State-Space Approach to the Time-Fractional Maxwell’s Equations under Caputo Fractional Derivative of an Electromagnetic Half-Space under Four Different Thermoelastic Theorems

Abstract

This paper introduces a new mathematical modelling method of a thermoelastic and electromagnetic half-space in the context of four different thermoelastic theorems: Green–Naghdi type-I, and type-III; Lord–Shulman; and Moore–Gibson–Thompson. The bunding plane of the half-space surface is subjected to ramp-type heat and traction-free. We consider that Maxwell’s time-fractional equations have been under Caputo’s fractional derivative definition, which is the novelty of this work. Laplace transform techniques are utilized to obtain solutions using the state-space approach. Laplace transform’s inversions were calculated using Tzou’s iteration method. The temperature increment, strain, displacement, stress, induced electric field, and induced magnetic field distributions were obtained numerically and are illustrated in figures. The time-fraction parameter of Maxwell’s equations had a major impact on all the studied functions. The time-fractional parameter of Maxwell’s equations worked as resistant to the changing of temperature, particle movement, and induced magnetic field, while it acted as a catalyst to the induced electric field through the material. Moreover, all the studied functions have different values in the context of the four studied theorems.

1. Introduction

The link between strain and magnetic fields in thermoelastic substances is a subject of increasing interest due to its many important applications in geophysics, plasma physics, and related disciplines. Temperature gradients, very high temperatures, and internal magnetic fields of nuclear reactors all affect the field’s comprehension of how these devices operate. This is the domain of magneto-thermoelasticity. This hybrid theory combines the domains of thermoelasticity and electromagnetism [1]. As a result, the governing system of equations of the electromagnetic theory consists of hyperbolic partial differential equations, which ensure finite wave propagation rates. Biot’s theory of linked thermoelasticity is based on the parabolic partial differential equation of heat conduction and the hyperbolic partial differential equation of motion [2]. The features of the second equation suggest that heat waves may move at infinite speeds, despite evidence to the contrary. Lord and Shulman developed generalized thermoelasticity, with one relaxation time theory to overcome this restriction, and suggested a new and distinct rule of heat transfer to replace the traditional Fourier’s law of heat conduction (L-S) [3]. Both the heat flow vector and its time derivative are included in this law. Moreover, it has a new parameter, the relaxation time. There are limited propagation speeds for both heat and elastic waves since this theory’s heat equation is of the wave type. Like coupled and uncoupled theories, the other governing differential equations of this theory are the differential equations of motion and the constitutive relations of stress and strain. This theory was expanded upon by Dhaliwal and Sherief to include the general anisotropic case [4]. Green and Naghdi (G-N) presented three theories for investigating thermoelasticity. Instead of inequality, these models are based on entropy equality. Furthermore, the heat flow vector’s focus shifts [5,6,7]. The Green–Naghdi theorem is divided into three types: type I, type II, and type III, which represent three distinct assumptions. Type I uses the same traditional thermoelasticity mechanism and has a linear form. The waste of energy is prohibited under Type II. Type II is a subset of type III. Type III allows for the waste of energy [6,7,8,9,10,11,12]. In recent years, there has been significant interest in the Moore–Gibson–Thompson (MGT) heat conduction equation, and several articles have been published on it [13,14,15,16].

Whether one is discussing coupled or uncoupled versions, the generalized heat equation is often disregarded while discussing magneto-thermoelastic problems. This method is often justified since the outcomes of solving any of these equations vary very little quantitatively. However, when the short-term effects are considered, a significant degree of accuracy is lost unless the whole generalized system of differential equations is employed. This concept comes in quite handy for many problems involving heat gradients and for searching for fast-acting effects [1,17,18].

Leibniz presented the derivative of order half for the first time, initiating the extensive history of fractional calculus [19,20]. Leibniz, Liouville, Grunwald, Letnikov, and Riemann are credited with establishing the theory of fractional-order derivatives and integrals. Regarding fractional calculus and fractional differential equations with solutions, there are many fascinating sources and books available [19,21]. Recent physics research has used fractional-order derivatives and integrals, as well as fractional integro-differential equations, for a variety of purposes [22,23]. In scholarly circles, fractional-order electrodynamics is a relatively young subject. Among the many techniques, the addition of fractional-order derivatives to Maxwell’s equations in both space and time stands out [24,25,26,27,28]. These expansions of Maxwell’s equations, which make use of fractional-order derivatives, may be used to characterize the dynamics of electromagnetic systems with memory and energy dissipation [28]. The electric potential may be expressed in terms of fractional-order poles, which is an additional advantage of using fractional-order derivatives in electrodynamics [28,29]. Youssef developed the theory of fractional-order generalized thermoelasticity, which is based on the fractional-order non-Fourier heat conduction law [30] and the theory of generalized thermoelasticity with fractional-order strain based on fractional-order equations of motion, which is related to fractional-order stress–strain relations [31]. Neslihan examined the chaotic and regular dynamics of classical and fractional Gross–Pitaevskii equations (GPE) for interacting boson systems subjected to combined harmonic and optical lattice potentials through Poincaré sections of phase space, Lyapunov exponents, power spectrum analysis, and bifurcation techniques. They also studied chaotic and regular behaviours of classical and fractional Gross–Pitaevskii equations, including two-body, three-body, and higher-order interactions [32]. Al-Raeei simulated the spatial form of the fractional Schrödinger equation for the electrical screening potential using the Riemann–Liouville definition of the fractional derivatives and the numerical simulation methods [33].

State space approaches provide the fundamental basis of contemporary control theory, using differential equations rather than transfer functions to model processes. This technique is more encompassing than traditional Laplace and Fourier transform theory, allowing it to be applied to any system that can be examined using integral transforms in time. Additionally, it offers a distinct perspective on the temporal dynamics of linear systems. The state space approach is advantageous due to its ability to analyse linear systems with time-varying parameters; facilitate problem programming; analyse high-order linear systems; handle multiple input–multiple output systems; and serve as a basis for studying nonlinear systems, stochastic systems, and optimal control. The benefits stem from the use of state space, which enhances the generality and rigour of classical transform theory. In summary, state space methods are a crucial tool in modern control theory, providing a more general and rigorous approach to analysing the time behaviour of physical processes [34]. Recently, many authors have used the state-space approach in many applications [35,36,37,38,39,40,41].

The novelty of this study is that it addresses an application of electromagnetic generalized thermoelastic half-space in the context of the Green–Naghdi type I and type III, Lord–Shulman, and Moore–Gibson–Thompson theorems when the half-space surface is exposed to the time-fractional Maxwell electromagnetic effect. The time-fractional Maxwell’s equations are considered using the Caputo fractional derivative formulation. The main goal of this work is to investigate the effect of the fractional-order parameter of the time-fractional Maxwell’s equations, which has not been studied before, and to introduce new results where the time-fractional Maxwell’s equation has not been applied before. Moreover, this work introduces unified governing equations of four different theorems of thermoelasticity which are somehow close to each other, so one of the benefits of this work is that it compares these four theorems.

