# Noninteger Derivative Order Analysis on Plane Wave Reflection from Electro-Magneto-Thermo-Microstretch Medium with a Gravity Field within the Three-Phase Lag Model

Authors: S. H. Elhag; Fatimah S. Bayones; A. A. Kilany; S. M. Abo-Dahab; Emad A.-B. Abdel-Salam; M. Elsagheer; A. M. Abd-Alla
Journal: Advances in Mathematical Physics
 (2022)

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## Abstract

The present research paper illustrates how noninteger derivative order analysis affects the reflection of partial thermal expansion waves under the generalized theory of plane harmonic wave reflection from a semivacuum elastic solid material with both gravity and magnetic field in the three-phase lag model (3PHL). The main goal for this study is investigating the fractional order impact and the applications related to the orders, especially in biology, medicine, and bioinformatics, besides the integer order considering an external effect, such as electromagnetic, gravity, and phase lags in a microstretch medium. The problem fractional form was formulated, and the boundary conditions were applied. The results were displayed graphically, considering the 3PHL model with magnetic field, gravity, and relaxation time. These findings were an explicit comparison of the effect of the plane wave reflection amplitude with integer derivative order analysis and noninteger derivative order analysis. The fractional order was compared to the correspondence integer order that indicated to the difference between them and agreement with the applications in biology, medicine, and other related topics. This phenomenon has more applications in relation to the biology and biomathematics problems.


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## Body

## 1. Introduction

Recently, due attention has been made to study the interaction between electromagnetic field, thermal field, gravity, and the influence of microstretch due to its utilitarian features within different domains, such as physics, geophysics, geology, astronomy, and engineering. Choudhuri [1] proposed the problem of thermoelasticity theory with 3PHL. Kumar et al. [2] explored the plane strain problem within microstretch elastic solid. Kumar and Partap [3] examined the reflection of plane waves in a heat flux-dependent microstretch thermoelastic solid half space. Lord and Shulman [4] established generalized theories in which they replaced the Fourier law of thermal conductivity by Maxwell-Cattaneo law introducing a thermal relaxation time parameter within Fourier’s law. Green and Lindsay [5] illustrated binary relaxation parameters at the constitutive relationships regarding the entropy and stress tensor. After some years, Green and Naghdi [6–8] introduced new three models known as GN-I, GN-II, and GN-III. They argued that the linearized form of model-I agrees with the classical thermoelastic one. In model-II, the internal production rate of is typically zero, suggesting no dissipation of thermal energy. The model discloses undamped thermoelastic waves in a thermoelastic material. It is primarily indicated as the theory of thermoelasticity in the absence of the dissipation of energy. The former models are combined in model-III as special cases. The model discloses energy dissipation generally. Abbas and Kumar [9] discussed deformation because of the heat source in micropolar generalized thermoelastic half-space using the finite element technique. The authors of [10] explored the stabile solutions concerning the 3PHL thermal conductivity. Utilizing the theory, Kar and Kanoria [11] and Banik and Kanoria [12] resolved various issues.The initial stresses in solids considerably affect the mechanical response of the material from a primarily stressed configuration. They are widely applied in soft biological tissues’ performance, engineering structures, and geophysics. Elsagheer and Abo-Dahab [13] discussed the reflection of thermoelastic waves from insulated boundary fiber-reinforced half-space affected by the magnetic field and rotation. Abo-Dahab et al. [14] employed the Lord-Shulman and dual-phase lag models to explore how rotation and gravity affect an electro-magneto-thermoelastic medium in the presence of diffusion and voids. Using three thermoelastic theories, the authors of [15] discussed SV-waves occurrence at the interface between solid-liquid media under the initial stress and electromagnetic field. Abo-Dahab et al. [16] studied the impact of rotation and gravity on the reflection of P-waves from thermo-magneto-microstretch medium within the 3PHL model with initial stress.Several definitions were introduced concerning fractional derivatives. For instance, Riemann–Liouville’s definition is characterized by the fractional derivative of constant is not zero and applied for the non-differentiable functions. Caputo’s definition can be applied to differentiable functions, and zero is the value of the fractional derivative of constant [17].The fractional differential equations’ theory and applications were studied by Kilbas et al. [18]. Hilfer [17] discussed the uses of fractional calculus in physics. Katugampola [19] used a novel method to a generalized fractional integral. The authors of [20] discussed analytically solving the space-time fractional nonlinear Schrödinger equation. Abdel-Salam and Hassan [21] studied solving the class of linear and nonlinear fractional differential equations. Abdel-Salam and Yousif [22] discussed solving the nonlinear space-time fractional differential equations by the fractional Riccati expansion approach. The approximate solution to the fractional Lane-Emden kind equations was studied by Nouh and Abdel-Salam [23]. Examining the fractional derivative for natural convection in a slanted cavity having porous media was discussed by Ahmed et al. [24]. The authors of [25] discussed the impacts of the rotation, voids, magnetic field, and initial stress on plane waves in generalized thermoelasticity. Marin et al. [26] explained extending the domain of the influence theory concerning the generalized thermoelasticity of anisotropic material in the presence of voids. Saeed et al. [27] discussed the GL model on the thermoelastic interaction in a poroelastic material by the finite element approach. Abo-Dahab et al. [28] discussed the impact of the electromagnetic field in the fiber-reinforced micropolar thermoelastic medium in the context of four models.Many authors in [29–36] studied various applications of fractional calculus in mathematical modeling with a comparison to constant cases with physics properties. The analysis of the fractional derivative order and temperature-dependent characteristics on P- and SV-waves reflection influenced by initial stress and 3PHL model were considered by Abo-Dahab et al. [37]. Alotaibi et al. [38] studied the fractional calculus of thermoelastic P-wave reflection influenced by the electromagnetic fields and gravity. Hobiny and Abbas [39] analytically solved the fractional order photo-thermoelasticity within a nonhomogenous semiconductor medium. Povstenko and Kyrylych [40] illustrated the fractional thermoelasticity issue concerning a plane with a line crack influenced by heat flux loading.Many researchers in [41–47] investigated the effect of several variables in thermoelasticity with different models and archived all conditions to these models.In this paper, the reflection of the plane waves from a semivacuum elastic solid material with the electromagnetic field and gravitational under the influence of relaxation times was studied. The necessary comparisons were made to simplify and explain the phenomenon at the fractional and nonfractional differentiations. The reflection appeared in the effect of the amplitude of the plane waves where the presence of the fractional differential was evident in each of the different results of the phenomenon. The fractional order was compared to the correspondence integer order that indicated the application and agreement with the applications in biology, bioinformatics, medicine, and related topics.


