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Generalized model of thermoelasticity associated with fractional time-derivative operators and its applications to non-simple elastic materials

  • Adam Zakria , Ahmed E. Abouelregal ORCID logo EMAIL logo , Doaa Atta and Meshary Aleselmi
Published/Copyright: September 27, 2024
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Open Physics
From the journal Open Physics Volume 22 Issue 1

Abstract

This study proposes a comprehensive heat conduction model that incorporates fractional time derivatives and two-phase lags to describe the behavior of non-simple thermoelastic materials accurately. Generalized fractional differential operators with non-singular kernels are introduced. This type of fractional derivative includes the Caputo–Fabrizio and the Atangana–Baleanu fractional derivatives. The model also consists of the two-temperature idea, which considers the effect of microstructure through a two-stage delay approach. Interactions of a thermoelastic nature caused by the rapid heating of an isotropic substance under the influence of an external body force were studied as a practical application of the new concept. There has been some discussion about the effect of the discrepancy index and fractional differential operators. Finally, the graphical representations obtained from the numerical simulations were used to explain the behavior of the studied physical fields. The generalized fractional heat transfer model is demonstrated to be capable of producing a temperature forecast that is in close agreement with experimental data. As a result, the proposed model may be useful for solving difficulties in heat transfer, anomalous transport, and other branches of engineering analysis.

Keywords: fractional thermoelasticity; two-temperature; phaselags; non-singular kernels

Nomenclature

λ , μ

Lamé ’ s elastic parameters

α t

thermal expansion parameter

C s

specific heat

γ = ( 3 λ + 2 μ ) α t

thermal coupling parameter

T 0

reference temperature

θ = T − T 0

temperature change

T

absolute temperature

u

displacement vector

e = div u

dilatation

σ ij

stresses

e ij

strains

H

heat flow vector

K

thermal conductivity

ρ

material density

Q

heat source

t

instant time

δ ij

Kronecker ′ s delta

ϕ

conductive temperature

τ q

phase delay of heat flow

τ ϕ

phase deley of conductive temperature

D t ( α )

fractional operator

τ 0

thermal relaxation time

α

fractional orders

η

The entropy

1 Introduction

Generalized thermoelasticity is a subfield of continuum mechanics that builds on the traditional theory of thermoelasticity to include more realistic and complicated responses from materials. In the field of conventional thermoelasticity, the prevailing assumption is that heat transmission occurs immediately. However, in the context of extended thermoelasticity, this assumption is loosened. The present theory considers a limited thermal wave velocity, a factor that holds significance in specific materials and circumstances [1]. The concept of generalized thermoelasticity has found use in various fields, including investigating heat propagation in semiconductors, examining thermal shock, and modeling and simulating biological tissues.

In contrast to conventional thermoelasticity, which assumes instantaneous heat transfer as described by the parabolic heat transfer equation, extensional thermoelasticity involves hyperbolic heat transfer models. These models often include thermal delays, measuring the amount of time required for temperature changes to propagate through a given material [2]. Lord and Shulman [3] proposed a comprehensive theory of thermoelasticity that incorporates a single relaxation period. Consequently, the governing system for the heat equation has transformed into a hyperbolic form. In their study, Green and Lindsay [4] put forward an alternative model that incorporates two distinct relaxation phases. In their research, Tzou [5,6] introduced a thermoelastic model incorporating the dual-phase-lag (DPL) approach to both the heat flow vector and the temperature gradient. In his work, Choudhuri [7] expanded the DPL model to derive the equation for heat conduction’s three-phase lags (TPLs). This model allows for the estimation of TPL based on the heat flow component, temperature gradient, and thermal displacement, as opposed to relying on the classical Fourier transform law of heat transfer. The theories proposed by Green and Naghdi [8–10] encompass the fundamental principles of thermoelasticity, including both dissipative and non-dissipative systems. The aforementioned models introduce significant fundamental modifications to the constitutive equations, allowing the solution of a much wider range of heat flow problems, which can be classified as types I, II, and III. In recent times, several endeavors have been undertaken to alter the conventional heat transfer rule. Abouelregal et al. [11–15] made significant contributions by introducing generalized versions of heat conduction that incorporate higher-order time-derivative terms.

The two-temperature theory (2TT) approach is employed in examining thermal and heat transport processes in specific materials, necessitating the differentiation between two separate temperatures. This theory is especially relevant in materials where the movement of charge carriers, such as electrons, significantly affects the heat conduction process and where there may be a significant temperature difference between these carriers and the lattice or crystalline structure. Numerous researchers have investigated the linearized form of the two-temperature theory (2TT), focusing on two separate temperatures, namely, the conductive and thermodynamic temperatures [16–18]. The conductive temperature is a parameter linked to the heat conduction rate within a certain material. It refers to the local temperature at which heat conduction occurs. In the classical thermoelasticity, the temperature referred to is equivalent to the thermodynamic temperature, and heat conduction is believed to occur instantaneously [19]. Nevertheless, in the context of generalized thermoelasticity, the conductor may exhibit discrepancies with thermodynamics as a result of the limited velocity of heat conduction. This implies that the temperature associated with heat conduction, which is conductive, shows a delay in the thermodynamic temperature throughout the heat transfer process inside the material. The thermodynamic temperature refers to the typical temperature concept that is often employed in the field of thermodynamics. The parameter in question denotes the state of thermal equilibrium for a given substance and is the temperature variable utilized in equations of state and thermodynamic relationships. In the context of classical thermoelasticity, the thermodynamic temperature is employed to represent the temperature distribution inside the material [20]. Quintanilla [21] has engaged in a discussion over the existence, structural stability, and spatial behavior of the solution within the context of the 2TT model. Lesan [22] has proposed and proven the theorems of uniqueness and reciprocity in the context of the 2TT thermoelastic model. The two-temperature theory finds utility in several domains, such as investigating the dynamics of ultrafast laser interactions with materials, simulating semiconductor devices, and examining the response of materials under the influence of strong electromagnetic radiation. This facilitates a more precise characterization of temperature dynamics in such materials, wherein the electron temperature exhibits fast fluctuations in response to external influences.

Fractional derivatives are an extension of conventional derivatives, which are typically defined for integer orders to encompass non-integer orders. The concept of derivatives offers a means to quantify the rate of change of a function at a certain point or over a given interval, encompassing non-integer orders. The idea of fractional derivatives holds significant importance within the fractional calculus. Using integral operators commonly establishes fractional derivatives [23,24]. The Riemann–Liouville (RF) fractional and Caputo (C) fractional derivatives are widely recognized as the two most prevalent definitions in the field. The selection of a definition is contingent upon the particular problem under consideration. Fractional derivatives have been shown to have several applications in a wide range of disciplines, such as physics, engineering, economics, biology, and control theory. These models represent and examine phenomena characterized by intricate dynamics and memory effects, including, but not limited to, anomalous diffusion, viscoelasticity, and fractional Brownian motion [25,26].

Fractional derivatives involving non-singular kernels can be considered an extension of classical derivatives. In classical calculus, derivatives characterize the rate of change of a function at a certain location, typically denoted by integer-order derivatives. The definition of fractional derivatives expands the notion of differentiation to include non-integer orders, with a notable feature being the utilization of non-singular (or non-local) kernels. Researchers have recognized the significance of discovering novel fractional derivatives utilizing distinct unique or non-singular kernels to address the requirement for effectively simulating real-world phenomena across several disciplines, including hydrodynamics, viscoelasticity, physics, biological sciences, and mechanical engineering [27]. Non-singular kernels sometimes incorporate transcendental functions such as gamma, exponential, or more intricate functions such as Mittag–Leffler functions. The selection of these kernels is based on ensuring that the fractional derivatives exhibit well-defined mathematical features and hold practical importance in many applications.

Caputo and Fabrizio [28] proposed a solution to deal with the single kernel problem that occurs in Liouville–Caputo and RL fractional derivative definitions of fractional-order derivatives, among others. The solution is provided through the use of an exponential function. Nevertheless, this particular operator had some challenges, one of which pertains to its non-local nature. Furthermore, it should be noted that the integral corresponding to the fractional-order derivative is not a fractional integral. Atangana and Baleanu [29] successfully addressed these challenges. Two fractional derivatives, namely, the Liouville–Caputo and RL derivatives, were presented. These derivatives are defined using the extended Mittag–Leffler function. A comprehensive formulation encompassing all previously established fractional derivative operators featuring non-singular kernels was recently introduced in the study by Hattaf [30]. The definitions discussed earlier have been included in various physical and engineering models [31–52].