2. The Basic Steps of the Work

Now, we will describe the steps of the work to shaw the formulation of the problem and obtain the solutions in the Laplace transform domain, then figure out the results and discussions as in Scheme 1.

3. Formulation of the Problem

Assuming that a generalized thermoelastic and electromagnetic half-space occupies the space (see Figure 1):

$$\Omega =\left\{\left(x,0,0\right):0\le x<\infty \right\}.$$

(1)

$\mathbf{H}$ is a magnetic vector field with constant intensity which acts tangent to the bounding plane of the half-space $x=0$. All the studied functions of the material are functions only of the distance $x$ and the time variable $t$.

An induced magnetic vector field $\mathbf{h}=\left(0,h\left(x,t\right),0\right)$ is the result of the effect of the primary magnetic vector field ${\mathbf{H}}_0=\left(0,H_0,0\right)$. As a sequence, an induced electric vector field $\mathbf{E}=\left(E_x,E_y,E_z\right)$ is generated as shown in Figure 1 [17,42,43,44,45]:

We consider $\mathbf{h}$ and $\mathbf{E}$ to have small magnitudes based on the linear theory of Green–Naghdi thermoelasticity. Therefore, the vector of displacement will possess the following components:

$$\mathbf{U}=\left(U_x,U_y,U_z\right)=\left(u\left(x,t\right),0,0\right).$$

(2)

The magnetic intensity vector will have the following components [17,42,43,44,45]:

$${\mathbf{H}}_0=\left(H_x,H_y,H_z\right)=\left(0,H_0+h\left(x,t\right),0\right).$$

(3)

According to the left-hand rule, the electric intensity vector field must be perpendicular to both the magnetic intensity vector and the displacement vectors. Then, $\mathbf{E}$ has the following components:

$$\mathbf{E}=\left(0,0,E\left(x,t\right)\right).$$

(4)

The current density vector $\mathbf{J}$ must be parallel to the electric intensity vector $\mathbf{E}$; hence, we have:

$$\mathbf{J}=\left(0,0,J\right).$$

(5)

The time-fractional Maxwell’s equations in general vector forms are as follows [28]:

$$\nabla \times \mathbf{h}=\mathbf{J}+{\epsilon }_0^{\alpha }{D}_t^{\alpha }\mathbf{E},$$

(6)

$$\nabla \times \mathbf{E}=-{\mu }_0^{\alpha }{D}_t^{\alpha }\mathbf{h},$$

(7)

$$\nabla \cdot \mathbf{h}=0,$$

(8)

$$\nabla \cdot \mathbf{E}=0,$$

(9)

and

$$\mathbf{B}={\mu }_0\left({\mathbf{H}}_0+\mathbf{h}\right),$$

(10)

where any symbol which is written in bold is a vector. ${\mu }_0$ and ${\epsilon }_0$ are the magnetic and electric permeabilities, respectively [5,17,28,42,45,46,47,48].

The Equations (6)–(10) are supplemented by Ohm’s law, namely [5,17,28,42,45,46,47,48]:

$$\mathbf{J}={\sigma }_0\left({\mu }_0{\displaystyle \frac{\partial \mathbf{U}}{\partial t}}\times {\mathbf{H}}_0+\mathbf{E}\right).$$

(11)

Ohm’s law gives [5,17,28,42,45,46,47,48]:

$$\mathbf{J}={\sigma }_0\left(0,0,H_0{\mu }_0{\displaystyle \frac{\partial u}{\partial t}}+E\right),$$

(12)

where ${\sigma }_0$ is the electric conductivity.

The well-known Lorentz force $\overrightarrow{F}$ is given by the following law [5,17,28,42,45,46,47,48]:

$$\mathbf{F}=\mathbf{J}\times \mathbf{B}={\sigma }_0{\mu }_0\left(-\left(H_0+h\right)\left(H_0{\mu }_0{\displaystyle \frac{\partial u}{\partial t}}+E\right),0,0\right).$$

(13)

After linearization, we obtain:

$$\mathbf{F}=-{\sigma }_0{\mu }_0{H}_0\left({\mu }_0{H}_0{\displaystyle \frac{\partial u}{\partial t}}+E,0,0\right).$$

(14)

The strain components have the following forms [17]:

$$e_{xx}={\displaystyle \frac{\partial u}{\partial x}},e_{yy}=e_{zz}=e_{xz}=e_{yz}=e_{xy}=0,$$

(15)

and

$$e=e_{xx}+e_{yy}+e_{zz}={\displaystyle \frac{\partial u}{\partial x}}$$

(16)

The stress components are given by the following constitutive relation [17]:

$${\sigma }_{ij}=2\mu e_{ij}+\lambda e{\delta }_{ij}-\gamma \left(T-T_0\right){\delta }_{ij},$$

(17)

which has the components [17]:

$${{\displaystyle \sigma }}_{xx}=\sigma =\left(\lambda +2\mu \right){\displaystyle \frac{\partial u}{\partial x}}-\gamma \left(T-T_0\right),$$

(18)

$${{\displaystyle \sigma }}_{yy}={\sigma }_{zz}=\lambda {\displaystyle \frac{\partial u}{\partial x}}-\gamma \left(T-T_0\right),$$

(19)

and

$${{\displaystyle \sigma }}_{xy}={{\displaystyle \sigma }}_{yz}={{\displaystyle \sigma }}_{zx}=0,$$

(20)

where $T$ is the absolute temperature, $T_0$ is a reference temperature such that ${\displaystyle \frac{\left|T-T_0\right|}{T_0}}\ll 1$, $\lambda$ and $\mu$ are Lamé’s moduli, $\gamma$ is given by $\gamma =\left(3\lambda +2\mu \right){\alpha }_T$, and ${\alpha }_T$ is the coefficient of the thermal linear expansion.

Equations of motion have the following form [1,2,3]:

$${\sigma }_{ij,j}+F_i=\rho {\ddot{u}}_i,i=1,2,3,$$

(21)

where $\rho$ is the density of the material.