### 1.1. Formulation of the Problem

We utilized the generalized fractional thermo-microstretch’s equations in a rectangular coordinate schemex,y,z with z-axis directed into the media. We take the constant magnetic field intensity H=0,H0,0 to represent the y-axis direction. We consider with linearized equations of electrodynamics in the presence of displacement current due to motion as (Ezzat [48])
(1)J=curlαh−ε0DtαE,∇α=Dxαi¯+Dyαj¯+Dzαk¯,curlαh=∇α×h=i¯j¯k¯DxαDyαDzα0h20,curlαE=−μ0Dtαh,curlαE=∇α×E=i¯j¯k¯DxαDyαDzαE10E3,E=−μ0Dtαu×H,Dtαu×H=i¯j¯k¯Dtαu10Dtαu30H00,∇α⋅h=0,Dtαh2=0.The fractional motion equation with gravitational and Lorentz’s body forces as (Paria [49], Lord and Shulman [4])
(2)Dℓασℓi+Fi+Gi=ρDtααui,where
(3)Fi=μ0J×Hi,Gi=ρgDxαu3,0,−Dzαu1.Using the previous equations, the following equations are obtained:(4)E=μ0H0Dtαu3,0,−Dtαu1,(5)h=0,h2,0=0,h,0=0,−H0e,0e=Dxαu1+Dzαu3,(6)J=−Dzαh−ε0μ0H0Dtααu3,0,Dxαh+ε0μ0H0Dtααu1=H0−DzαDxαu1−Dzααu3−ε0μ0Dtααu3,0,Dxααu1+DxαDzαu3+ε0μ0Dtααu1.From equations (3) and (5), we obtain
(7)F=F1α,F2α,F3α=μ0H02Dxαe−ε0μ02H02Dtααu1,0,μ0H02Dzαe−ε0μ02H02Dtααu3=μ0H02Dxααu1+DxαDzαu3−ε0μ0Dtααu1,0,DzαDxαu1+Dzααu3−ε0μ0Dtααu3.So, the displacement vectoru=u1,u2,u3 has the components u1=u1x,z,t,u2=0,u3=u3x,z,t.The basic governing equations become(8)λ+μDxααu1+DxαDzαu3+μ+kDxααu1+Dzααu1−kDzαφ2+λ0Dxαφ∗−γ^DxαT+F1+ρgDxαu3=ρDtααu1,(9)λ+μDxααu3+DxαDzαu1+μ+kDxααu3+Dzααu3+kDxαφ2+λ0Dzαφ∗−γ^DzαT+F3−ρgDzαu1=ρDtααu3,(10)a+b+c∇α∇α⋅φ−c∇α×∇α×φ+k∇α×u−2kφ=ρjDtααφ,(11)α0∇α2φ∗−13b1φ∗−13b0∇α⋅u+13γ^1T=32ρjDtααφ∗,(12)K∗+τν∗Dtα+KτTDtαα∇α2T=1+τqDtα+τq22DtααρCeDtααT+γ^T0Dtααe+γ^1T0Dtααφ∗.Such thatτν∗=K+K∗τν,∇α2=Dxαα+Dzαα.(13)σiℓ=b0φ∗+λDrαurδiℓ+μ+kDiαuℓ+μDℓαui−kεiℓrφr−γ^Tδiℓ,miℓ=aDrαφrδiℓ+bDℓαφi+cDiαφℓ,λi=a0Diαφ∗.The constitutive relations take the following form:(14)σxx=λ+2μ+kDxαu1+λDzαu3+b0φ∗−γ^T,σzz=λ+2μ+kDzαu3+λDxαu1+b0φ∗−γ^T,σxz=μDzαu1+μ+kDxαu3+kφ2,σzx=μDxαu3+μ+kDzαu1−kφ2,mxy=cDxαφ2,mzy=cDzαφ2,λx=a0Dxαφ∗,λz=a0Dzαφ∗.Moreover, we introduce these dimensionless quantities to make the solution easier(15)xi′=ω∗c0xi,ui′=ρc0ω∗γ^T0ui,Θ′=γ^ρc02T−T0,t′,τT′,τν′,τq′=ω∗t,τT,τν,τq,σij′=σijγ^T0,mij′=ω∗c0γ^T0mij,λi′=ω∗c0γ^T0λi,φ′∗=ρc02γ^T0φ∗,φ2′=ρc02γ^T0φ2,g′=gc0ω∗,ω∗=ρCec02k,h′=hH0,(16)c02=λ+2μ+kρ,i,j=1,2,3.Utilizing Equation (16), the governing Equations (8)–(12) recast in this form (after suppressing the primes)
(17)μ+kρc02∇α2u1+μ+λρc02+RHDxαe−kρc02Dzαφ2+λ0ρc02Dxαφ∗−ρc02γ^T0DxαΘ+gDxαu3=β2Dtααu1,(18)μ+kρc02∇α2u3+μ+λρc02+RHDzαe+kρc02Dxαφ2+λ0ρc02Dzαφ∗−ρc02γ^T0DzαΘ−gDxαu1=β2Dtααu3,(19)∇α2φ2−2kc02c2ω∗2φ2+kc02c2ω∗2Dzαu1−Dxαu3=ρjc02c2Dtααφ2,(20)C12c02∇α2−C22ω∗2−Dtααφ∗−C32ω∗2e+C4Θ=0,(21)Ck+CνDtα+CTDtαα∇α2Θ=1+τqDtα+τq22DtααDtααΘ+ε1Dtααe+ε2Dtααφ∗,(22)σxx=a1φ∗+Dxαu1+a2Dzαu3−a3Θ−1,(23)σzz=a1φ∗+Dzαu3+a2Dxαu1−a3Θ−1,(24)σxz=a4Dzαu1+a5Dxαu3+a6φ2,(25)σzx=a4Dzαu3+a5Dxαu1−a6φ2,(26)mxy=a7Dxαφ2,(27)mzy=a7Dzαφ2,(28)λx=a8Dxαφ∗,(29)λz=a8Dzαφ∗,where(30)RH=μ0H02ρc02,b2=1+ε0μ02H02ρ,C12,C22,C32=29ρj3a0,b1,b0,C4=2ρc04γ^19jγ∧2T0ω∗2c2=2a+2b+c,C=2c02γ^19jγ^ω∗2,Ck,Cv,CT=1ρc02CeΚ∗,τv∗,ΚτTω∗,ε1=γ∧3T0ρ3c04Ce,ε2=γ^1γ∧2T02ρ3c04Ce,a1,a2=1ρc02b0,λ,a3=ρc02γ^T,a4,a5,a6=1ρc02μ,μ+k,k,a7,a8=ω∗2ρc04c,a0.The components of the displacementu1x,z,t and u3x,z,t take the following form concerning the potential functions Φx,z,t and Ψx,z,t(31)u1=DxαΦ+DzαΨ,u3=DzαΦ−DxαΨ,Ψ⟶=0,−Ψ,0.Using Equation (31) in Equations (17)–(21), we get
(32)∇α2−ζ0DtααΦ−ζ1DxαΨ−ζ2Θ+ζ3φ∗=0,(33)ζ4DxαΦ+∇α2−ζ5DtααΨ−ζ6φ2=0,(34)kc02cω∗2∇α2Ψ+∇α2−2kc02cω∗2−jρc02cDtααφ2=0,(35)C12c02∇α2−C22ω∗2−Dtααφ∗−C32ω∗2∇α2Φ+a0Θ=0,(36)Ck∇α2Θ+Cν∇α2DtαΘ+CT∇α2DtααΘ=1+τqDtα+τq22DtααDtααΘ+ε1∇α2DtααΦ+ε2Dtααφ∗,where
(37)ζ0,ζ1,ζ2,ζ3=11+RHb2,g,ρc02γ^T0,b0ρc02,ζ4,ζ5,ζ6=ρc02μ+kg,b2,kρc02.