In the fields of material science and continuum mechanics, the idea of fractional thermoelasticity with phase delays and two temperatures is relatively new. The new aspect of this method is that it can help us better understand the thermal processes occurring within materials, even ones with complex and non-local behavior. Non-local and memory-dependent impacts, which standard models of thermoelasticity do not capture, can be modeled using this framework thanks to the introduction of fractional derivatives. To accurately characterize the dynamic behavior of materials, it is necessary to account for the time lag between the application of a thermal stimulus and the resulting mechanical response, and this can be done by integrating phase delays. The accuracy of temperature forecasts and thermal analysis can be improved by considering two temperatures, a technique known as dual-phase-lag modeling. This allows for a more accurate description of the thermal interactions between distinct components or phases inside a material. The evaluation of the suggested model involves the utilization of fractional derivatives, incorporating both singular and non-singular kernels. This proposed approach also allows extracting some previous cases from the thermoelasticity model, whether with one temperature or two temperatures, both when fractional differentiation is present and when it is not.

The current work utilizes the newly developed model to investigate a thermoelasticity problem involving two temperatures. Specifically, the thermodynamic response of a one-dimensional elastic medium with a surface free from traction under the influence of thermal shock and a decreasing external force is examined. Discussions were conducted to determine the effect of temperature difference coefficients, phase delay coefficients, and fractional factors on the propagation behavior of thermal and mechanical waves within the material. The numerical results in this study were compared with those in previous literature and were determined to be comparable and consistent. In addition, the results and observations showed a high degree of agreement between the analytical solutions and their numerical counterparts.

2 Derivation of the fractional thermoelasticity model

Fourier’s law holds significant importance within heat conduction, serving as a foundational concept for comprehending the mechanisms by which heat is transmitted across solid and liquid mediums. This statement elucidates the correlation between heat flux H , which represents the rate of heat transfer per unit area, and thermal gradient ∇ θ , which denotes the spatial change of temperature within a material. The mathematical expression of Fourier’s law can be represented as:

(1) H ( r , t ) = − K ∇ θ ( r , t ) .

The laws of energy conservation are essential notions within the fields of physics and engineering. These concepts articulate the fundamental law that energy is conserved, meaning it cannot be generated or destroyed. Instead, energy can only transform or be moved between other systems. The aforementioned concepts are encompassed under the laws of thermodynamics, which are widely recognized as fundamental and firmly established principles in physics. The equations that follow may be obtained using the increment in entropy denoted by η :

(2) div ( H ) + Q = − ρ η ̇ T 0 ,

(3) η = ρ C s ρ T 0 θ + γ ρ div ( u ) .

We may obtain the energy equation using Eqs. (2) and (3) as follows:

(4) ρ C e ∂ θ ∂ t + γ T 0 ∂ ∂ t div ( u ) = − div ( H ) + Q .

The following has changed the conventional form of Fourier’s law (1), according to Quintanilla [21]:

(5) H ( r , t ) = − K ∇ ϕ ( r , t ) .

Chen and Gurtin [17] and Chen et al. [18,19] developed the theory of heat transfer in flexible structures. This theory depends on two different temperatures: the conductive temperature, ϕ , and the thermodynamical temperature, θ . The equation that describes the relationship between the two temperatures, θ and ϕ , may be written as [17–19]:

(6) θ = ϕ − β ∇ 2 ϕ ,

where β is the characteristic parameter to distinguish between temperatures.

Tzou [5,6] included the influence of microstructural influences on the delayed temporal response in the macroscopic formulation. This was achieved by considering the delayed rise in lattice temperature resulting from phonon–electron interactions at the macroscopic scale. A macroscopic lagging, or delayed, reaction between the temperature gradient and the heat flow vector seems like it could happen because of interactions that happen over time.

In this case, the usual Fourier law is replaced by a rough approximation that uses a changed version of the law. This approximation has two separate phase lags for the heat flow vector ( τ q ) and the conductive temperature gradient ( τ ϕ ):

(7) H ( r , t + τ q ) = − K ∇ ϕ ( r , t + τ ϕ ) .

In order to obtain a close approximation of the modified heat transfer rule, the Taylor series expansion of Eq. (7) is utilized at a specific position r and time t , using only the first-order variables in τ q and τ ϕ :

(8) 1 + τ q ∂ ∂ t H = − K 1 + τ ϕ ∂ ∂ t ∇ ϕ .

According to Quintanilla [21], if 2 τ θ > τ q holds, then the system is (even exponentially) stable, but if 2 τ θ < τ q holds, then it is not (exponentially) stable.

The fractional heat equation is a partial differential equation (PDE) that extends the traditional heat equation by incorporating fractional derivatives. The given sentence explains how the temperature of a substance changes over time when non-local or fractional-order diffusion phenomena are considered. The fractional heat equation represents the heat conduction in materials with intricate characteristics and behavior.

The fractional heat equation is a mathematical model that describes the phenomenon of heat conduction, in which the local temperature gradient and previous temperature behavior in the surrounding area both impact the heat transfer rate at a given location. This phenomenon has special significance in materials characterized by fractal or multifractal features, as well as materials that display anomalous diffusion or sub-diffusion.

The established formulation for the C time-fractional derivative is expressed as follows:

(9) D C t ( α ) f ( t ) = 1 Γ ( 1 − α ) ∫ 0 t 1 ( t − s ) α d f ( s ) d s d s , 0 < α < 1 .

The representation of the fractional derivative of fractional order α , as proposed by Caputo and Fabrizio [28], may be expressed as follows:

(10) D CF t ( α ) f ( t ) = 1 1 − α ∫ 0 t exp − α ( t − s ) 1 − α d f ( s ) d s d s , 0 < α < 1 .

The investigation into the development of fractional differentiation in latex has also brought to light the notion that the derivative introduced by Caputo and Fabrizio can be considered a filter rather than a true fractional derivative. This conclusion is based on the observation that the employed kernel is localized and may not adequately capture the intricate dynamics of the heat flow process. The idea of elasticity can be characterized by employing a non-local framework, exemplified by the extended Mittag–Leffler function.

The mathematical expression for the Atangana–Baleanu (AB) fractional time derivative is given by the following formula [29]:

(11) D AB t ( α ) f ( t ) = 1 1 − α ∫ 0 t E α − α 1 − α ( t − s ) α d f ( s ) d s d s , 0 < α < 1 .

The function E α ( z ) represents a Mittag–Leffler function, which may be mathematically represented as

(12) E α ( z ) = ∑ r = 0 ∞ z r Γ ( 1 + r α ) , z ∈ C , 0 < α < 1 .

By substituting a derivative of fractional order in place of the time derivative in Eq. (8), we may convert the Fourier law (8) into an improved formula of fractional order as

(13) ( 1 + τ q α D t ( α ) ) H = − K ( 1 + τ ϕ α D t ( α ) ) ∇ ϕ .

By performing the divergence operation on both sides of Eq. (4), the resulting equation may be obtained:

(14) ( 1 + τ q α D t ( α ) ) div ( H ) = − K ( 1 + τ ϕ α D t ( α ) ) ∇ 2 ϕ .

By combining Eq. (9) with the energy Eq. (4), we obtain the fractional two-temperature heat transfer equation:

(15) K ( 1 + τ ϕ α D t ( α ) ) ∇ 2 ϕ = ( 1 + τ q α D t ( α ) ) ( ρ C s θ ̇ + γ T 0 e ̇ − ρ Q ) .

The fractional heat equation has utility in several domains, such as the representation of heat propagation in intricate substances, including fractal media, porous materials, biological tissues, and polymers. Also, understanding heat transport in systems that display phenomena outside the scope of conventional diffusion models has considerable importance.

Furthermore, the stress–strain–temperature, the relationships describing the material properties, and the relation between strain and displacement for thermoelastic isotropic materials at a constant ambient temperature T 0 are given by

(16) σ kl = λ div ( u ) δ kl + μ ( u l , k + u k , l ) − γ θ δ kl , 2 e kl = u l , k + u k , l ,

(17) μ u k , ll + ( λ + μ ) u l , kl − γ θ , k + F k = ρ u ̈ k .

In this article, we define the equilibrium point and give stability concepts and accompanying conclusions for fractional-order systems. Now, let us look at a generic kind of fractional differential equation that uses the AB fractional time derivative

(18) D AB t ( α ) f ( t ) = g ( t , f ( t ) ) , g ( 0 ) = g 0 ,

where 0 < α < 1 , and g ∈ R n .

The equilibrium point P is defined as a point of stability for the AB fractional-order system (18) if and only if the function g ( t , P ) = 0 . Also, the local asymptotic stability of the equilibrium points of the fractional-order system (18) is determined by the satisfaction of the condition | arg ( λ i ) | = α π / 2 for all eigenvalues of the Jacobian matrix of the system, which are assessed at said equilibrium sites [53].