Substituting from Equation (12) into Equation (9), we obtain the partial differential equation of the motion in the following form [5,17,42,43,45,46,47,48,49]:

$$\left(\lambda +2\mu \right){\displaystyle \frac{{{\displaystyle \partial }}^{2}u}{\partial x^2}}-\gamma {\displaystyle \frac{\partial T}{\partial x}}-{\sigma }_0{\mu }_0{H}_{0}E-{\sigma }_0{H}_0^2{\mu }_0^2{\displaystyle \frac{\partial u}{\partial t}}=\rho {\displaystyle \frac{{{\displaystyle \partial }}^{2}u}{\partial t^2}},$$

(22)

By exciting the partial derivative with respect to the variable $x$ and using Equation (16), Equation (22) takes the following new form:

$$\left(\lambda +2\mu \right){\displaystyle \frac{{{\displaystyle \partial }}^{2}e}{\partial x^2}}-\gamma {\displaystyle \frac{{\partial }^{2}T}{\partial x^2}}-{\sigma }_0{\mu }_0{H}_0{\displaystyle \frac{\partial E}{\partial x}}-{\sigma }_0{H}_0^2{\mu }_0^2{\displaystyle \frac{\partial e}{\partial t}}=\rho {\displaystyle \frac{{{\displaystyle \partial }}^{2}e}{\partial t^2}}.$$

(23)

The one-dimensional heat conduction equation of Green–Naghdi type I and type III, Lord–Shulman, and Moore–Gibson–Thompson takes the following unified form [50,51,52]:

$${\displaystyle \frac{{\partial }^{3}T}{\partial x^2\partial t}}+{\displaystyle \frac{K^{*}}{K}}{\displaystyle \frac{{\partial }^{2}T}{\partial x^2}}=\left(1+{\tau }_q{\displaystyle \frac{\partial }{\partial t}}\right)\left({\displaystyle \frac{\rho C_E}{K}}{\displaystyle \frac{{{\displaystyle \partial }}^{2}T}{\partial t^2}}+{\displaystyle \frac{T_0\gamma }{K}}{\displaystyle \frac{{{\displaystyle \partial }}^{2}e}{\partial t^2}}\right).$$

(24)

The unified Equation (24) can be applied to the MGT theory, the Lord and Shulman theory (LS), and the type I (GN-I), and type III (GN-III) Green–Naghdi equations [53]. Hence, we have:

(i)

$K^{*}=0\mathrm{and}{\tau }_q=0$ provides the Green–Naghdi (GN-I) heat conduction equation.

(ii)

$K^{*}=0\mathrm{and}{\tau }_q\ne 0$ provides the Lord–Shulman (LS) model heat conduction equation [54].

(iii)

$K^{*}\ne 0\mathrm{and}{\tau }_q=0$ provides the Green–Naghdi (GN-III) model heat conduction equation [53,54].

(iv)

$K^{*}\ne 0\mathrm{and}{\tau }_q\ne 0$ provides the Moore–Gibson–Thompson (MGT) model heat conduction equation [15].

We may take $K^{*}={\displaystyle \frac{C_E\left(\lambda +2\mu \right)}{4}}$ as the main character of the Green–Naghdi theory, where $K$ gives the thermal conductivity and $C_E$ denotes the specific heat with constant deformation.

The above four theorems are used because they are well known in applications and they are close to each other in terms of explaining the propagation behaviour of thermal, mechanical, and magnetic waves and can be unified into an easily assembled thermal conductivity equation.

Equation (6) for the current model will take the following form:

$${\displaystyle \frac{\partial h}{\partial x}}=J+{\epsilon }_0^{\alpha }{D}_t^{\alpha }E.$$

(25)

Substituting from Equation (12), we obtain [28]:

$${\displaystyle \frac{\partial h}{\partial x}}=\left({\sigma }_0+{\epsilon }_0^{\alpha }{D}_t^{\alpha }\right)E+{\sigma }_0{H}_0{\mu }_0{\displaystyle \frac{\partial u}{\partial t}}.$$

(26)

Moreover, Equation (7) will take the following form [28]:

$${\displaystyle \frac{\partial E}{\partial x}}={\mu }_0^{\alpha }{D}_t^{\alpha }h.$$

(27)

By executing the partial derivative with respect to the variable $x$ for Equation (26) and using Equation (27), we obtain the following equation [28]:

$${\displaystyle \frac{{\partial }^{2}h}{\partial x^2}}=\left({\sigma }_0+{\epsilon }_0^{\alpha }{D}_t^{\alpha }\right){\mu }_0^{\alpha }{D}_t^{\alpha }h+{\sigma }_0{H}_0{\mu }_0{\displaystyle \frac{\partial e}{\partial t}}.$$

(28)

Moreover, Equation (23) takes the following form [28]:

$$\left(\lambda +2\mu \right){\displaystyle \frac{{{\displaystyle \partial }}^{2}e}{\partial x^2}}-\gamma {\displaystyle \frac{{\partial }^{2}T}{\partial x^2}}-{\sigma }_0{H}_0{\mu }_0^{\alpha +1}{D}_t^{\alpha }h-{\sigma }_0{H}_0^2{\mu }_0^2{\displaystyle \frac{\partial e}{\partial t}}=\rho {\displaystyle \frac{{{\displaystyle \partial }}^{2}e}{\partial {{\displaystyle \mathrm{t}}}^2}}.$$

(29)

The following dimensionless variables are used for simplifications, [17,42,43,45,46]:

$$\begin{array}{l}\left\{x,u\right\}={\displaystyle \frac{\left\{x^{\prime },u^{\prime }\right\}}{c_0\eta }},t={\displaystyle \frac{t^{\prime }}{c_0^2\eta }},{\sigma }_{ij}=\left(\lambda +2\mu \right){{\sigma }^{\prime }}_{ij},\left(T-T_0\right)={\displaystyle \frac{\left(\lambda +2\mu \right)}{\gamma }}\theta ,E={\displaystyle \frac{{\sigma }_0{H}_0{c}_0{\mu }_0^2}{\eta }}{E}^{\prime },\\ h={\displaystyle \frac{{\sigma }_0{H}_0{\mu }_0}{\eta }}{h}^{\prime }.\end{array}$$

Equations (24) and (28)–(29) are reduced to the following system of differential equations (dropping the primes for convenience):

$${\displaystyle \frac{{{\displaystyle \partial }}^{2}e}{\partial x^2}}-{\displaystyle \frac{{\partial }^2\theta }{\partial x^2}}-\nu \epsilon {\epsilon }_1{D}_t^{\alpha }h-{\epsilon }_1{\displaystyle \frac{\partial e}{\partial t}}={\displaystyle \frac{{{\displaystyle \partial }}^{2}e}{\partial {{\displaystyle \mathrm{t}}}^2}},$$

(30)

$${\displaystyle \frac{{\partial }^{3}T}{\partial x^2\partial t}}+{\epsilon }_2{\displaystyle \frac{{\partial }^{2}T}{\partial x^2}}=\left(1+{\tau }_q{\displaystyle \frac{\partial }{\partial t}}\right)\left({\displaystyle \frac{{{\displaystyle \partial }}^{2}T}{\partial t^2}}+{\epsilon }_3{\displaystyle \frac{{{\displaystyle \partial }}^{2}e}{\partial t^2}}\right),$$

(31)

$${\displaystyle \frac{{\partial }^{2}h}{\partial x^2}}=\nu \epsilon D_t^{\alpha }h+V^2\epsilon {\epsilon }_4{D}_t^{\alpha }{D}_t^{\alpha }h+{\displaystyle \frac{\partial e}{\partial t}},$$