### 1.2. Solution of the Problem

In this section, solving Equations (32)–(36) is assumed as
(38)Φ,Ψ,Θ,φ∗,φ2=Φ¯,Ψ¯,Θ¯,φ∗¯,φ2¯expiξxαsinθ+zαcosθ−iωtα.Substituting from Equation (38) into Equtions (32)–(36), the result is
(39)α2−ξ2+ζ0ω2Φ¯−iαζ1ξsinθΨ¯−ζ2Θ¯+ζ3φ∗¯=0,(40)iαξζ4sinθΦ¯+α2−ξ2+ω2ζ5Ψ¯−ζ6φ2¯=0,(41)−kα2ξ2c02cω∗2Ψ¯+−α2ξ2+jρc02ω2α2c−2kc02cω∗2φ2¯=0,(42)α2ξ2C32ω∗2Φ¯+a0Θ¯+−α2ξ2C12c02−C22ω∗2+α2ω2φ∗¯=0,(43)−ε1α4ξ2ω2τq∗Φ¯+−Ck+iCναω+CTα2ω2α2ξ2+α2ω2τq∗Θ¯+ε2α2ω2τq∗φ¯∗=0.where τq∗=1−iαωτq−α2ω2τq2/2.To help identify the nontrivial solution, we change from Equation (39) into Equations (33)–(38) and get
(44)α2ζ0ω2−ξ2−iαζ1ξsinθ−ζ2ζ30iαξζ4sinθα2−ξ2+ω2ζ500−ζ60−kα2ξ2c02cω∗200α2jρc02ω2b−ξ2−2kc02bω∗2α2ξ2C32ω∗20a0α2ω2−C12ξ2C02−C22ω∗20−ε1α4ξ2ω2τq∗0α2ξ2iαωCv+α2ω2CT−Ck+α2ω2τq∗ε2α2ω2τq∗0=0,which tends to
(45)Lα10ξ10+Mα8ξ8+Nα6ξ6+Oα4ξ4+Pα2ξ2+Q=0.where L, M, N, O, P, and Q are in the Appendix.From Equation (45), we can identify five diverse velocities accompanying five waves.After that,Φ,Ψ,Θ,φ∗,andφ2 are written as
(46)Φ,Ψ,Θ,φ∗,φ2=1,η0,κ0,χ0,ϑ0A0expiξxαsinθ+zαcosθ−iωtα∑j=151,ηj,κj,χj,ϑjAjexpiξjxαsinθj−zαcosθj−iωtα.Based on Equations (40) and (41), the result is
(47)ηj=−ιξjζ4sinθjξj2−jρc02α2ω2/γ^+2kc02/cω∗2ξj2−α2ω2ζ5ξj2−jρc02α2ω2/γ^+2kc02/cω∗2−p∗ξj2ζ6,κj=ip∗ξj3ζ4sinθjξj2−α2ω2ζ5ξj2−jρc02α2ω2/γ^+2kc02/cω∗2−p∗ξj2ζ6,where p∗=kc02/γω∗2.Also, from Equations (42) and (43), we get
(48)χj=−ε2α2ω2τq∗ξj2C32/ω∗2−ε1α2ω2τq∗ξj2−C12/c02ξj2−C22/ω∗2+α2ω2a0ε2α2ω2τq∗−−Ck−iαωCv+α2ω2CTξj2+ω2τq∗−C12/c02ξj2−C22/ω∗2+α2ω2,ϑj=a0ε1α2ω2τq∗ξj2+C32/ω∗2ξj2−Ck−iαωCv+α2ω2CTξj2+α2ω2τq∗a0ε2α2ω2τq∗−−Ck−iαωCv+α2ω2CTξj2+α2ω2τq∗−C12/c02ξj2−C22/ω∗2+α2ω2.