3 Special cases

Three previous models of thermal elasticity with one temperature can be derived without the discrepancy coefficient and fractional derivatives ( β = 0 , α = 1 ). These models include the classical thermoelasticity (CTE) model ( τ q = τ ϕ = 0 ), the Lord and Shulman (LS) model with a single delay coefficient ( τ ϕ = 0 ), and the dual-phase lag model (DPL). Furthermore, in the event of the discrepancy factor being absent ( β = 0 ) and considering the inclusion of fractional differential derivatives ( 0 < α < 1 ), two distinct fractional models of thermoelasticity may be derived at a given temperature. These models are fractional Lord and Shulman (FLS) and fractional dual-phase lag (FDPL). Finally, when considering the incorporation of the model of thermoelasticity at two temperatures ( β > 0 ) and the utilization of fractional operators with orders ( 0 < α < 1 ), two further models for fractional thermoelasticity emerge. These models are two temperature fractional Lord and Shulman (2TFLS) and two temperature fractional dual-phase lag (2TFDPL).

4 Problem formulation

In order to assess the precision of the suggested model, its utilization in the context of a thermoelastic problem is being examined. This study will discuss the thermoelastic behavior of a half-space with infinite area ( x ≥ 0 ), where the x -axis is directed inward into the medium and is perpendicular to the surface x = 0 (Figure 1). The medium under investigation is initially at rest without pressure or strain and is at an initial temperature of T 0 . A laser pulse of uniform intensity is directed into the medium at a level defined by the surface x = 0 . In this study, a one-dimensional problem can be conceptualized as a dynamic problem within a half-space. Thus, all areas under investigation depend only on time and space variables, x and t .

Figure 1 Representation diagram of an infinite half-space.
Figure 1

Representation diagram of an infinite half-space.

The expression for the displacement components can be described as follows:

(19) u 1 = u ( x , t ) , u 2 = 0 , u 3 = 0 .

Also, the expression for the only non-zero strain is provided by

(20) e = ∂ u ( x , t ) ∂ x .

In mechanics and physics, external body forces are those that come from outside of a body as opposed to those that come from internal tensions or interactions within the body. The forces in question may possess diverse components contingent upon the particular problem and the underlying physical phenomena at play. Forces can be considered to be exponentially fading with time in several different physical systems. In the present study, the selection of the components of the external body force can be determined as

(21) F 1 = e − b x , F 2 = 0 , F 3 = 0 ,

where the parameter b > 0 is responsible for determining the rate of decay.

Eqs. (6), (15), (16), and (18) can therefore be reduced to the following forms:

(22) θ = ϕ − β ∂ 2 ϕ ∂ x 2 ,

(23) σ xx = σ = ( λ + 2 μ ) ∂ u ∂ x − γ ϕ − β ∂ 2 ϕ ∂ x 2 ,

(24) ( λ + 2 μ ) ∂ 2 u ∂ x 2 − γ ∂ θ ∂ x + ρ e − b x = ρ ∂ 2 u ∂ t 2 ,

(25) K ( 1 + τ ϕ α D t ( α ) ) ∂ 2 ϕ ∂ x 2 = ( 1 + τ q α D t ( α ) ) ρ C s ∂ θ ∂ t + γ T 0 ∂ 2 u ∂ t ∂ x .

In physical and mathematical modeling, dimensionless quantities are essential for simplifying and comparing systems or circumstances. The subsequent dimensionless amounts are introduced to streamline the resolution of the problem at hand:

(26) x ′ = χ c 0 x , u ′ = χ c 0 u , σ ′ = 1 ρ c 0 2 σ , { τ q ′ , τ ϕ ′ } = c 0 2 χ { τ q , τ ϕ } , t ′ = c 0 2 χ t , β ′ = χ 2 c 0 2 β , { θ ′ , ϕ ′ } = γ ρ c 0 2 { θ , ϕ } , F ′ = ρ χ ρ c 0 3 F ,

where c 0 2 = λ + 2 μ ρ and χ = ρ C s K .

Eqs. (22)–(25) can be rewritten in non-dimensional forms by utilizing Eq. (26) while disregarding the primes

(27) θ = ϕ − β ∂ 2 ϕ ∂ x 2 ,

(28) σ = ∂ u ∂ x − ϕ + β ∂ 2 ϕ ∂ x 2 ,

(29) ∂ 2 u ∂ x 2 − ∂ 2 u ∂ t 2 = ∂ θ ∂ x − e − b x ,

(30) ( 1 + τ ϕ α D t ( α ) ) ∂ 2 ϕ ∂ x 2 = ( 1 + τ q α D t ( α ) ) ∂ θ ∂ t + ε ∂ 2 u ∂ t ∂ x ,

where ε = γ 2 T 0 ρ 2 c 0 2 C s .

The initial conditions, along with the conditions of regularity, are provided by

(31) θ = ϕ = u = σ = 0 , at t = 0 , ∂ θ ∂ t = ∂ ϕ ∂ t = ∂ u ∂ t = 0 , at t = 0 , θ = ϕ = u = σ = 0 , at x → ∞ .

Since the upper part of the medium is not subjected to traction and is loaded with heat, the following boundary conditions are applicable:

(32) σ ( x , t ) = 0 , at x = 0 ,

(33) ϕ ( x , t ) = h ( t ) , at x = 0 ,

where

(34) h ( t ) = 0 , t ≤ 0 , ϕ 0 t 0 t , 0 < t ≤ t 0 , ϕ 0 , t > t 0 ,

where ϕ 0 and t 0 are the constants.

5 Methodology used to solve the problem

The Laplace transform is a useful mathematical tool for analyzing and solving differential equations in various scientific and engineering disciplines, especially for time-varying linear systems. This method allows complex systems to be solved by transferring them from the space–time domain to the space domain only. The integral representation for the Laplace transforms of a function g ( x , t ) , designated as g ̅ ( x , s ) , is expressed as follows:

(35) g ̅ ( x , s ) = ∫ 0 ∞ g ( x , t ) e − st d t .

Transforming Eqs. (23)–(26), we obtain

(36) θ ̅ = ϕ ̅ − β d 2 ϕ ̅ d x 2 ,

(37) σ ̅ = d u ̅ d x − ϕ ̅ + β d 2 ϕ ̅ d x 2 ,

(38) d 2 u ̅ d x 2 − s 2 u ̅ = d θ d x − 1 s e − b x ,

(39) l ϕ d 2 ϕ ̅ d x 2 = l q ϕ ̅ − β d 2 ϕ ̅ d x 2 + ε d u ̅ d x ,

where

(40) l q = 1 + τ q α s α α + s α ( 1 − α ) , l ϕ = 1 + τ ϕ α s α α + s α ( 1 − α ) .

Eqs. (38) and (39) can be rewritten as

(41) d 2 d x 2 − β d 4 d x 4 ϕ ̅ = d 2 d x 2 − s 2 e ̅ − α 1 e − b x ,

(42) d 2 d x 2 − α 2 ϕ ̅ = α 3 e ̅ ,

where

(43) α 1 = b s , α 2 = l q β l q + l ϕ , α 3 = ε l q β l q + l ϕ .

By eliminating the variable e ̅ from Eqs. (41) and (42), the following DF may be derived:

(44) d 4 d x 4 − l d 2 d x 2 + m ϕ ̅ = α 4 e − b x ,

where

(45) l = s 2 + α 2 + α 3 1 + β α 3 , m = s 2 α 2 1 + β α 3 , α 4 = α 1 α 3 1 + β α 3 .

The solution of Eq. (44) that is guaranteed to remain finite as x approaches infinity is expressed as follows:

(46) ϕ ̅ ( x , s ) = M 1 e − h 1 x + M 2 e − h 2 x + M 3 e − b x ,

where M 3 = α 4 / ( b 4 − l b 2 + m ) and M 1 and M 2 are the integration coefficients.

Moreover, the coefficients h 1 and h 1 represent the solutions to the equation

(47) k 4 − l k 2 + m = 0 .

By substituting the value of (46) into Eqs. (36) and (42), we can obtain the following expressions:

(48) e ̅ ( x , s ) = ∑ i = 1 2 ( h i 2 − α 2 ) α 3 M i e − h i x + ( b 2 − α 2 ) α 3 M 3 e − b x ,

(49) θ ̅ ( x , s ) = ∑ i = 1 2 ( 1 − β h i 2 ) M i e − k i x + ( 1 − β b 2 ) M 3 e − b x .

By substituting Eq. (48) into Eq. (20), the resulting expression yields the converted displacement:

(50) u ̅ ( x , s ) = − ∑ i = 1 2 ( k i 2 − α 2 ) k i α 3 M i e − k i x − ( b 2 − α 2 ) b α 3 M 3 e − b x .

Furthermore, by substituting Eqs. (46) and (48) into Eq. (37), we obtain

(51) σ ̅ ( x , s ) = ∑ i = 1 2 ( k i 2 − α 2 ) α 3 − ( 1 − β k i 2 ) M i e − k i x + ( b 2 − α 2 ) α 3 − ( 1 − β b 2 ) M 3 e − b x .