(32)

and

$${\displaystyle \frac{\partial E}{\partial x}}=\epsilon D_t^{\alpha }h,$$

(33)

where

$$\begin{array}{l}\eta ={\displaystyle \frac{\rho C_E}{K}},c^2={\displaystyle \frac{1}{{\epsilon }_0{\mu }_0}},\nu ={\displaystyle \frac{{\sigma }_0{\mu }_0}{\eta }},V={\displaystyle \frac{c_0}{c}},c_0^2={\displaystyle \frac{\lambda +2\mu }{\rho }},\epsilon ={\left({\mu }_0{c}_0^2\eta \right)}^{\alpha -1},{\epsilon }_1={\displaystyle \frac{H_0^2{\mu }_0\nu }{\left(\lambda +2\mu \right)}},\\ {\epsilon }_2={\displaystyle \frac{K^{*}}{K{c}_0^2\eta }},{\epsilon }_3={\displaystyle \frac{{\gamma }^2{T}_0}{\eta K\left(\lambda +2\mu \right)}},\mathrm{and}{\epsilon }_4={\left({\epsilon }_0{c}_0^2\eta \right)}^{\alpha -1}.\end{array}$$

The constitutive equations are also reduced to the following forms:

$${{\displaystyle \sigma }}_{xx}=\sigma =e-\theta ,$$

(34)

and

$${{\displaystyle \sigma }}_{yy}={\sigma }_{zz}=\beta e-\theta$$

(35)

where $\beta ={\displaystyle \frac{\lambda }{\lambda +2\mu }}$.

Now, we define the operator $D_t^{\alpha }={\displaystyle \frac{d^{\alpha }}{d{t}^{\alpha }}}$, which is a fractional-order derivative and is given by the normal derivative when $\alpha =1$, and a Caputo fractional derivative when $1<\alpha <2$, respectively, as in the following unified form [19,20,21,22,23,24,25,26,27,28,29,30,31]:

$$D_t^{\alpha }f\left(t\right)={\displaystyle \frac{1}{\Gamma \left(2-\alpha \right)}}{\displaystyle \underset{0}{\overset{t}{\int }}{\left(t-\xi \right)}^{1-\alpha }\frac{d^{2}f\left(\xi \right)}{d{t}^2}}d\xi ,1<\alpha <2,t>0,$$

(36)

and the Laplace transform of the above formula and its derivatives is given by [19,20,21,22,23,24,25,26,27,28,29,30,31]:

$$L\left\{D_t^{\alpha }f\left(t\right)\right\}=s^{\alpha }\overline{f}\left(s\right)-s^{\alpha -1}f\left(0^{+}\right)-s^{\alpha }{f}^{(1)}\left(0^{+}\right),1\le \alpha <2.$$

(37)

The initial conditions of the current model are given as follows:

$$\theta \left(x,0^{+}\right)=e\left(x,0^{+}\right)=h\left(x,0^{+}\right)={\displaystyle \frac{\partial \theta \left(x,0^{+}\right)}{\partial t}}={\displaystyle \frac{\partial e\left(x,0^{+}\right)}{\partial t}}={\displaystyle \frac{\partial h\left(x,0^{+}\right)}{\partial t}}=0.$$

(38)

After applying the initial conditions, (37) is in the following simple form [19,20,21,22,23,24,25,26,27,28,29,30,31]:

$$L\left\{D_t^{\alpha }f\left(t\right)\right\}=s^{\alpha }\overline{f}\left(s\right),1\le \alpha <2.$$

(39)

Then, we have:

$$\left({\displaystyle \frac{{{\displaystyle d}}^2}{d{x}^2}}-{\beta }_2\right)\overline{e}-{\displaystyle \frac{d^2\overline{\theta }}{d{x}^2}}-{\beta }_1\overline{h}=0,$$

(40)

$$\left({\displaystyle \frac{{{\displaystyle d}}^2}{d{x}^2}}-{\beta }_3\right)\overline{\theta }-{\beta }_4\overline{e}=0,$$

(41)

$$\left({\displaystyle \frac{d^2}{d{x}^2}}-{\beta }_5\right)\overline{h}-{\beta }_6\overline{e}=0,$$

(42)

and

$${\displaystyle \frac{d\overline{E}}{dx}}=\epsilon s^{\alpha }\overline{h}$$

(43)

where $\begin{array}{l}{\beta }_1=\epsilon {\epsilon }_1\nu s^{\alpha },{\beta }_2=\left({\epsilon }_{1}s+s^2\right),{\beta }_3={\displaystyle \frac{s^2\left(1+{\tau }_{q}s\right)}{\left(s+{\epsilon }_2\right)}},{\beta }_4={\displaystyle \frac{s^2\left(1+{\tau }_{q}s\right){\epsilon }_3}{\left(s+{\epsilon }_2\right)}},{\beta }_5=\epsilon s^{\alpha }\left(\nu +V^2{\epsilon }_4{s}^{\alpha }\right),\end{array}$ and ${\beta }_6=s.$

The constitutive equations of stress–strain are given by:

$${{\displaystyle \overline{\sigma }}}_{xx}=\overline{\sigma }=\overline{e}-\overline{\theta },$$

(44)

and

$${{\displaystyle \overline{\sigma }}}_{yy}={\overline{\sigma }}_{zz}=\beta \overline{e}-\overline{\theta }$$

(45)

4. State-Space Formulation

We re-write the Equations (40)–(42) in the following forms:

$${\displaystyle \frac{{{\displaystyle d}}^2\overline{e}}{d{x}^2}}=\left({\beta }_2+{\beta }_4\right)\overline{e}+{\beta }_3\overline{\theta }+{\beta }_1\overline{h},$$

(46)

$${\displaystyle \frac{{{\displaystyle d}}^2\overline{\theta }}{d{x}^2}}={\beta }_4\overline{e}+{\beta }_3\overline{\theta },$$

(47)

$${\displaystyle \frac{d^2\overline{h}}{d{x}^2}}={\beta }_6\overline{e}+{\beta }_5\overline{h}.$$

(48)

Taking as the state variable the functions $\overline{e},\overline{\theta },\overline{h},{\displaystyle \frac{d\overline{e}}{dx}},{\displaystyle \frac{d\overline{\theta }}{dx}},{\displaystyle \frac{d\overline{h}}{dx}}$ in the x-direction, Equations (46)–(48) can be written in matrix form using the Bahar–Hetnarski method [34,55,56]:

$${\displaystyle \frac{d\overline{\psi }(x,s)}{dx}}=A(s)\overline{\psi }(x,s),$$

(49)

where

$$\overline{\psi }\left(x,s\right)=\left[\begin{array}{c}\overline{e}\left(x,s\right)\\ \overline{\theta }\left(x,s\right)\\ \overline{h}\left(x,s\right)\\ {\overline{e}}^{\prime }\left(x,s\right)\\ {\overline{\theta }}^{\prime }\left(x,s\right)\\ {\overline{h}}^{\prime }\left(x,s\right)\end{array}\right]$$