### 1.3. The Boundary Conditions

The problem’s boundary conditions are written as follows:(49)σzz+τzz=0,σxz+τxz=0,DzαT=0,myz=0,λx=0atz=0,(50)∑j=15a1ϑj−μeH02ξj2−a3ξj2sin2θj+ηj2sin2θj−a2ξj2cos2θj−ηj2sin2θj+kjξj2Aj=a1ϑ1−μeH02ξ12−a3ξ12sin2θ0+η12sin2θ0−a2ξ12cos2θ0−η12sin2θ0+k1ξ12A0,(51)∑j=15a4+a5ξj22sin2θj+a4ξj2ηjcos2θj−a5ξj2ηjsin2θj+a6ξjAj=−a4+a5ξ122sin2θ0+a4ξ12η1cos2θ0−a5ξ12η1sin2θ0+a6ξ1A0,(52)∑j=15kjξjcosθjAj=k1ξ1cosθ0A0,(53)∑j=15ζjξjcosθjAj=ζ1ξ1cosθ0A0.Finally,(54)∑j=15ϑjξjcosθjAj=−ϑ1ξ1cosθ0A0.Based on Equations (50)–(54), the result is
(55)aijZj=Bi,i,j=1,2,⋯,5,Zj=AjA0,a1j=a1ϑj−ξj2a3sin2θj+ηj2sin2θj−a2cos2θj−ηj2sin2θj+kjξj2+μeH02,a2j=a4+a5ξj22sin2θj+a4ξj2ηjcos2θj−a5ξj2ηjsin2θj+a6ξj,a3j=kjξjcosθj,a4j=ζjξjcosθjAj,a5j=ϑjξjcosθj,B1=−a1ϑ1+ξj2a3sin2θ0+η12sin2θ0−a2cos2θ0−η12sin2θ0+kjξj2+μeH02,B2=−a4+a5ξ122sin2θ0−a4ξ12η1cos2θ1+a5ξ12η1sin2θ1−a6ξ1,B3=k1ξ1cosθ0,B4=ζ1ξ1cosθ0,B5=−ϑ1ξ1cosθ0.