In Eqs. (32) and (33), the Laplace transform is employed to obtain the modified boundary conditions as

(52) σ ̅ ( x , s ) = 0 at x = 0 , ϕ ̅ ( x , s ) = ϕ 0 ( 1 − e − s t 0 ) s 2 t 0 = R ̅ ( s ) at x = 0 .

By substituting the expressions for ϕ ̅ and σ ̅ provided in Eqs. (41) and (46) into the boundary conditions (47), the integral coefficients M 1 and M 2 can be determined.

6 Numerical methods for Laplace transform inversions

The inverse Laplace transform is employed to obtain the initial function from its corresponding Laplace transform. Numerical methods for Laplace transform inversions are used to approximate the inverse Laplace transform of a given function. There exist multiple techniques for quantitatively comparing the inverse Laplace transform. When conducting numerical inversions of Laplace transforms, it is crucial to consider the selection of numerical methods, the desired level of precision, and the processing resources at hand. The choice of a suitable methodology is contingent upon the particular situation and the characteristics exhibited by the function that has undergone the Laplace transformation.

The Bromwich integral is a widely used technique employed in the numerical inversion of Laplace transforms. The process encompasses the utilization of contour integration inside the complicated plane. The inverse Laplace transform can be approximated by picking a suitable contour and employing numerical integration algorithms. Further information regarding these strategies can be located in the work of Honig and Hirdes [54].

7 Numerical results

The present section aims to present a realistic illustration to substantiate the precision of the existing fractional thermoelastic model incorporating phase lags and two temperatures. In order to facilitate numerical analyses, we have adopted the copper material coefficients as the values to be utilized [55]:

C E = 383.1 J kg − 1 K − 1 , T 0 = 293 K , α t = 1.78 × 10 − 5 K − 1 , K = 386 W m − 1 K − 1 , t = 0.12 s , λ = 7.76 × 10 10 N m − 2 , μ = 3.86 × 10 10 N m − 2 , φ 0 = 1 , ρ = 8 , 954 kg m − 3 .

We provide a selection of illustrations to illustrate and enhance our understanding of observable physical events. To analyze the effect of phase lag and the coefficient of variation of the two temperatures on measurable thermophysical quantities in the range of 0 ≤ x ≤ 5 , the corresponding graphs are plotted. In addition, the results obtained can be arranged in a tabular manner to facilitate the use of these results by other researchers, enabling them to conduct comparative analyses and verify the accuracy of their results. Based on the aforementioned physical coefficients, the field quantities are calculated using MATHEMATICA programming and visually shown through graphical means. The simulations were conducted for three different scenarios.

Case I: The effect of the temperature discrepancy

Thermoelasticity with dual temperatures represents a non-classical framework within the field of thermo-mechanics, specifically addressing the behavior of elastic solids. The main difference between this theory and the traditional model is its dependence on temperature. This analysis will examine the impact of temperature discrepancies on the thermo-physical variables under study. In this particular scenario, the utilization of a FDPL thermoelastic model is employed, which incorporates the AB fractional derivative operator. It should be noted that the value of β = 0 represents the one-temperature fractional DPL model (1TFDPL), whereas a value of β > 0 represents the 2TFDPL. Figures 2–5 exhibit the variances of thermodynamic and conductive temperatures, θ and ϕ , as well as displacement u and normal stress σ , respectively, for various values of distances x . In addition, for the purpose of computation, we have assumed that τ ϕ = 0.02 , τ q = 0.05 , t 0 = 0.1 , and α = 0.85 .

Figure 2 Influence of discrepancy index β \beta on the variation of conductive temperature ϕ \phi .
Figure 2

Influence of discrepancy index β on the variation of conductive temperature ϕ .

Figure 3 Influence of discrepancy index β \beta on the variation of thermodynamic temperature θ \theta .
Figure 3

Influence of discrepancy index β on the variation of thermodynamic temperature θ .

Figure 4 Influence of discrepancy index β \beta on the variation of displacement u {\mathscr{u}} .
Figure 4

Influence of discrepancy index β on the variation of displacement u .

Figure 5 Influence of discrepancy index β \beta on the variation of stress σ \sigma .
Figure 5

Influence of discrepancy index β on the variation of stress σ .

The observation of limited transmission of wave speeds can be deduced from the analysis of tables and figures. Furthermore, it can be observed from the presented tables and figures that the various models exhibit notable variations in values close to the surface borders. These discrepancies gradually diminish as the distance from the boundaries increases. This trend can be attributed to the influence of thermal shock exerted on the stress-free boundary. The conductive temperature pattern exhibits a peak value at the surface x = 0 , indicating thermal loading, and subsequently diminishes progressively until reaching zero. It is observed that the thermal stress σ becomes negligible at the surface x = 0 , consistent with the limited mechanical condition described in Eq. (32) for the discussed problem.

Figure 2 shows a constant decreasing trend of the field ϕ with increasing depth into the medium. The figure also shows that the amplitude of the function ϕ in the case of the 1TFDPL model at β = 0 exceeds the amplitude of the 2TFDPL model with two temperatures. In addition, the decay rate of the ϕ curve associated with the 1TFDPL model is higher compared to the histogram representing the 2FDPL model. Furthermore, it has been observed that the parameter “ β ,” which represents the two-temperature index, exerts a substantial impact on the fields under investigation. Therefore, based on the findings, it is crucial to distinguish between thermodynamic and conductive temperatures.

Figure 3 shows the effect of the two temperature difference indexes on the thermodynamic temperature change θ . Figure 3 shows the relationship between θ and the spatial coordinate x in both the 1FDPL and 2FDPL models, considering two different values of β = 0 (representing a single-type temperature) and β > 0 (representing a two-type temperature). The observed actions in this figure exhibit similarities to those depicted in Figure 2. As one moves deeper into the medium, the dynamic and conductive temperatures decrease from their maximum values at the surface. These results are consistent with the physical properties of thermoelasticity and confirm the validity of the generalized models [56]. Based on the proposed novel framework, it is imperative to establish a revised categorization scheme for materials based on their two temperature parameters. These parameters serve as a novel metric for evaluating the thermal conductivity of materials [57].

Figure 4 depicts the graphical representations of the displacement u as a function of x for the 1TFDPL and 2TFDPL systems. The displacement field exhibits both expansion and compression deformations. This phenomenon arises due to the occurrence of thermal shock, leading to the expansion of the half-space areas next to the boundary edge in an unconstrained direction. First, this results in negative displacement, which gradually transitions from negative to positive. Eventually, the displacement reaches a steady state and becomes zero at the heat wavefront [58]. The smallest displacement values are obtained using the 2TFDPL model with ( β = 0 ), whereas the biggest displacement values are obtained using the 1TFDPL theory (Figure 4).

Figure 4 illustrates the fluctuation of the stress σ with respect to the variable x . The normal stress σ exhibits a high cost across all values of t . The stress differences have similar characteristics for both models, while disparities become more pronounced at higher temporal values. The stress σ first assumes a zero value due to the boundary condition, then experiences a rapid increase, and ultimately approaches zero. It is worth mentioning that the radial stress increases as the parameter β grows, regardless of the values of x . The provided graphic illustrates that the stress experienced at the boundary is compressive.

Case II: Comparison of fractional differentiation operators

Various mathematical strategies are employed to solve non-integer-order models of PDEs in mathematical analysis. The mathematical theory of fractional calculus deals with nonlocal integration and differentiation. Fractional calculus has found extensive application in mathematical modeling and several physical phenomena. Numerous scholars have recently addressed thermoelastic difficulties in heat transmission under diverse mechanical and thermal circumstances by utilizing fractional C derivatives and Riemann–Liouville (RL) fractional derivatives. The AB time-fractional derivative is a recent and adapted version of the Caputo–Fabrizio (CF) technique. It possesses desirable characteristics such as non-singularity, non-locality of the kernel, prominent memory influences, and transmission effects.

This study introduces a thermal conductivity model incorporating fractional differential operators ( D t ( α ) ) and two distinct temperatures. In the present section, we will conduct a comparative analysis of fractional derivatives AB, CF, and C, focusing on the potential benefits associated with fractional derivative AB in relation to fractional derivatives CF and C. The impact of certain fractional differentiation operators ( D t ( α ) ) and the fractional parameter α on system fields such as temperature and displacement was analyzed graphically and tabularly, respectively. The study also included a comparison of thermoelastic models incorporating fractional derivatives and those incorporating integer derivatives ( α = 1 ). Furthermore, for the purpose of computational analysis, it has been postulated that the values of τ ϕ , τ q , t 0 , and β are 0.02, 0.05, 0.1, and 0.001, respectively.

  • The new FDPL thermoelastic model appears to improve the temperature field, as shown in Table 1. This improvement can be attributed to the incorporation of fractional operators, which effectively decrease thermal diffusion within the medium. This observation aligns with both experimental findings and established physical facts.