(50)

and

$$A\left(s\right)=\left[\begin{array}{cccccc}0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\\ {\beta }_2+{\beta }_4& {\beta }_3& {\beta }_1& 0& 0& 0\\ {\beta }_4& {\beta }_3& 0& 0& 0& 0\\ {\beta }_6& 0& {\beta }_5& 0& 0& 0\end{array}\right]$$

(51)

The formal and bounded solution in the domain (1) is given by:

$$\overline{\psi }\left(x,s\right)=\exp \left[A\left(s\right)x\right]\overline{\psi }\left(0,s\right),$$

(52)

where

$$\overline{\psi }\left(0,s\right)=\left[\begin{array}{c}\overline{e}\left(0,s\right)\\ \overline{\theta }\left(0,s\right)\\ \overline{h}\left(0,s\right)\\ {\overline{e}}^{\prime }\left(0,s\right)\\ {\overline{\theta }}^{\prime }\left(0,s\right)\\ {\overline{h}}^{\prime }\left(0,s\right)\end{array}\right]$$

(53)

The characteristic equation of the matrix $A\left(s\right)$ takes the following form:

$$k^6-L{k}^4+M{k}^2-N=0,$$

(54)

where $L={\beta }_2+{\beta }_3+{\beta }_4+{\beta }_5,M={\beta }_3{\beta }_5+{\beta }_2{\beta }_3+{\beta }_2{\beta }_5+{\beta }_4{\beta }_5-{\beta }_1{\beta }_6,N={\beta }_3\left({\beta }_2{\beta }_5-{\beta }_1{\beta }_6\right)$.

The expansion of the Taylor series for the matrix exponential $\exp \left[A\left(s\right)x\right]$ is given by:

$$\exp \left[A\left(s\right)x\right]={\displaystyle \sum _{i=0}^{\infty }\frac{{\left[A\left(s\right)x\right]}^i}{i!}}.$$

(55)

Applying the Cayley–Hamilton theorem [55,56,57], this infinite series can be truncated to the following form:

$$\exp \left[A\left(s\right)x\right]=a_0{I}_6+a_{1}A+a_2{A}^2+a_3{A}^3+a_4{A}^4+a_5{A}^5,$$

(56)

where $I_6$ is the unit matrix of order 6 and $a_0-a_5$ are some parameters which depend on s and x to be determined.

Applying the Cayley–Hamilton theorem again, we obtain [55,56,57]:

$$\exp \left[k_{1}x\right]=a_0{I}_6+a_1{k}_1+a_2{k}_1^2+a_3{k}_1^3+a_4{k}_1^4+a_5{k}_1^5,$$

(57)

$$\exp \left[-k_{1}x\right]=a_0{I}_6-a_1{k}_1+a_2{k}_1^2-a_3{k}_1^3+a_4{k}_1^4-a_5{k}_1^5,$$

(58)

$$\exp \left[k_{2}x\right]=a_0{I}_6+a_1{k}_2+a_2{k}_2^2+a_3{k}_2^3+a_4{k}_2^4+a_5{k}_2^5,$$

(59)

$$\exp \left[-k_{2}x\right]=a_0{I}_6-a_1{k}_2+a_2{k}_2^2-a_3{k}_2^3+a_4{k}_2^4-a_5{k}_2^5,$$

(60)

$$\exp \left[k_{3}x\right]=a_0{I}_6+a_1{k}_3+a_2{k}_3^2+a_3{k}_3^3+a_4{k}_3^4+a_5{k}_3^5,$$

(61)

$$\exp \left[-k_{3}x\right]=a_0{I}_6-a_1{k}_3+a_2{k}_3^2-a_3{k}_3^3+a_4{k}_3^4-a_5{k}_3^5,$$

(62)

where $\pm k_1,\pm k_2,\pm k_3$ are the roots of the characteristic Equation (54).

By solving the system of Equations (57)–(62), we obtain the parameters $a_0,a_1,a_2,a_3,a_4,a_5$ as follows:

$$a_0=-\xi \left(c_1{k}_2^2{k}_3^2+c_2{k}_1^2{k}_3^2+c_3{k}_2^2{k}_1^2\right),$$

(63)

$$a_1=-\xi \left(s_1{k}_2^2{k}_3^2+s_2{k}_1^2{k}_3^2+s_3{k}_2^2{k}_1^2\right),$$

(64)

$$a_2=\xi \left(c_1\left(k_2^2+k_3^2\right)+c_2\left(k_1^2+k_3^2\right)+c_3\left(k_2^2+k_1^2\right)\right),$$

(65)

$$a_3=\xi \left(s_1\left(k_2^2+k_3^2\right)+s_2\left(k_1^2+k_3^2\right)+s_3\left(k_2^2+k_1^2\right)\right),$$

(66)

$$a_4=-\xi \left(c_1+c_2+c_3\right),$$

(67)

$$a_5=-\xi \left(s_1+s_2+s_3\right),$$

(68)

where $c_1=\left(k_2^2-k_3^2\right)\cosh \left(k_{1}x\right)$, $c_2=\left(k_3^2-k_1^2\right)\cosh \left(k_{2}x\right)$, $c_3=\left(k_1^2-k_2^2\right)\cosh \left(k_{3}x\right)$, $s_1={\displaystyle \frac{\left(k_2^2-k_3^2\right)}{k_1}}\sinh \left(k_{1}x\right)$, $s_2={\displaystyle \frac{\left(k_3^2-k_1^2\right)}{k_2}}\sinh \left(k_{2}x\right)$, $s_3={\displaystyle \frac{\left(k_1^2-k_2^2\right)}{k_3}}\sinh \left(k_{3}x\right)$, $\xi ={\left(k_1^2-k_2^2\right)}^{-1}{\left(k_2^2-k_3^2\right)}^{-1}{\left(k_3^2-k_1^2\right)}^{-1}$.

Substituting Equations (63)–(68) into Equation (56), we obtain the matrix exponential in the form:

$$\exp \left[A\left(s\right)x\right]=\mathcal{l}\left(x,s\right)=\left[{\mathcal{l}}_{ij}\left(x,s\right)\right],i,j=1,2,3,4,5,6,$$

(69)

where the elements of the matrix $\left[{\mathcal{l}}_{ij}\left(x,s\right)\right],i,j=1,2,3,4,5,6$ are defined in the Appendix A.

Thus, the solution of the Equation (52) is given by:

$$\overline{\psi }\left(x,s\right)=\left[{\mathcal{l}}_{ij}\left(x,s\right)\right]\overline{\psi }\left(0,s\right).$$

(70)

Now, to obtain the column matrix $\overline{\psi }\left(0,s\right)$, we must apply the following boundary conditions:

(i)

The thermal boundary condition: the bounding surface of the half-space $x=0$ is thermally loaded by a ramp-type heat, i.e.,

$$\theta \left(0,x\right)={\theta }_0\left\{\begin{array}{cc}t/t_0& 0\le t<t_0\\ 1& t\ge t_0\end{array}\right\}=g\left(t\right),$$

(71)

where $t_0$ is the parameter of the ramp-time heat. ${\theta }_0$ is constant and gives the thermal loading intensity.