### 1.4. Numerical Results and Discussion

In this section, some numerical findings are discussed based on the illustration of the results obtained in the former sections and a comparison of such results in different cases.Therefore, we chose these materials:(56)i=−1,a0=0.779×10−4,b0=0.5×1011,b1=0.5×1011,j=0.2×10−15,ρ=8954,Ce=383.1,k=386,T0=293,λ=7.76×1010,μ=3.86×1010,c=0.779×10−4,Κ∗=2.97×1013,μ0=0.1,ε0=0.1,ω=0.034,τT=0.2,τv=0.1,τq=0.5.Figure1 highlights the various values of the amplitudes z1 concerning the angle of incidence of the reflected waves θ for various values of the fractional parameter of two different values of gravity g. Clearly, the amplitudes of incidence waves z1 decline when rising the gravity field. They decline then rise with the increase of θ to reach the unit at θ=90°.Figure 1
The various values of the amplitudesz1 concerning the angle of incidence θ for various values of the fractional parameter of the diverse values of gravity g.In the case ofg=0.1, the amplitude of the wave is less than in the case of g=0.25, indicating that the greater the effect of gravity leads to an increase in the wave’s amplitude.Figure2 displays the various values of amplitudes z2 concerning the angle of incidence of the reflected waves θ for various values of the fractional parameter for the two diverse values of gravity g that shows the oscillatory performance in the entire range of angle θ. Clearly, the amplitudes of reflection waves z2 rise when rising the gravity field, which rises then declines with increasing θ approaches zero at θ=90°.Figure 2
The various values of the amplitudesz2 concerning the angle of incidence θ for various values of the fractional parameter of the diverse values of the values of gravityg.In the case ofz2, the wave amplitude started to increase significantly than z1 at g=0.1, and the effect of the fractional differential appears more clearly in the case of α=0.90 or α=0.94, both at g=0.1 or g=0.25.Figure3 highlights the various values of amplitudes z3 concerning the angle of incidence of the reflected waves θ for various values of the fractional parameter for the two values of gravity g that shows the oscillatory performance in the entire range of angle θ. Clearly, the amplitudes of reflection waves z3 rise when rising the gravity field and decline when rising θ until it equals zero at θ=90°.Figure 3
The various values of the amplitudesz3 concerning the angle of incidence θ for various values of the fractional parameter of the diverse values of gravity g.In the case ofz3, the effect of the wave amplitude shows a slight change in 0≤θ≤10, and it is not affected by the gravity at the lowest value of the fractional differential parameter at α=0.90, while the rest of the values for the fractional differential parameter are clear.Figures4 and 5 display the various values of amplitudes z4,z5 concerning the angle of incidence of the reflected waves θ regarding various values of the fractional parameter for the two diverse values of gravity g that escalates in the whole entire of angle θ. Clearly, the amplitudes of the reflection waves z4,z5 increase with a higher gravity field. They rise then decline when rising θ until it equals zero at θ=90°. In the wave amplitude z4, the effect of the fractional differential parameter at α=0.90 takes the same behavior whether gravitational g=0.1 or g=0.25. In the case of z5, the effect of the wave amplitude shows a slight change in 10≤θ≤20, and it is not affected by the gravity at the lowest value of the fractional differential parameter at α=0.90, while the rest of the values for the fractional differential parameter are clear.Figure 4
The various values of the amplitudesz4 concerning the angle of incidence θ for various values of the fractional parameter of the diverse values of gravity g.Figure 5
The various values of the amplitudesz5 concerning the angle of incidence θ for various values of the fractional parameter of the diverse values of gravity g.Figure6 illustrates the magnitude of amplitude ratios z1 concerning the angle of incidence of the reflected waves θ regarding various values of the fractional parameter α of two diverse values of the magnetic field H0. The amplitudes of incidence waves z1 are lower when rising the magnetic field H0 that decline then grows with rising θ to reach the unit at θ=90°. It appears that the amplitude of the wave is lower in the case of the fractional differential parameter at α=0.90 along 0≤θ≤90, while it increases with an increase of α=0.94 and is more amplitude in the case of α=1.Figure 6
The various values of the amplitudesz1 concerning the angle of incidence θ for various values of the fractional parameter of the diverse values of the magnetic field H0.Figures7–10 display the various values of amplitudes z2,z3,z4,z5 concerning the angle of incidence of the reflected waves θ for various values of the fractional parameter α of the two different values of the magnetic field H0, which escalates in the entire range of angle θ. Obviously, the amplitude reflection waves z2,z3,z4,z5 decrease with an increase of the magnetic field H0 and decrease with an increase of θ until equal zero at θ=90°. In the case of amplitudes z2,z3,z5, the behavior of the waves appears to be the same in all cases for different alpha values. At the same time, the effect of the fractal calculus parameter is less amplitude than the absence of fractional differential parameter. In the case of z4, the impact of the magnetic field, whether H0=8×106 or H0=8.5×106 is not affected by the wave amplitude, as we see the effect of fractional differentiation in all the fractional differential parameter values.Figure 7
The various values of the amplitudesz2 concerning the angle of incidence θ for various values of the fractional parameter of the diverse values of the magnetic field H0.Figure 8
The various values of the amplitudesz3 concerning the angle of incidence θ for various values of the fractional parameter of the diverse values of the magnetic field H0.Figure 9
The various values of the amplitudesz4 concerning the angle of incidence θ for various values of the fractional parameter of the diverse values of the magnetic field H0.Figure 10
The various values of the amplitudesz5 concerning the angle of incidence θ for various values of the fractional parameter of the diverse values of the magnetic field H0.In sum, the method of the fractional derivative technique applies to other relevant issues in geology, geophysics, physics, astronomy, and engineering, agreeing with the integer derivative ifα=1.If the fractional order considers only an integer without an electric field, the results obtained are deduced to the results given in Othman et al. [46].