  • It was found that the two-temperature thermoelastic C fractional model with phase delays (2TFDPL-C) has much higher thermodynamic and conductive temperatures as the fractional parameter is raised. This is in contrast to the CF fractional model (2TFDPL-CF) and the AB fractional model (2TFDPL-AB).

  • It is imperative to acknowledge that the transmission of thermal and mechanical waves within an elastic medium is contingent upon the specific fractional-order derivative operator utilized. Hence, it can be inferred that the fractional arrangement considers the variability of the elastic medium, as each operator and fractional order signifies a distinct characteristic of elasticity.

  • The aforementioned theoretical findings possess potential utility in addressing some practical issues. Based on the theoretical findings and empirical evidence, selecting a fractional mathematical model that exhibits congruence between the experimental findings and theoretical outcomes is possible.

  • Tables 1–4 showcase the empirical results, which suggest that the power-law kernel exhibits a resilient memory effect in long-term historical data. On the other hand, non-singular kernels may be more suitable for describing relaxation or diffusion processes characterized by exceptionally strong memory.

  • It is demonstrated that the decay rate of the model with a stretched exponential kernel, which is of the CF type, is much quicker than that of the model with a power-law kernel. This indicates that a CF-type derivative model with a stretched exponential kernel, as opposed to one with a power-law kernel, may be used to describe a wider variety of relaxation events.

  • It is important to acknowledge that a mathematical model employing the AB-type derivative with a stretched Mittag–Leffler function kernel exhibits a slower diffusive motion compared to a model utilizing an exponential kernel.

  • Furthermore, it is highly anticipated that the novel fractional derivatives of AB and CF will significantly contribute to investigating the macroscopic characteristics of particular materials associated with nonlocal exchanges. These exchanges predominantly govern the properties of these materials.

Table 1

Dynamic temperature distribution under the influence of different fractional differential operators

x 2TDPL 2TFDPL-C 2TFDPL-CF 2TFDPL-AB
α = 1 α = 0.85 α = 0.75 α = 0.85 α = 0.75 α = 0.85 α = 0.75
0.0 0.826009 0.792505 0.7609570 0.731233 0.703209 0.676771 0.651815
0.5 0.423119 0.405957 0.3897960 0.374570 0.360215 0.346673 0.333889
1.0 0.215114 0.206389 0.1981730 0.190432 0.183134 0.176249 0.169750
1.5 0.109364 0.104928 0.1007510 0.096816 0.093105 0.089605 0.086301
2.0 0.055601 0.053346 0.0512222 0.049222 0.047335 0.045556 0.043876
2.5 0.028268 0.027122 0.0260421 0.025025 0.024066 0.023161 0.022307
3.0 0.014373 0.013790 0.0132407 0.012724 0.012236 0.011776 0.011342
3.5 0.007308 0.007012 0.00673259 0.006470 0.006222 0.005988 0.005767
4.0 0.003716 0.003566 0.00342388 0.003290 0.003164 0.003045 0.002933
4.5 0.001890 0.001814 0.00174174 0.001674 0.001610 0.001549 0.001492
5.0 0.000962 0.000923 0.000886547 0.000852 0.000819 0.000789 0.000760
Table 2

Conductive temperature distribution under the influence of different fractional differential operators

x 2TDPL 2TFDPL-C 2TFDPL-CF 2TFDPL-AB
α = 1 α = 0.85 α = 0.75 α = 0.85 α = 0.75 α = 0.85 α = 0.75
0.0 0.874464 0.838995 0.8055960 0.774128 0.744460 0.716472 0.690051
0.5 0.443852 0.425848 0.4088960 0.392924 0.377866 0.363660 0.350249
1.0 0.225652 0.216499 0.2078810 0.199761 0.192105 0.184883 0.178065
1.5 0.114721 0.110068 0.1056860 0.101558 0.097666 0.093994 0.090528
2.0 0.058325 0.055959 0.0537314 0.051633 0.049654 0.047787 0.046025
2.5 0.029653 0.028450 0.0273178 0.026251 0.025245 0.024296 0.023400
3.0 0.015076 0.014465 0.0138893 0.013347 0.012835 0.012353 0.011898
3.5 0.007666 0.007355 0.00706229 0.006787 0.006527 0.006281 0.006050
4.0 0.003898 0.003740 0.00359149 0.003451 0.003319 0.003194 0.003077
4.5 0.001983 0.001903 0.00182696 0.001756 0.001688 0.001625 0.001565
5.0 0.001009 0.000968 0.00092987 0.000894 0.000859 0.000827 0.000797
Table 3

Displacement distribution under the influence of different fractional differential operators

x 2TDPL 2TFDPL-C 2TFDPL-CF 2TFDPL-AB
α = 1 α = 0.85 α = 0.75 α = 0.85 α = 0.75 α = 0.85 α = 0.75
0.0 −0.12601 −0.12505 −0.1095700 −0.091230 −0.08032 −0.06768 −0.05182
0.5 0.379483 0.364091 0.3495980 0.335943 0.323068 0.310923 0.299458
1.0 0.193224 0.185387 0.1780080 0.171055 0.164500 0.158316 0.152478
1.5 0.098241 0.094256 0.0905047 0.086970 0.083637 0.080494 0.077526
2.0 0.049951 0.047926 0.0460186 0.044222 0.042527 0.040929 0.039420
2.5 0.025402 0.024372 0.0234020 0.022488 0.021627 0.020815 0.020048
3.0 0.012920 0.012397 0.0119039 0.011439 0.011002 0.010588 0.010199
3.5 0.006575 0.006309 0.0060583 0.005822 0.005600 0.005390 0.005191
4.0 0.003349 0.003214 0.00308639 0.002966 0.002853 0.002746 0.002646
4.5 0.001709 0.001640 0.00157544 0.001514 0.001457 0.001403 0.001351
5.0 0.000875 0.000840 0.000807237 0.000776 0.000747 0.000719 0.000693
Table 4

Thermal stress distribution under the influence of different fractional differential operators

x 2TDPL 2TFDPL-C 2TFDPL-CF 2TFDPL-AB
α = 1 α = 0.85 α = 0.75 α = 0.85 α = 0.75 α = 0.85 α = 0.75
0.0 0 0 0 0 0 0 0
0.5 −0.4218 −0.40469 −0.388578 −0.37340 −0.35909 −0.34559 −0.33285
1.0 −0.21474 −0.20603 −0.197825 −0.19010 −0.18281 −0.17594 −0.16945
1.5 −0.10918 −0.10475 −0.100580 −0.09665 −0.09295 −0.08945 −0.08616
2.0 −0.05551 −0.05326 −0.0511408 −0.04914 −0.04726 −0.04548 −0.04381
2.5 −0.02823 −0.02708 −0.0260063 −0.02499 −0.02403 −0.02313 −0.02228
3.0 −0.01436 −0.01378 −0.0132280 −0.01271 −0.01223 −0.01177 −0.01133
3.5 −0.00731 −0.00701 −0.00673156 −0.00647 −0.00622 −0.00599 −0.00577
4.0 −0.00372 −0.00357 −0.00342877 −0.00330 −0.00317 −0.00305 −0.00294
4.5 −0.0019 −0.00182 −0.00174961 −0.00168 −0.00162 −0.00156 −0.00150
5.0 −0.00097 −0.00093 −0.000895892 −0.00086 −0.00083 −0.00080 −0.00077

8 Conclusions

The physical aspects of relaxation and thermoelastic heat transfer models are studied in this work. It focuses on the newly introduced temporal derivative that uses a non-power function kernel. The use of analytical analysis shows that the C-type derivative idea, when combined with a power function kernel, is not good at describing the non-exponential changes that are often seen in asymmetrical heat conduction. Because of this, a new version of the thermoelasticity theory has been made that includes two temperatures, two separate phase delays, and fractional-order differential operators that are not in a single position. The revised definition’s primary benefit is replacing the singular power-law kernel with a Mittag–Leffler function and a non-singular exponential kernel. This replacement facilitates using the revised definition in theoretical analysis, numerical calculations, practical scenarios, and real-world applications.

The results show that the field variables, such as temperature, displacement, and thermal stress, are affected by the two-temperature parameter ( β ) and fractional-order differential operators, as well as the position ( x ) and time ( t ). The impact of fractional differential operators on various field quantities is particularly important in academic research. All the investigated field quantities are considerably impacted by fractional differential operators. The fractional coefficient characterizes the thermal conductivity of a thermoelastic material, necessitating the development of new categories of materials. Moreover, there is a great expectation that the unique fractional derivatives proposed by AB and CF will make a substantial contribution to the study of the macroscopic properties of specific materials that involve nonlocal interactions. These interactions primarily regulate the characteristics of these materials.