Using the Laplace transform, the above condition takes the form:

$$\overline{\theta }\left(0,s\right)={\displaystyle \frac{1-e^{-t_{0}s}}{t_0{s}^2}}=\overline{g}\left(s\right).$$

(72)

(ii)

The mechanical boundary condition: the bounding plane of the half-space $x=0$ traction free, i.e.,

$$\sigma \left(0,t\right)=0\mathrm{which}\mathrm{gives}\overline{\sigma }\left(0,s\right)=0.$$

(73)

Hence, from Equations (44) and (72), we obtain:

$$\overline{e}\left(0,s\right)=\overline{g}\left(s\right).$$

(74)

(iii)

The electromagnetic boundary condition: the magnetic and electric functions $h$ and $E$ must satisfy the continuity conditions as follows [17,45]:

$${\left.h\left(x,t\right)\right|}_{x=0}={\left.h_0\left(x,t\right)\right|}_{x\le 0}\mathrm{and}{\left.E\left(x,t\right)\right|}_{x=0}={\left.E_0\left(x,t\right)\right|}_{x\le 0},$$

(75)

where $E_0\left(x,t\right)\mathrm{and}{h}_0\left(x,t\right)$ are the electric and magnetic intensities in the free space, respectively.

In the free space, using the non-dimensional Maxwell’s equations in the Laplace transform domain, we can establish $\nu =0\mathrm{and}\alpha =1$, which give the following equations:

$${\displaystyle \frac{\partial {\overline{h}}_0}{\partial x}}=s{V}^2{\overline{E}}_0,$$

(76)

and

$${\displaystyle \frac{\partial {\overline{E}}_0}{\partial x}}=s{\overline{h}}_0$$

(77)

Eliminating ${\overline{E}}_0$ between the above two equations, we obtain:

$${\displaystyle \frac{d^2{\overline{h}}_0}{d{x}^2}}={\delta }^2{\overline{h}}_0\mathrm{for}x\le 0,$$

(78)

where $\delta =Vs$.

The general and bounded solution of the Equation (78) is given by:

$${\overline{h}}_0\left(x,s\right)=A{e}^{\delta x},x\le 0.$$

(79)

From Equations (79) and (77), we obtain:

$${\overline{E}}_0={\displaystyle \frac{s}{\delta }}A{e}^{\delta x},x\le 0.$$

(80)

From Equations (80) and (76), we obtain the condition when $x=0$ as [17,45]:

$${\left.\left({\displaystyle \frac{d\overline{h}\left(x,s\right)}{dx}}-\delta \overline{h}\left(x,s\right)\right)\right|}_{x=0}=0.$$

(81)

Using the boundary conditions (73), (72), and (81), we obtain the following:

$$\left[\begin{array}{c}\overline{e}\left(x,s\right)\\ \overline{\theta }\left(x,s\right)\\ \overline{h}\left(x,s\right)\\ {\overline{e}}^{\prime }\left(x,s\right)\\ {\overline{\theta }}^{\prime }\left(x,s\right)\\ {\overline{h}}^{\prime }\left(x,s\right)\end{array}\right]={\left[{\mathcal{l}}_{ij}\left(x,s\right)\right]}_{6\times 6}\left[\begin{array}{c}\overline{g}\left(s\right)\\ \overline{g}\left(s\right)\\ \overline{h}\left(0,s\right)\\ {\overline{e}}^{\prime }\left(0,s\right)\\ {\overline{\theta }}^{\prime }\left(0,s\right)\\ {\overline{h}}^{\prime }\left(0,s\right)\end{array}\right].$$

(82)

To obtain ${\overline{e}}^{\prime }\left(0,s\right),{\overline{\theta }}^{\prime }\left(0,s\right),{\overline{h}}^{\prime }\left(0,s\right)$, we can use the Equations (81) and (82) when $x=0$ in the following form:

$$\left[\begin{array}{c}\overline{e}\left(0,s\right)\\ \overline{\theta }\left(0,s\right)\\ \overline{h}\left(0,s\right)\\ {\overline{e}}^{\prime }\left(0,s\right)\\ {\overline{\theta }}^{\prime }\left(0,s\right)\\ \delta \overline{h}\left(0,s\right)\end{array}\right]={\left[{\mathcal{l}}_{ij}\left(0,s\right)\right]}_{6\times 6}\left[\begin{array}{c}\overline{g}\left(s\right)\\ \overline{g}\left(s\right)\\ \overline{h}\left(0,s\right)\\ {\overline{e}}^{\prime }\left(0,s\right)\\ {\overline{\theta }}^{\prime }\left(0,s\right)\\ \delta \overline{h}\left(0,s\right)\end{array}\right],$$

(83)

which gives the following equations:

$$\left(1-{\mathcal{l}}_{44}^0\right){\overline{e}}^{\prime }\left(0,s\right)-{\mathcal{l}}_{45}^0{\overline{\theta }}^{\prime }\left(0,s\right)-\left({\mathcal{l}}_{43}^0+{\mathcal{l}}_{46}^0\delta \right)\overline{h}\left(0,s\right)=\left({\mathcal{l}}_{41}^0+{\mathcal{l}}_{42}^0\right)\overline{g}\left(s\right),$$

(84)

$${\mathcal{l}}_{54}^0{\overline{e}}^{\prime }\left(0,s\right)-\left(1-{\mathcal{l}}_{55}^0\right){\overline{\theta }}^{\prime }\left(0,s\right)+\left({\mathcal{l}}_{53}^0+{\mathcal{l}}_{56}^0\delta \right)\overline{h}\left(0,s\right)=-\left({\mathcal{l}}_{51}^0+{\mathcal{l}}_{52}^0\right)\overline{g}\left(s\right),$$

(85)

$${\mathcal{l}}_{64}^0{\overline{e}}^{\prime }\left(0,s\right)+{\mathcal{l}}_{65}^0{\overline{\theta }}^{\prime }\left(0,s\right)-\left(1-{\mathcal{l}}_{63}^0-\delta {\mathcal{l}}_{66}^0\right)\overline{h}\left(0,s\right)=-\left({\mathcal{l}}_{61}^0+{\mathcal{l}}_{62}^0\right)\overline{g}\left(s\right),$$

(86)

where ${\mathcal{l}}_{ij}^0={\mathcal{l}}_{ij}\left(0,s\right),i,j=1,2,3,4,5,6$.

After obtaining the solution of the above system of equations and using Equation (81), we can obtain the known ${\overline{e}}^{\prime }\left(0,s\right)$, ${\overline{\theta }}^{\prime }\left(0,s\right)$, $\overline{h}\left(0,s\right)$, and ${\overline{h}}^{\prime }\left(0,s\right)$.

Now, the complete solutions are obtained in the domain of Laplace transform.