## 1.1. Formulation of the Problem

We utilized the generalized fractional thermo-microstretch’s equations in a rectangular coordinate schemex,y,z with z-axis directed into the media. We take the constant magnetic field intensity H=0,H0,0 to represent the y-axis direction. We consider with linearized equations of electrodynamics in the presence of displacement current due to motion as (Ezzat [48])
(1)J=curlαh−ε0DtαE,∇α=Dxαi¯+Dyαj¯+Dzαk¯,curlαh=∇α×h=i¯j¯k¯DxαDyαDzα0h20,curlαE=−μ0Dtαh,curlαE=∇α×E=i¯j¯k¯DxαDyαDzαE10E3,E=−μ0Dtαu×H,Dtαu×H=i¯j¯k¯Dtαu10Dtαu30H00,∇α⋅h=0,Dtαh2=0.The fractional motion equation with gravitational and Lorentz’s body forces as (Paria [49], Lord and Shulman [4])
(2)Dℓασℓi+Fi+Gi=ρDtααui,where
(3)Fi=μ0J×Hi,Gi=ρgDxαu3,0,−Dzαu1.Using the previous equations, the following equations are obtained:(4)E=μ0H0Dtαu3,0,−Dtαu1,(5)h=0,h2,0=0,h,0=0,−H0e,0e=Dxαu1+Dzαu3,(6)J=−Dzαh−ε0μ0H0Dtααu3,0,Dxαh+ε0μ0H0Dtααu1=H0−DzαDxαu1−Dzααu3−ε0μ0Dtααu3,0,Dxααu1+DxαDzαu3+ε0μ0Dtααu1.From equations (3) and (5), we obtain
(7)F=F1α,F2α,F3α=μ0H02Dxαe−ε0μ02H02Dtααu1,0,μ0H02Dzαe−ε0μ02H02Dtααu3=μ0H02Dxααu1+DxαDzαu3−ε0μ0Dtααu1,0,DzαDxαu1+Dzααu3−ε0μ0Dtααu3.So, the displacement vectoru=u1,u2,u3 has the components u1=u1x,z,t,u2=0,u3=u3x,z,t.The basic governing equations become(8)λ+μDxααu1+DxαDzαu3+μ+kDxααu1+Dzααu1−kDzαφ2+λ0Dxαφ∗−γ^DxαT+F1+ρgDxαu3=ρDtααu1,(9)λ+μDxααu3+DxαDzαu1+μ+kDxααu3+Dzααu3+kDxαφ2+λ0Dzαφ∗−γ^DzαT+F3−ρgDzαu1=ρDtααu3,(10)a+b+c∇α∇α⋅φ−c∇α×∇α×φ+k∇α×u−2kφ=ρjDtααφ,(11)α0∇α2φ∗−13b1φ∗−13b0∇α⋅u+13γ^1T=32ρjDtααφ∗,(12)K∗+τν∗Dtα+KτTDtαα∇α2T=1+τqDtα+τq22DtααρCeDtααT+γ^T0Dtααe+γ^1T0Dtααφ∗.Such thatτν∗=K+K∗τν,∇α2=Dxαα+Dzαα.(13)σiℓ=b0φ∗+λDrαurδiℓ+μ+kDiαuℓ+μDℓαui−kεiℓrφr−γ^Tδiℓ,miℓ=aDrαφrδiℓ+bDℓαφi+cDiαφℓ,λi=a0Diαφ∗.The constitutive relations take the following form:(14)σxx=λ+2μ+kDxαu1+λDzαu3+b0φ∗−γ^T,σzz=λ+2μ+kDzαu3+λDxαu1+b0φ∗−γ^T,σxz=μDzαu1+μ+kDxαu3+kφ2,σzx=μDxαu3+μ+kDzαu1−kφ2,mxy=cDxαφ2,mzy=cDzαφ2,λx=a0Dxαφ∗,λz=a0Dzαφ∗.Moreover, we introduce these dimensionless quantities to make the solution easier(15)xi′=ω∗c0xi,ui′=ρc0ω∗γ^T0ui,Θ′=γ^ρc02T−T0,t′,τT′,τν′,τq′=ω∗t,τT,τν,τq,σij′=σijγ^T0,mij′=ω∗c0γ^T0mij,λi′=ω∗c0γ^T0λi,φ′∗=ρc02γ^T0φ∗,φ2′=ρc02γ^T0φ2,g′=gc0ω∗,ω∗=ρCec02k,h′=hH0,(16)c02=λ+2μ+kρ,i,j=1,2,3.Utilizing Equation (16), the governing Equations (8)–(12) recast in this form (after suppressing the primes)
(17)μ+kρc02∇α2u1+μ+λρc02+RHDxαe−kρc02Dzαφ2+λ0ρc02Dxαφ∗−ρc02γ^T0DxαΘ+gDxαu3=β2Dtααu1,(18)μ+kρc02∇α2u3+μ+λρc02+RHDzαe+kρc02Dxαφ2+λ0ρc02Dzαφ∗−ρc02γ^T0DzαΘ−gDxαu1=β2Dtααu3,(19)∇α2φ2−2kc02c2ω∗2φ2+kc02c2ω∗2Dzαu1−Dxαu3=ρjc02c2Dtααφ2,(20)C12c02∇α2−C22ω∗2−Dtααφ∗−C32ω∗2e+C4Θ=0,(21)Ck+CνDtα+CTDtαα∇α2Θ=1+τqDtα+τq22DtααDtααΘ+ε1Dtααe+ε2Dtααφ∗,(22)σxx=a1φ∗+Dxαu1+a2Dzαu3−a3Θ−1,(23)σzz=a1φ∗+Dzαu3+a2Dxαu1−a3Θ−1,(24)σxz=a4Dzαu1+a5Dxαu3+a6φ2,(25)σzx=a4Dzαu3+a5Dxαu1−a6φ2,(26)mxy=a7Dxαφ2,(27)mzy=a7Dzαφ2,(28)λx=a8Dxαφ∗,(29)λz=a8Dzαφ∗,where(30)RH=μ0H02ρc02,b2=1+ε0μ02H02ρ,C12,C22,C32=29ρj3a0,b1,b0,C4=2ρc04γ^19jγ∧2T0ω∗2c2=2a+2b+c,C=2c02γ^19jγ^ω∗2,Ck,Cv,CT=1ρc02CeΚ∗,τv∗,ΚτTω∗,ε1=γ∧3T0ρ3c04Ce,ε2=γ^1γ∧2T02ρ3c04Ce,a1,a2=1ρc02b0,λ,a3=ρc02γ^T,a4,a5,a6=1ρc02μ,μ+k,k,a7,a8=ω∗2ρc04c,a0.The components of the displacementu1x,z,t and u3x,z,t take the following form concerning the potential functions Φx,z,t and Ψx,z,t(31)u1=DxαΦ+DzαΨ,u3=DzαΦ−DxαΨ,Ψ⟶=0,−Ψ,0.Using Equation (31) in Equations (17)–(21), we get
(32)∇α2−ζ0DtααΦ−ζ1DxαΨ−ζ2Θ+ζ3φ∗=0,(33)ζ4DxαΦ+∇α2−ζ5DtααΨ−ζ6φ2=0,(34)kc02cω∗2∇α2Ψ+∇α2−2kc02cω∗2−jρc02cDtααφ2=0,(35)C12c02∇α2−C22ω∗2−Dtααφ∗−C32ω∗2∇α2Φ+a0Θ=0,(36)Ck∇α2Θ+Cν∇α2DtαΘ+CT∇α2DtααΘ=1+τqDtα+τq22DtααDtααΘ+ε1∇α2DtααΦ+ε2Dtααφ∗,where
(37)ζ0,ζ1,ζ2,ζ3=11+RHb2,g,ρc02γ^T0,b0ρc02,ζ4,ζ5,ζ6=ρc02μ+kg,b2,kρc02.