The AB and CF fractional operators offer reliable mathematical frameworks for comprehensively capturing nonlocal interactions inside materials. These operators empower researchers to construct advanced models that account for the impact of remote events on the macroscopic features of the material. By integrating these fractional operators into the governing equations, scholars are able to adequately consider the long-range correlations and spatial dependencies that are inherent in materials with nonlocal interactions. This enables a more precise depiction of their macroscopic behavior.

  1. Funding information: The authors express their gratitude to the Deanship of Scientific Research at Jouf University for providing financial support for this study through research grant No. (DSR-2021-03-03172).

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Data availability statement: All data generated or analysed during this study are included in this published article.

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Received: 2023-09-19
Revised: 2024-04-25
Accepted: 2024-05-04
Published Online: 2024-09-27

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  12. Thermal proficiency of magnetized and radiative cross-ternary hybrid nanofluid flow induced by a vertical cylinder
  13. Enhanced heat transfer and fluid motion in 3D nanofluid with anisotropic slip and magnetic field
  14. Numerical analysis of thermophoretic particle deposition on 3D Casson nanofluid: Artificial neural networks-based Levenberg–Marquardt algorithm
  15. Analyzing fuzzy fractional Degasperis–Procesi and Camassa–Holm equations with the Atangana–Baleanu operator
  16. Bayesian estimation of equipment reliability with normal-type life distribution based on multiple batch tests
  17. Chaotic control problem of BEC system based on Hartree–Fock mean field theory
  18. Optimized framework numerical solution for swirling hybrid nanofluid flow with silver/gold nanoparticles on a stretching cylinder with heat source/sink and reactive agents
  19. Stability analysis and numerical results for some schemes discretising 2D nonconstant coefficient advection–diffusion equations
  20. Convective flow of a magnetohydrodynamic second-grade fluid past a stretching surface with Cattaneo–Christov heat and mass flux model
  21. Analysis of the heat transfer enhancement in water-based micropolar hybrid nanofluid flow over a vertical flat surface
  22. Microscopic seepage simulation of gas and water in shale pores and slits based on VOF
  23. Model of conversion of flow from confined to unconfined aquifers with stochastic approach
  24. Study of fractional variable-order lymphatic filariasis infection model
  25. Soliton, quasi-soliton, and their interaction solutions of a nonlinear (2 + 1)-dimensional ZK–mZK–BBM equation for gravity waves
  26. Application of conserved quantities using the formal Lagrangian of a nonlinear integro partial differential equation through optimal system of one-dimensional subalgebras in physics and engineering
  27. Nonlinear fractional-order differential equations: New closed-form traveling-wave solutions
  28. Sixth-kind Chebyshev polynomials technique to numerically treat the dissipative viscoelastic fluid flow in the rheology of Cattaneo–Christov model
  29. Some transforms, Riemann–Liouville fractional operators, and applications of newly extended M–L (p, s, k) function
  30. Magnetohydrodynamic water-based hybrid nanofluid flow comprising diamond and copper nanoparticles on a stretching sheet with slips constraints
  31. Super-resolution reconstruction method of the optical synthetic aperture image using generative adversarial network
  32. A two-stage framework for predicting the remaining useful life of bearings
  33. Influence of variable fluid properties on mixed convective Darcy–Forchheimer flow relation over a surface with Soret and Dufour spectacle
  34. Inclined surface mixed convection flow of viscous fluid with porous medium and Soret effects
  35. Exact solutions to vorticity of the fractional nonuniform Poiseuille flows
  36. In silico modified UV spectrophotometric approaches to resolve overlapped spectra for quality control of rosuvastatin and teneligliptin formulation
  37. Numerical simulations for fractional Hirota–Satsuma coupled Korteweg–de Vries systems
  38. Substituent effect on the electronic and optical properties of newly designed pyrrole derivatives using density functional theory
  39. A comparative analysis of shielding effectiveness in glass and concrete containers
  40. Numerical analysis of the MHD Williamson nanofluid flow over a nonlinear stretching sheet through a Darcy porous medium: Modeling and simulation
  41. Analytical and numerical investigation for viscoelastic fluid with heat transfer analysis during rollover-web coating phenomena
  42. Influence of variable viscosity on existing sheet thickness in the calendering of non-isothermal viscoelastic materials
  43. Analysis of nonlinear fractional-order Fisher equation using two reliable techniques
  44. Comparison of plan quality and robustness using VMAT and IMRT for breast cancer
  45. Radiative nanofluid flow over a slender stretching Riga plate under the impact of exponential heat source/sink
  46. Numerical investigation of acoustic streaming vortices in cylindrical tube arrays
  47. Numerical study of blood-based MHD tangent hyperbolic hybrid nanofluid flow over a permeable stretching sheet with variable thermal conductivity and cross-diffusion
  48. Fractional view analytical analysis of generalized regularized long wave equation
  49. Dynamic simulation of non-Newtonian boundary layer flow: An enhanced exponential time integrator approach with spatially and temporally variable heat sources
  50. Inclined magnetized infinite shear rate viscosity of non-Newtonian tetra hybrid nanofluid in stenosed artery with non-uniform heat sink/source
  51. Estimation of monotone α-quantile of past lifetime function with application
  52. Numerical simulation for the slip impacts on the radiative nanofluid flow over a stretched surface with nonuniform heat generation and viscous dissipation
  53. Study of fractional telegraph equation via Shehu homotopy perturbation method
  54. An investigation into the impact of thermal radiation and chemical reactions on the flow through porous media of a Casson hybrid nanofluid including unstable mixed convection with stretched sheet in the presence of thermophoresis and Brownian motion
  55. Establishing breather and N-soliton solutions for conformable Klein–Gordon equation
  56. An electro-optic half subtractor from a silicon-based hybrid surface plasmon polariton waveguide
  57. CFD analysis of particle shape and Reynolds number on heat transfer characteristics of nanofluid in heated tube
  58. Abundant exact traveling wave solutions and modulation instability analysis to the generalized Hirota–Satsuma–Ito equation
  59. A short report on a probability-based interpretation of quantum mechanics
  60. Study on cavitation and pulsation characteristics of a novel rotor-radial groove hydrodynamic cavitation reactor
  61. Optimizing heat transport in a permeable cavity with an isothermal solid block: Influence of nanoparticles volume fraction and wall velocity ratio
  62. Linear instability of the vertical throughflow in a porous layer saturated by a power-law fluid with variable gravity effect
  63. Thermal analysis of generalized Cattaneo–Christov theories in Burgers nanofluid in the presence of thermo-diffusion effects and variable thermal conductivity
  64. A new benchmark for camouflaged object detection: RGB-D camouflaged object detection dataset
  65. Effect of electron temperature and concentration on production of hydroxyl radical and nitric oxide in atmospheric pressure low-temperature helium plasma jet: Swarm analysis and global model investigation
  66. Double diffusion convection of Maxwell–Cattaneo fluids in a vertical slot
  67. Thermal analysis of extended surfaces using deep neural networks
  68. Steady-state thermodynamic process in multilayered heterogeneous cylinder
  69. Multiresponse optimisation and process capability analysis of chemical vapour jet machining for the acrylonitrile butadiene styrene polymer: Unveiling the morphology
  70. Modeling monkeypox virus transmission: Stability analysis and comparison of analytical techniques
  71. Fourier spectral method for the fractional-in-space coupled Whitham–Broer–Kaup equations on unbounded domain
  72. The chaotic behavior and traveling wave solutions of the conformable extended Korteweg–de-Vries model
  73. Research on optimization of combustor liner structure based on arc-shaped slot hole
