laplace transforms are obtained numerically. Numerical results for the temperature, the stress, the strain, and the displacement distributions are represented graphically. The predictions of the theory are discussed and compared with those for the coupled and generalized theories of thermoelasticity.

The constitutive equations are derived for the thermoelastic diffusion in anisotropic and isotropic solids, in the context of a new generalized thermoelasticity theory with two time delays and kernel functions. The coupled thermoelastic diffusion and the Lord–Shulman theories result from the given theory as particular cases. For anisotropic solid, the reciprocity theorem is proved; the convolutional variational principle is given and the uniqueness theorem based on the variational principle is proved.

A new mathematical model of thermoelectric MHD theory has been constructed in the context of a new consideration of heat conduction law with time-delay and kernel function. The memory-dependent differential equation of heat transfer is established. This model is applied to Stokes' flow of unsteady incompressible fluid due to a moving flat plate in the presence of both heat sources and a transverse magnetic field. Laplace transforms and state-space techniques will be used to obtain the general solution for any set of boundary conditions. According to the numerical results and its graphs, conclusion about the new theory has been constructed. The predictions of the theory are discussed and compared with dynamic coupled theory for different forms of kernel functions. The thermoelectric effects with time-delay parameter on the temperature and velocity fields are analyzed and discussed in detail with the aid of graphical illustrations.

A fractional model of the equations of generalized magneto-thermoelasticity for a perfect conducting isotropic thermoelastic media is given. This model is applied to solve a problem of an infinite body with a cylindrical cavity in the presence of an axial uniform magnetic field. The boundary of the cavity is subjected to a combination of thermal and mechanical shock acting for a finite period of time. The solution is obtained by a direct approach by using the thermoelastic potential function. Laplace transform techniques are used to derive the solution in the Laplace transform domain. The inversion process is carried out using a numerical method based on Fourier series expansions. Numerical computations for the temperature, the displacement and the stress distributions as well as for the induced magnetic and electric fields are carried out and represented graphically. Comparisons are made with the results predicted by the generalizations, Lord–Shulman theory, and Green–Lindsay theory as well as to the coupled theory.

In this work, we proposed three models of generalized thermoelasticity: a single –phase - lag Green–Naghdi theory of type III, a dual-phase–lag Green–Naghdi theory of type II and of type III. The solid is assumed to be linear anisotropic inhomogeneous. A unified heat conduction law and heat transport equation are given, which consolidate the three theories and also the Lord–Shulman theory and the Green–Naghdi theory of type II. Dissipative inequality is derived, constitutive equations and thermodynamic restrictions are given. The speed of thermal wave propagation is given, for each of the three theories in case of isotropic solid. A uniqueness theorem is proved for the considered theories. Variational characterization of solution is established. An application is given for isotropic thermoelastic solid and the results are presented graphically.

A new mathematical model of two-temperature electro-thermo viscoelasticity theory has been constructed in the context of a new consideration of heat conduction with memory-dependent derivative. The resulting formulation is applied to several concrete problems, a thermal shock problem and a problem for a half-space subjected to ramp-type heating as well as a problem of a layer media. The exact solutions for temperature, stress and displacement are obtained in the Laplace transform domain for each problem. According to the numerical results and its graphs, conclusion about the new theory has been constructed. The predictions of the theory are discussed and compared with dynamic classical coupled theory. The result provides a motivation to investigate conducting thermoelectric viscoelastic materials as a new class of applicable materials.

The new model of the equations of the linear theory of magneto-thermo-viscoelasticity with two relaxation times and fractional heat transfer involving fractional relaxation operator is given. The resulting formulation is applied to thermal shock problems for a perfect electrically conducting half-space in the presence of a transverse magnetic field. Laplace transform techniques are used. Some essential theorems on the linear coupled and generalized theories of thermo-viscoelasticity with two relaxation times are established. According to the numerical results and its graphs, conclusion about the new theory has been constructed. The effects of the fractional relaxation operator on viscoelastic material like poly (methyl methacrylate) (Perspex) are discussed.

A mathematical model of electro-thermoelasticity has been constructed in the context of a new consideration of heat conduction with memory-dependent derivative. The governing coupled equations with time-delay and kernel function, which can be chosen freely according to the necessity of applications, are applied to several concrete problems: (i) a problem of time-dependent thermal shock; (ii) a problem for a half-space subjected to ramp-type heating and (iii) a problem of a layer media. Laplace transform techniques are used. According to the numerical results and its graphs, conclusion about the new theory has been constructed. The predictions of the theory are discussed and compared with dynamic classical coupled theory. The result provides a motivation to investigate conducting thermoelectric materials as a new class of applicable materials.

In this study, the constitutive relation for the heat flux vector is derived to be the Fourier's law of heat conduction with a variable thermal conductivity and time-fractional order. The Stokes' flow of unsteady incompressible thermoelectric fluid due to a moving plate in the presence of a transverse magnetic field is molded. Stokes' first problem is solved by applying Laplace transform with respect to time variable and evaluating the inverse transform integrals by using a numerical approach. Numerical results for the temperature and the velocity distributions are given and illustrated graphically for given problem. The results indicate that the thermal conductivity and time-fractional order play a major role in the temperature and velocity distributions.