## 1.2. Solution of the Problem

In this section, solving Equations (32)–(36) is assumed as
(38)Φ,Ψ,Θ,φ∗,φ2=Φ¯,Ψ¯,Θ¯,φ∗¯,φ2¯expiξxαsinθ+zαcosθ−iωtα.Substituting from Equation (38) into Equtions (32)–(36), the result is
(39)α2−ξ2+ζ0ω2Φ¯−iαζ1ξsinθΨ¯−ζ2Θ¯+ζ3φ∗¯=0,(40)iαξζ4sinθΦ¯+α2−ξ2+ω2ζ5Ψ¯−ζ6φ2¯=0,(41)−kα2ξ2c02cω∗2Ψ¯+−α2ξ2+jρc02ω2α2c−2kc02cω∗2φ2¯=0,(42)α2ξ2C32ω∗2Φ¯+a0Θ¯+−α2ξ2C12c02−C22ω∗2+α2ω2φ∗¯=0,(43)−ε1α4ξ2ω2τq∗Φ¯+−Ck+iCναω+CTα2ω2α2ξ2+α2ω2τq∗Θ¯+ε2α2ω2τq∗φ¯∗=0.where τq∗=1−iαωτq−α2ω2τq2/2.To help identify the nontrivial solution, we change from Equation (39) into Equations (33)–(38) and get
(44)α2ζ0ω2−ξ2−iαζ1ξsinθ−ζ2ζ30iαξζ4sinθα2−ξ2+ω2ζ500−ζ60−kα2ξ2c02cω∗200α2jρc02ω2b−ξ2−2kc02bω∗2α2ξ2C32ω∗20a0α2ω2−C12ξ2C02−C22ω∗20−ε1α4ξ2ω2τq∗0α2ξ2iαωCv+α2ω2CT−Ck+α2ω2τq∗ε2α2ω2τq∗0=0,which tends to
(45)Lα10ξ10+Mα8ξ8+Nα6ξ6+Oα4ξ4+Pα2ξ2+Q=0.where L, M, N, O, P, and Q are in the Appendix.From Equation (45), we can identify five diverse velocities accompanying five waves.After that,Φ,Ψ,Θ,φ∗,andφ2 are written as
(46)Φ,Ψ,Θ,φ∗,φ2=1,η0,κ0,χ0,ϑ0A0expiξxαsinθ+zαcosθ−iωtα∑j=151,ηj,κj,χj,ϑjAjexpiξjxαsinθj−zαcosθj−iωtα.Based on Equations (40) and (41), the result is
(47)ηj=−ιξjζ4sinθjξj2−jρc02α2ω2/γ^+2kc02/cω∗2ξj2−α2ω2ζ5ξj2−jρc02α2ω2/γ^+2kc02/cω∗2−p∗ξj2ζ6,κj=ip∗ξj3ζ4sinθjξj2−α2ω2ζ5ξj2−jρc02α2ω2/γ^+2kc02/cω∗2−p∗ξj2ζ6,where p∗=kc02/γω∗2.Also, from Equations (42) and (43), we get
(48)χj=−ε2α2ω2τq∗ξj2C32/ω∗2−ε1α2ω2τq∗ξj2−C12/c02ξj2−C22/ω∗2+α2ω2a0ε2α2ω2τq∗−−Ck−iαωCv+α2ω2CTξj2+ω2τq∗−C12/c02ξj2−C22/ω∗2+α2ω2,ϑj=a0ε1α2ω2τq∗ξj2+C32/ω∗2ξj2−Ck−iαωCv+α2ω2CTξj2+α2ω2τq∗a0ε2α2ω2τq∗−−Ck−iαωCv+α2ω2CTξj2+α2ω2τq∗−C12/c02ξj2−C22/ω∗2+α2ω2.


## 1.3. The Boundary Conditions

The problem’s boundary conditions are written as follows:(49)σzz+τzz=0,σxz+τxz=0,DzαT=0,myz=0,λx=0atz=0,(50)∑j=15a1ϑj−μeH02ξj2−a3ξj2sin2θj+ηj2sin2θj−a2ξj2cos2θj−ηj2sin2θj+kjξj2Aj=a1ϑ1−μeH02ξ12−a3ξ12sin2θ0+η12sin2θ0−a2ξ12cos2θ0−η12sin2θ0+k1ξ12A0,(51)∑j=15a4+a5ξj22sin2θj+a4ξj2ηjcos2θj−a5ξj2ηjsin2θj+a6ξjAj=−a4+a5ξ122sin2θ0+a4ξ12η1cos2θ0−a5ξ12η1sin2θ0+a6ξ1A0,(52)∑j=15kjξjcosθjAj=k1ξ1cosθ0A0,(53)∑j=15ζjξjcosθjAj=ζ1ξ1cosθ0A0.Finally,(54)∑j=15ϑjξjcosθjAj=−ϑ1ξ1cosθ0A0.Based on Equations (50)–(54), the result is
(55)aijZj=Bi,i,j=1,2,⋯,5,Zj=AjA0,a1j=a1ϑj−ξj2a3sin2θj+ηj2sin2θj−a2cos2θj−ηj2sin2θj+kjξj2+μeH02,a2j=a4+a5ξj22sin2θj+a4ξj2ηjcos2θj−a5ξj2ηjsin2θj+a6ξj,a3j=kjξjcosθj,a4j=ζjξjcosθjAj,a5j=ϑjξjcosθj,B1=−a1ϑ1+ξj2a3sin2θ0+η12sin2θ0−a2cos2θ0−η12sin2θ0+kjξj2+μeH02,B2=−a4+a5ξ122sin2θ0−a4ξ12η1cos2θ1+a5ξ12η1sin2θ1−a6ξ1,B3=k1ξ1cosθ0,B4=ζ1ξ1cosθ0,B5=−ϑ1ξ1cosθ0.