  74. Construction of M-shaped solitons for a modified regularized long-wave equation via Hirota's bilinear method
  75. Effectiveness of microwave ablation using two simultaneous antennas for liver malignancy treatment
  76. Discussion on optical solitons, sensitivity and qualitative analysis to a fractional model of ion sound and Langmuir waves with Atangana Baleanu derivatives
  77. Reliability of two-dimensional steady magnetized Jeffery fluid over shrinking sheet with chemical effect
  78. Generalized model of thermoelasticity associated with fractional time-derivative operators and its applications to non-simple elastic materials
  79. Migration of two rigid spheres translating within an infinite couple stress fluid under the impact of magnetic field
  80. A comparative investigation of neutron and gamma radiation interaction properties of zircaloy-2 and zircaloy-4 with consideration of mechanical properties
  81. New optical stochastic solutions for the Schrödinger equation with multiplicative Wiener process/random variable coefficients using two different methods
  82. Physical aspects of quantile residual lifetime sequence
  83. Synthesis, structure, I–V characteristics, and optical properties of chromium oxide thin films for optoelectronic applications
  84. Smart mathematically filtered UV spectroscopic methods for quality assurance of rosuvastatin and valsartan from formulation
  85. A novel investigation into time-fractional multi-dimensional Navier–Stokes equations within Aboodh transform
  86. Homotopic dynamic solution of hydrodynamic nonlinear natural convection containing superhydrophobicity and isothermally heated parallel plate with hybrid nanoparticles
  87. A novel tetra hybrid bio-nanofluid model with stenosed artery
  88. Propagation of traveling wave solution of the strain wave equation in microcrystalline materials
  89. Innovative analysis to the time-fractional q-deformed tanh-Gordon equation via modified double Laplace transform method
  90. A new investigation of the extended Sakovich equation for abundant soliton solution in industrial engineering via two efficient techniques
  91. New soliton solutions of the conformable time fractional Drinfel'd–Sokolov–Wilson equation based on the complete discriminant system method
  92. Irradiation of hydrophilic acrylic intraocular lenses by a 365 nm UV lamp
  93. Inflation and the principle of equivalence
  94. The use of a supercontinuum light source for the characterization of passive fiber optic components
  95. Optical solitons to the fractional Kundu–Mukherjee–Naskar equation with time-dependent coefficients
  96. A promising photocathode for green hydrogen generation from sanitation water without external sacrificing agent: silver-silver oxide/poly(1H-pyrrole) dendritic nanocomposite seeded on poly-1H pyrrole film
  97. Photon balance in the fiber laser model
  98. Propagation of optical spatial solitons in nematic liquid crystals with quadruple power law of nonlinearity appears in fluid mechanics
  99. Theoretical investigation and sensitivity analysis of non-Newtonian fluid during roll coating process by response surface methodology
  100. Utilizing slip conditions on transport phenomena of heat energy with dust and tiny nanoparticles over a wedge
  101. Bismuthyl chloride/poly(m-toluidine) nanocomposite seeded on poly-1H pyrrole: Photocathode for green hydrogen generation
  102. Infrared thermography based fault diagnosis of diesel engines using convolutional neural network and image enhancement
  103. On some solitary wave solutions of the Estevez--Mansfield--Clarkson equation with conformable fractional derivatives in time
  104. Impact of permeability and fluid parameters in couple stress media on rotating eccentric spheres
  105. Review Article
  106. Transformer-based intelligent fault diagnosis methods of mechanical equipment: A survey
  107. Special Issue on Predicting pattern alterations in nature - Part II
  108. A comparative study of Bagley–Torvik equation under nonsingular kernel derivatives using Weeks method
  109. On the existence and numerical simulation of Cholera epidemic model
  110. Numerical solutions of generalized Atangana–Baleanu time-fractional FitzHugh–Nagumo equation using cubic B-spline functions
  111. Dynamic properties of the multimalware attacks in wireless sensor networks: Fractional derivative analysis of wireless sensor networks
  112. Prediction of COVID-19 spread with models in different patterns: A case study of Russia
  113. Study of chronic myeloid leukemia with T-cell under fractal-fractional order model
  114. Accumulation process in the environment for a generalized mass transport system
  115. Analysis of a generalized proportional fractional stochastic differential equation incorporating Carathéodory's approximation and applications
  116. Special Issue on Nanomaterial utilization and structural optimization - Part II
  117. Numerical study on flow and heat transfer performance of a spiral-wound heat exchanger for natural gas
  118. Study of ultrasonic influence on heat transfer and resistance performance of round tube with twisted belt
  119. Numerical study on bionic airfoil fins used in printed circuit plate heat exchanger
  120. Improving heat transfer efficiency via optimization and sensitivity assessment in hybrid nanofluid flow with variable magnetism using the Yamada–Ota model
  121. Special Issue on Nanofluids: Synthesis, Characterization, and Applications
  122. Exact solutions of a class of generalized nanofluidic models
  123. Stability enhancement of Al2O3, ZnO, and TiO2 binary nanofluids for heat transfer applications
  124. Thermal transport energy performance on tangent hyperbolic hybrid nanofluids and their implementation in concentrated solar aircraft wings
  125. Studying nonlinear vibration analysis of nanoelectro-mechanical resonators via analytical computational method
  126. Numerical analysis of non-linear radiative Casson fluids containing CNTs having length and radius over permeable moving plate
  127. Two-phase numerical simulation of thermal and solutal transport exploration of a non-Newtonian nanomaterial flow past a stretching surface with chemical reaction
  128. Natural convection and flow patterns of Cu–water nanofluids in hexagonal cavity: A novel thermal case study
  129. Solitonic solutions and study of nonlinear wave dynamics in a Murnaghan hyperelastic circular pipe
  130. Comparative study of couple stress fluid flow using OHAM and NIM
  131. Utilization of OHAM to investigate entropy generation with a temperature-dependent thermal conductivity model in hybrid nanofluid using the radiation phenomenon
  132. Slip effects on magnetized radiatively hybridized ferrofluid flow with acute magnetic force over shrinking/stretching surface
  133. Significance of 3D rectangular closed domain filled with charged particles and nanoparticles engaging finite element methodology
  134. Robustness and dynamical features of fractional difference spacecraft model with Mittag–Leffler stability
  135. Characterizing magnetohydrodynamic effects on developed nanofluid flow in an obstructed vertical duct under constant pressure gradient
  136. Study on dynamic and static tensile and puncture-resistant mechanical properties of impregnated STF multi-dimensional structure Kevlar fiber reinforced composites
  137. Thermosolutal Marangoni convective flow of MHD tangent hyperbolic hybrid nanofluids with elastic deformation and heat source
  138. Investigation of convective heat transport in a Carreau hybrid nanofluid between two stretchable rotatory disks
  139. Single-channel cooling system design by using perforated porous insert and modeling with POD for double conductive panel
  140. Special Issue on Fundamental Physics from Atoms to Cosmos - Part I
  141. Pulsed excitation of a quantum oscillator: A model accounting for damping
  142. Review of recent analytical advances in the spectroscopy of hydrogenic lines in plasmas
  143. Heavy mesons mass spectroscopy under a spin-dependent Cornell potential within the framework of the spinless Salpeter equation
  144. Coherent manipulation of bright and dark solitons of reflection and transmission pulses through sodium atomic medium
  145. Effect of the gravitational field strength on the rate of chemical reactions
  146. The kinetic relativity theory – hiding in plain sight
  147. Special Issue on Advanced Energy Materials - Part III
  148. Eco-friendly graphitic carbon nitride–poly(1H pyrrole) nanocomposite: A photocathode for green hydrogen production, paving the way for commercial applications
https://doi.org/10.1515/phys-2024-0031
Keywords for this article
fractional thermoelasticity; two-temperature; phaselags; non-singular kernels
Creative Commons
BY 4.0