The inversions of the Laplace transforms can be obtained by applying the following iteration form of Tzou [58]:

$$f(t)=L^{-1}\left\{\overline{f}\left(s\right)\right\}\approx {\displaystyle \frac{e^{\kappa t}}{t}}\left[{\displaystyle \frac{1}{2}}\overline{f}\left(\kappa \right)+\mathrm{Re}{\displaystyle \sum _{r=1}^R{\left(-1\right)}^r\overline{f}\left(\kappa +\frac{ir\pi }{t}\right)}\right],$$

(87)

where $i=\sqrt{-1}$ is the imaginary number unit, “$\mathrm{Re}$” denotes the real part of a complex function, and $R$ is an integer parameter that can be chosen such that:

$$\left|g{\left(t\right)}_{R+1}-g{\left(t\right)}_R\right|<{10}^{-6}.$$

(88)

To obtain a faster convergence for the above iteration, some experiments verify that the parameter “$\kappa$” may satisfy the following relation $\kappa t\approx 4.7$ [57,58,59].

5. The Numerical Results

The copper material was used to obtain the numerical calculations. The parameters and the material properties’ constants were taken as follows [4,5,17,42,43,44,45,46,47,49]:

$$\begin{array}{l}\lambda =7.76\times {10}^{10},\mu =3.86\times {10}^{10},T_0=293,\rho =9854,{\alpha }_T=1.78\times {10}^{-5},K=386,C_E=381.0,\\ {\sigma }_0=5.7\times {10}^7,{\epsilon }_0={\displaystyle \frac{{10}^{-9}}{36\pi }},{\mu }_0=4\pi \times {10}^{-7}.\end{array}$$

The dimensionless variables were taken as follows: $t=t_0=1.0$ and $0.0\le \mathrm{x}\le 2.0$.

The calculations by using MAPLE software will be go on as in the Scheme 2 as follows:

Figure 2 contains four figures showing the temperature increment distributions under the theorems GN-I in Figure 2a, LS in Figure 2b, GN-III in Figure 2c, and MGT in Figure 2d, with variance values of the time-fractional parameter of Maxwell’s equations to establish this parameter of the temperature increment. We can see that the temperature increment distributions almost are the same under the four studied models. Moreover, the effect of the time-fractional parameter of Maxwell’s equations is limited.

Figure 3 contains four figures showing the volumetric dilatation distributions under the theorems GN-I in Figure 3a, LS in Figure 3b, GN-III in Figure 3c, and MGT in Figure 3d, along with variance values of the time-fractional parameter of Maxwell’s equations to establish this parameter of the volumetric dilatation. We can see that the volumetric dilatation distributions have the same behaviour with different values under the four studied models. Moreover, the effect of the time-fractional parameter of Maxwell’s equations is significant; increasing the values of the time-fractional parameter of Maxwell’s equations leads to a decrease in the value of the volumetric dilatation. In other words, the time-fractional parameter of Maxwell’s equations works as a resistor to the volumetric dilatation. In addition, the maximum values of the volumetric dilatation distributions take the following order:

$$e_{\mathrm{Max}}\left(\mathrm{LS}\right)>e_{\mathrm{Max}}\left(\mathrm{GN}\text{-}\mathrm{I}\right)>e_{\mathrm{Max}}\left(\mathrm{GN}\text{-}\mathrm{III}\right)=e_{\mathrm{Max}}\left(\mathrm{MGT}\right)$$

(89)

Also, we can say that the behaviour of the volumetric dilatation distributions under the two models of Green–Naghdi type III and Moore–Gibson–Thompson are too close to each other.

Figure 4 presents four figures showing the displacement distributions under the theorems GN-I in Figure 4a, LS in Figure 4b, GN-III in Figure 4c, and MGT in Figure 4d, along with variance values of the time-fractional parameter of Maxwell’s equations to establish this parameter of the displacement. We can see that the displacement distributions have the same behaviour with different values under the four studied models. Moreover, the effects of the time-fractional parameter of Maxwell’s equations are significant; increasing the values of the time-fractional parameter of Maxwell’s equations leads to a decrease in the absolute value of the displacement. This means that the time-fractional parameter of Maxwell’s equations works as a resistor to the displacement. In addition, the maximum values of the absolute value of the displacement distributions take the following order:

$$\left|u_{\mathrm{Max}}\left(\mathrm{LS}\right)\right|>\left|u_{\mathrm{Max}}\left(\mathrm{GN}\text{-}\mathrm{I}\right)\right|>\left|u_{\mathrm{Max}}\left(\mathrm{GN}\text{-}\mathrm{III}\right)\right|=\left|u_{\mathrm{Max}}\left(\mathrm{MGT}\right)\right|$$

(90)

Thus, the behaviour of the displacement distributions under the two models of Green–Naghdi type III and Moore–Gibson–Thompson are too close to each other.

Figure 5 presents four figures showing the stress distributions under the theorems GN-I in Figure 5a, LS in Figure 5b, GN-III in Figure 5c, and MGT in Figure 5d, along with variance values of the time-fractional parameter of Maxwell’s equations to establish this parameter of the stress. We can see that the stress distributions have the same behaviour with different values under the four studied models. Also, the effects of the time-fractional parameter of Maxwell’s equations are significant; increasing the values of the time-fractional parameter of Maxwell’s equations leads to a decrease in the value of the stress. This means that the time-fractional parameter of Maxwell’s equations works as a resistor to the stress. In addition, the maximum values of the stress distributions take the following order:

$${\sigma }_{\mathrm{Max}}\left(\mathrm{LS}\right)>{\sigma }_{\mathrm{Max}}\left(\mathrm{GN}\text{-}\mathrm{I}\right)>{\sigma }_{\mathrm{Max}}\left(\mathrm{GN}\text{-}\mathrm{III}\right)={\sigma }_{\mathrm{Max}}\left(\mathrm{MGT}\right)$$

(91)

Thus, the behaviour of the stress distributions under the two models of Green–Naghdi type III and Moore–Gibson–Thompson are too close to each other.

Figure 6 presents four figures showing the induced magnetic field distributions under the theorems GN-I in Figure 6a, LS in Figure 6b, GN-III in Figure 6c, and MGT in Figure 6d, along with variance values of the time-fractional parameter of Maxwell’s equations to establish this parameter on the induced magnetic field. We can see that the induced magnetic field distributions have the same behaviour with different values under the four studied models. Also, the effects of the time-fractional parameter of Maxwell’s equations are significant; increasing the values of the time-fractional parameter of Maxwell’s equations leads to an increase in the absolute value of the induced magnetic field. In other words, the time-fractional parameter of Maxwell’s equations works as a resistor to the induced magnetic field. In addition, the maximum values of the absolute value of the induced magnetic field distributions in the boundary $x=0$ take the following order:

$$\left|h_{\mathrm{Max}}\left(\mathrm{LS}\right)\right|=\left|h_{\mathrm{Max}}\left(\mathrm{GN}\text{-}\mathrm{I}\right)\right|>\left|h_{\mathrm{Max}}\left(\mathrm{GN}\text{-}\mathrm{III}\right)\right|>\left|h_{\mathrm{Max}}\left(\mathrm{MGT}\right)\right|$$

(92)

Thus, the behaviour of the displacement distributions under the two models of Lord–Shulman and Green–Naghdi type I are too close to each other.