## 1.4. Numerical Results and Discussion

In this section, some numerical findings are discussed based on the illustration of the results obtained in the former sections and a comparison of such results in different cases.Therefore, we chose these materials:(56)i=−1,a0=0.779×10−4,b0=0.5×1011,b1=0.5×1011,j=0.2×10−15,ρ=8954,Ce=383.1,k=386,T0=293,λ=7.76×1010,μ=3.86×1010,c=0.779×10−4,Κ∗=2.97×1013,μ0=0.1,ε0=0.1,ω=0.034,τT=0.2,τv=0.1,τq=0.5.Figure1 highlights the various values of the amplitudes z1 concerning the angle of incidence of the reflected waves θ for various values of the fractional parameter of two different values of gravity g. Clearly, the amplitudes of incidence waves z1 decline when rising the gravity field. They decline then rise with the increase of θ to reach the unit at θ=90°.Figure 1
The various values of the amplitudesz1 concerning the angle of incidence θ for various values of the fractional parameter of the diverse values of gravity g.In the case ofg=0.1, the amplitude of the wave is less than in the case of g=0.25, indicating that the greater the effect of gravity leads to an increase in the wave’s amplitude.Figure2 displays the various values of amplitudes z2 concerning the angle of incidence of the reflected waves θ for various values of the fractional parameter for the two diverse values of gravity g that shows the oscillatory performance in the entire range of angle θ. Clearly, the amplitudes of reflection waves z2 rise when rising the gravity field, which rises then declines with increasing θ approaches zero at θ=90°.Figure 2
The various values of the amplitudesz2 concerning the angle of incidence θ for various values of the fractional parameter of the diverse values of the values of gravityg.In the case ofz2, the wave amplitude started to increase significantly than z1 at g=0.1, and the effect of the fractional differential appears more clearly in the case of α=0.90 or α=0.94, both at g=0.1 or g=0.25.Figure3 highlights the various values of amplitudes z3 concerning the angle of incidence of the reflected waves θ for various values of the fractional parameter for the two values of gravity g that shows the oscillatory performance in the entire range of angle θ. Clearly, the amplitudes of reflection waves z3 rise when rising the gravity field and decline when rising θ until it equals zero at θ=90°.Figure 3
The various values of the amplitudesz3 concerning the angle of incidence θ for various values of the fractional parameter of the diverse values of gravity g.In the case ofz3, the effect of the wave amplitude shows a slight change in 0≤θ≤10, and it is not affected by the gravity at the lowest value of the fractional differential parameter at α=0.90, while the rest of the values for the fractional differential parameter are clear.Figures4 and 5 display the various values of amplitudes z4,z5 concerning the angle of incidence of the reflected waves θ regarding various values of the fractional parameter for the two diverse values of gravity g that escalates in the whole entire of angle θ. Clearly, the amplitudes of the reflection waves z4,z5 increase with a higher gravity field. They rise then decline when rising θ until it equals zero at θ=90°. In the wave amplitude z4, the effect of the fractional differential parameter at α=0.90 takes the same behavior whether gravitational g=0.1 or g=0.25. In the case of z5, the effect of the wave amplitude shows a slight change in 10≤θ≤20, and it is not affected by the gravity at the lowest value of the fractional differential parameter at α=0.90, while the rest of the values for the fractional differential parameter are clear.Figure 4
The various values of the amplitudesz4 concerning the angle of incidence θ for various values of the fractional parameter of the diverse values of gravity g.Figure 5
The various values of the amplitudesz5 concerning the angle of incidence θ for various values of the fractional parameter of the diverse values of gravity g.Figure6 illustrates the magnitude of amplitude ratios z1 concerning the angle of incidence of the reflected waves θ regarding various values of the fractional parameter α of two diverse values of the magnetic field H0. The amplitudes of incidence waves z1 are lower when rising the magnetic field H0 that decline then grows with rising θ to reach the unit at θ=90°. It appears that the amplitude of the wave is lower in the case of the fractional differential parameter at α=0.90 along 0≤θ≤90, while it increases with an increase of α=0.94 and is more amplitude in the case of α=1.Figure 6
The various values of the amplitudesz1 concerning the angle of incidence θ for various values of the fractional parameter of the diverse values of the magnetic field H0.Figures7–10 display the various values of amplitudes z2,z3,z4,z5 concerning the angle of incidence of the reflected waves θ for various values of the fractional parameter α of the two different values of the magnetic field H0, which escalates in the entire range of angle θ. Obviously, the amplitude reflection waves z2,z3,z4,z5 decrease with an increase of the magnetic field H0 and decrease with an increase of θ until equal zero at θ=90°. In the case of amplitudes z2,z3,z5, the behavior of the waves appears to be the same in all cases for different alpha values. At the same time, the effect of the fractal calculus parameter is less amplitude than the absence of fractional differential parameter. In the case of z4, the impact of the magnetic field, whether H0=8×106 or H0=8.5×106 is not affected by the wave amplitude, as we see the effect of fractional differentiation in all the fractional differential parameter values.Figure 7
The various values of the amplitudesz2 concerning the angle of incidence θ for various values of the fractional parameter of the diverse values of the magnetic field H0.Figure 8
The various values of the amplitudesz3 concerning the angle of incidence θ for various values of the fractional parameter of the diverse values of the magnetic field H0.Figure 9
The various values of the amplitudesz4 concerning the angle of incidence θ for various values of the fractional parameter of the diverse values of the magnetic field H0.Figure 10
The various values of the amplitudesz5 concerning the angle of incidence θ for various values of the fractional parameter of the diverse values of the magnetic field H0.In sum, the method of the fractional derivative technique applies to other relevant issues in geology, geophysics, physics, astronomy, and engineering, agreeing with the integer derivative ifα=1.If the fractional order considers only an integer without an electric field, the results obtained are deduced to the results given in Othman et al. [46].


## 2. Conclusion

The plane harmonic waves’ reflection from a semi-infinite elastic solid of thermo-microstretch was studied in this article under the thermoelasticity theory with three-phase lag. Expressing the reflection coefficients that represent the relationships of the amplitudes of the reflected waves to the amplitude of the incidence waves obtained the reflection coefficient ratio variations with the angle of incidence with changing the dielectric constant, magnetic field, and gravity field. The results were compared in the cases of the existence and negligence of the parameter fractional.Due to their practical issues, we recommend utilizing the research findings in other fields, such as geophysics, engineering, geology, volcanoes, earthquakes, and structures. Future works will consider taking into account the effect of radiation, rotation, and other external parameters related to the phenomenon topics applicable in the environment.



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*Source: 6559779-2022-01-06.xml*