Figure 7 presents four figures showing the induced electric field distributions under the theorems GN-I in Figure 7a, LS in Figure 7b, GN-III in Figure 7c, and MGT in Figure 7d, along with variance values of the time-fractional parameter of Maxwell’s equations to establish this parameter on the induced electric field. We can see that the induced electric field distributions show the same behaviour with different values under the four studied models. Also, the effects of the time-fractional parameter of Maxwell’s equations are significant; increasing the values of the time-fractional parameter of Maxwell’s equations leads to an increase in the value of the induced electric field. In other words, the time-fractional parameter of Maxwell’s equations works as a catalyst to the induced electric field. In addition, the maximum values of the induced electric field distributions in the boundary $x=0$ take the following order:

$$E_{\mathrm{Max}}\left(\mathrm{LS}\right)=E_{\mathrm{Max}}\left(\mathrm{GN}\text{-}\mathrm{I}\right)>E_{\mathrm{Max}}\left(\mathrm{GN}\text{-}\mathrm{III}\right)=E_{\mathrm{Max}}\left(\mathrm{MGT}\right)$$

(93)

Thus, the behaviour of the stress distributions under the two models of Lord–Shulman and Green–Naghdi type I are too close to each other.

Figure 8, Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13 show the temperature increment, volumetric dilatation, displacement, stress, induced magnetic field, and induced electric field distributions, respectively, when $\alpha =1.2$ under the theorems of GN-I, LS, GN-III, and MGT.

Figure 8 shows that the temperature increment distributions under the four studied theorems have the same behaviour, but with different values. The temperature increment values have the following order:

$$\theta \left(\mathrm{GN}\text{-}\mathrm{III}\right)>\theta \left(\mathrm{MGT}\right)>\theta \left(\mathrm{GN}\text{-}\mathrm{I}\right)>\theta \left(\mathrm{LS}\right)$$

(94)

This means that the speed of propagation of the thermal wave under the Lord–Shulman theorem has a limited value, more than the other theorems, and goes to zero faster.

Figure 9 shows that the volumetric dilatation distributions under the four studied theorems have the same behaviour, but with different values. The volumetric dilatation values have the following order:

$$e\left(\mathrm{LS}\right)>e\left(\mathrm{GN}\text{-}\mathrm{I}\right)>e\left(\mathrm{MGT}\right)>e\left(\mathrm{GN}\text{-}\mathrm{III}\right)$$

(95)

Figure 10 shows that the displacement distributions under the four studied theorems have the same behaviour, but with different values. The absolute values of displacement have the following order:

$$\left|u\left(\mathrm{LS}\right)\right|>\left|u\left(\mathrm{GN}\text{-}\mathrm{I}\right)\right|>\left|u\left(\mathrm{MGT}\right)\right|>\left|u\left(\mathrm{GN}\text{-}\mathrm{III}\right)\right|$$

(96)

Figure 11 shows that the stress distributions under the four studied theorems have the same behaviour, but with different values. The values of stress have the following order:

$$\rho \left(\mathrm{LS}\right)>\rho \left(\mathrm{GN}\text{-}\mathrm{I}\right)>\sigma \left(\mathrm{MGT}\right)>\sigma \left(\mathrm{GN}\text{-}\mathrm{III}\right)$$

(97)

Figure 9, Figure 10 and Figure 11 show that the speed of the mechanical wave under the theorem of GN-III has a limited value, which is higher than its values under the other theorems.

Figure 12 shows that the induced magnetic field distributions under the four studied theorems have the same behaviour, but with different values. The values of the induced magnetic field have the following order on the bounding plane of the half-space $x=0$:

$$\left|h\left(\mathrm{LS}\right)\right|>\left|h\left(\mathrm{GN}\text{-}\mathrm{I}\right)\right|>\left|h\left(\mathrm{MGT}\right)\right|>\left|h\left(\mathrm{GN}\text{-}\mathrm{III}\right)\right|$$

(98)

Figure 13 shows that the induced electric field distributions under the four studied theorems have the same behaviour, but with different values. The values of the induced electric field have the following order on the bounding plane of the half-space $x=0$:

$$E\left(\mathrm{LS}\right)>E\left(\mathrm{GN}\text{-}\mathrm{I}\right)>E\left(\mathrm{MGT}\right)>E\left(\mathrm{GN}\text{-}\mathrm{III}\right)$$

(99)

Figure 12 and Figure 13 show that the speed of the electromagnetic waves under the theorem of GN-III has a limited value, which is higher than its values under the other theorems.

For the validation of the current results, the current results of the case $\alpha =1.0$ show the same behaviour regarding the temperature increment, volumetric dilatation, displacement, stress, induced magnetic field, and induced electric field distributions seen in previously published papers [5,10,17,24,43,44,45,46,47,48].

6. Conclusions

This work introduced a new mathematical model of a thermoelastic and electromagnetic half-space based on Green–Naghdi theory type I (GN-I) and type III (GN-III), the Lord–Shulman (LS) theorem, and the Moore–Gibson–Thompson (MGT) theorem of thermoelasticity based on time-fractional Maxwell’s equations. The Caputo fractional derivative is the fractional derivative which was applied in this model.

The solutions were obtained directly using the state-space approach, with the Laplace transform and the general solutions for any set of boundary conditions obtained in the Laplace transform domain. Tzou’s iteration method was used to compute the inverse Laplace transforms. The distributions of temperature increment, volumetric dilatation, displacement, stress, induced magnetic field, and induced electric field were discovered and discussed.

The time-fractional Maxwell’s equation parameter based on the Caputo fractional derivative has significant influence on all the functions under investigation.

The time-fractional parameter of Maxwell’s equations functions as a hindrance to the particle’s displacement, deformation, and the induced magnetic field, but it catalyses the generated electric field inside the material.

The speed of propagation of the thermal wave under the Lord–Shulman theorem has a limited value, which is higher than the other theorems, and goes to zero faster, which makes the Lord–Shulman model the most successful model for the thermal transfer.

The speed of the mechanical and electromagnetic waves under the theorem of GN-III has a limited value, which is higher than its values under the other theorems. This means that the theorem of Green–Naghdi type III is the most successful model in explaining the mechanical and electromagnetic behaviour.

The time-fractional parameter of Maxwell’s equations works as a resistor to the volumetric dilatation, displacement, stress, and induced magnetic field, while it works as catalyst to the induced electric field.

Therefore, the time-fractional parameter of Maxwell’s equations could be used to control the thermomechanical and electromagnetic waves through thermoelastic materials.