Articles in the same Issue

  1. Regular Articles
  2. Numerical study of flow and heat transfer in the channel of panel-type radiator with semi-detached inclined trapezoidal wing vortex generators
  3. Homogeneous–heterogeneous reactions in the colloidal investigation of Casson fluid
  4. High-speed mid-infrared Mach–Zehnder electro-optical modulators in lithium niobate thin film on sapphire
  5. Numerical analysis of dengue transmission model using Caputo–Fabrizio fractional derivative
  6. Mononuclear nanofluids undergoing convective heating across a stretching sheet and undergoing MHD flow in three dimensions: Potential industrial applications
  7. Heat transfer characteristics of cobalt ferrite nanoparticles scattered in sodium alginate-based non-Newtonian nanofluid over a stretching/shrinking horizontal plane surface
  8. The electrically conducting water-based nanofluid flow containing titanium and aluminum alloys over a rotating disk surface with nonlinear thermal radiation: A numerical analysis
  9. Growth, characterization, and anti-bacterial activity of l-methionine supplemented with sulphamic acid single crystals
  10. A numerical analysis of the blood-based Casson hybrid nanofluid flow past a convectively heated surface embedded in a porous medium
  11. Optoelectronic–thermomagnetic effect of a microelongated non-local rotating semiconductor heated by pulsed laser with varying thermal conductivity
  12. Thermal proficiency of magnetized and radiative cross-ternary hybrid nanofluid flow induced by a vertical cylinder
  13. Enhanced heat transfer and fluid motion in 3D nanofluid with anisotropic slip and magnetic field
  14. Numerical analysis of thermophoretic particle deposition on 3D Casson nanofluid: Artificial neural networks-based Levenberg–Marquardt algorithm
  15. Analyzing fuzzy fractional Degasperis–Procesi and Camassa–Holm equations with the Atangana–Baleanu operator
  16. Bayesian estimation of equipment reliability with normal-type life distribution based on multiple batch tests
  17. Chaotic control problem of BEC system based on Hartree–Fock mean field theory
  18. Optimized framework numerical solution for swirling hybrid nanofluid flow with silver/gold nanoparticles on a stretching cylinder with heat source/sink and reactive agents
  19. Stability analysis and numerical results for some schemes discretising 2D nonconstant coefficient advection–diffusion equations
  20. Convective flow of a magnetohydrodynamic second-grade fluid past a stretching surface with Cattaneo–Christov heat and mass flux model
  21. Analysis of the heat transfer enhancement in water-based micropolar hybrid nanofluid flow over a vertical flat surface
  22. Microscopic seepage simulation of gas and water in shale pores and slits based on VOF
  23. Model of conversion of flow from confined to unconfined aquifers with stochastic approach
  24. Study of fractional variable-order lymphatic filariasis infection model
  25. Soliton, quasi-soliton, and their interaction solutions of a nonlinear (2 + 1)-dimensional ZK–mZK–BBM equation for gravity waves
  26. Application of conserved quantities using the formal Lagrangian of a nonlinear integro partial differential equation through optimal system of one-dimensional subalgebras in physics and engineering
  27. Nonlinear fractional-order differential equations: New closed-form traveling-wave solutions
  28. Sixth-kind Chebyshev polynomials technique to numerically treat the dissipative viscoelastic fluid flow in the rheology of Cattaneo–Christov model
  29. Some transforms, Riemann–Liouville fractional operators, and applications of newly extended M–L (p, s, k) function
  30. Magnetohydrodynamic water-based hybrid nanofluid flow comprising diamond and copper nanoparticles on a stretching sheet with slips constraints
  31. Super-resolution reconstruction method of the optical synthetic aperture image using generative adversarial network
  32. A two-stage framework for predicting the remaining useful life of bearings
  33. Influence of variable fluid properties on mixed convective Darcy–Forchheimer flow relation over a surface with Soret and Dufour spectacle
  34. Inclined surface mixed convection flow of viscous fluid with porous medium and Soret effects
  35. Exact solutions to vorticity of the fractional nonuniform Poiseuille flows
  36. In silico modified UV spectrophotometric approaches to resolve overlapped spectra for quality control of rosuvastatin and teneligliptin formulation
  37. Numerical simulations for fractional Hirota–Satsuma coupled Korteweg–de Vries systems
  38. Substituent effect on the electronic and optical properties of newly designed pyrrole derivatives using density functional theory
  39. A comparative analysis of shielding effectiveness in glass and concrete containers
  40. Numerical analysis of the MHD Williamson nanofluid flow over a nonlinear stretching sheet through a Darcy porous medium: Modeling and simulation
  41. Analytical and numerical investigation for viscoelastic fluid with heat transfer analysis during rollover-web coating phenomena
  42. Influence of variable viscosity on existing sheet thickness in the calendering of non-isothermal viscoelastic materials
  43. Analysis of nonlinear fractional-order Fisher equation using two reliable techniques
  44. Comparison of plan quality and robustness using VMAT and IMRT for breast cancer
  45. Radiative nanofluid flow over a slender stretching Riga plate under the impact of exponential heat source/sink
  46. Numerical investigation of acoustic streaming vortices in cylindrical tube arrays
  47. Numerical study of blood-based MHD tangent hyperbolic hybrid nanofluid flow over a permeable stretching sheet with variable thermal conductivity and cross-diffusion
  48. Fractional view analytical analysis of generalized regularized long wave equation
  49. Dynamic simulation of non-Newtonian boundary layer flow: An enhanced exponential time integrator approach with spatially and temporally variable heat sources
  50. Inclined magnetized infinite shear rate viscosity of non-Newtonian tetra hybrid nanofluid in stenosed artery with non-uniform heat sink/source
  51. Estimation of monotone α-quantile of past lifetime function with application
  52. Numerical simulation for the slip impacts on the radiative nanofluid flow over a stretched surface with nonuniform heat generation and viscous dissipation
  53. Study of fractional telegraph equation via Shehu homotopy perturbation method
  54. An investigation into the impact of thermal radiation and chemical reactions on the flow through porous media of a Casson hybrid nanofluid including unstable mixed convection with stretched sheet in the presence of thermophoresis and Brownian motion
  55. Establishing breather and N-soliton solutions for conformable Klein–Gordon equation
  56. An electro-optic half subtractor from a silicon-based hybrid surface plasmon polariton waveguide
  57. CFD analysis of particle shape and Reynolds number on heat transfer characteristics of nanofluid in heated tube
  58. Abundant exact traveling wave solutions and modulation instability analysis to the generalized Hirota–Satsuma–Ito equation
  59. A short report on a probability-based interpretation of quantum mechanics
  60. Study on cavitation and pulsation characteristics of a novel rotor-radial groove hydrodynamic cavitation reactor
  61. Optimizing heat transport in a permeable cavity with an isothermal solid block: Influence of nanoparticles volume fraction and wall velocity ratio
  62. Linear instability of the vertical throughflow in a porous layer saturated by a power-law fluid with variable gravity effect
  63. Thermal analysis of generalized Cattaneo–Christov theories in Burgers nanofluid in the presence of thermo-diffusion effects and variable thermal conductivity
  64. A new benchmark for camouflaged object detection: RGB-D camouflaged object detection dataset
  65. Effect of electron temperature and concentration on production of hydroxyl radical and nitric oxide in atmospheric pressure low-temperature helium plasma jet: Swarm analysis and global model investigation
  66. Double diffusion convection of Maxwell–Cattaneo fluids in a vertical slot
  67. Thermal analysis of extended surfaces using deep neural networks
  68. Steady-state thermodynamic process in multilayered heterogeneous cylinder
  69. Multiresponse optimisation and process capability analysis of chemical vapour jet machining for the acrylonitrile butadiene styrene polymer: Unveiling the morphology
  70. Modeling monkeypox virus transmission: Stability analysis and comparison of analytical techniques
  71. Fourier spectral method for the fractional-in-space coupled Whitham–Broer–Kaup equations on unbounded domain
  72. The chaotic behavior and traveling wave solutions of the conformable extended Korteweg–de-Vries model
  73. Research on optimization of combustor liner structure based on arc-shaped slot hole
  74. Construction of M-shaped solitons for a modified regularized long-wave equation via Hirota's bilinear method
  75. Effectiveness of microwave ablation using two simultaneous antennas for liver malignancy treatment
  76. Discussion on optical solitons, sensitivity and qualitative analysis to a fractional model of ion sound and Langmuir waves with Atangana Baleanu derivatives
  77. Reliability of two-dimensional steady magnetized Jeffery fluid over shrinking sheet with chemical effect
  78. Generalized model of thermoelasticity associated with fractional time-derivative operators and its applications to non-simple elastic materials
  79. Migration of two rigid spheres translating within an infinite couple stress fluid under the impact of magnetic field
  80. A comparative investigation of neutron and gamma radiation interaction properties of zircaloy-2 and zircaloy-4 with consideration of mechanical properties
  81. New optical stochastic solutions for the Schrödinger equation with multiplicative Wiener process/random variable coefficients using two different methods
  82. Physical aspects of quantile residual lifetime sequence
  83. Synthesis, structure, I–V characteristics, and optical properties of chromium oxide thin films for optoelectronic applications
  84. Smart mathematically filtered UV spectroscopic methods for quality assurance of rosuvastatin and valsartan from formulation
  85. A novel investigation into time-fractional multi-dimensional Navier–Stokes equations within Aboodh transform
  86. Homotopic dynamic solution of hydrodynamic nonlinear natural convection containing superhydrophobicity and isothermally heated parallel plate with hybrid nanoparticles
  87. A novel tetra hybrid bio-nanofluid model with stenosed artery
  88. Propagation of traveling wave solution of the strain wave equation in microcrystalline materials
  89. Innovative analysis to the time-fractional q-deformed tanh-Gordon equation via modified double Laplace transform method
  90. A new investigation of the extended Sakovich equation for abundant soliton solution in industrial engineering via two efficient techniques
  91. New soliton solutions of the conformable time fractional Drinfel'd–Sokolov–Wilson equation based on the complete discriminant system method
  92. Irradiation of hydrophilic acrylic intraocular lenses by a 365 nm UV lamp
  93. Inflation and the principle of equivalence
  94. The use of a supercontinuum light source for the characterization of passive fiber optic components
  95. Optical solitons to the fractional Kundu–Mukherjee–Naskar equation with time-dependent coefficients
  96. A promising photocathode for green hydrogen generation from sanitation water without external sacrificing agent: silver-silver oxide/poly(1H-pyrrole) dendritic nanocomposite seeded on poly-1H pyrrole film
  97. Photon balance in the fiber laser model
  98. Propagation of optical spatial solitons in nematic liquid crystals with quadruple power law of nonlinearity appears in fluid mechanics
  99. Theoretical investigation and sensitivity analysis of non-Newtonian fluid during roll coating process by response surface methodology
  100. Utilizing slip conditions on transport phenomena of heat energy with dust and tiny nanoparticles over a wedge
  101. Bismuthyl chloride/poly(m-toluidine) nanocomposite seeded on poly-1H pyrrole: Photocathode for green hydrogen generation
  102. Infrared thermography based fault diagnosis of diesel engines using convolutional neural network and image enhancement
  103. On some solitary wave solutions of the Estevez--Mansfield--Clarkson equation with conformable fractional derivatives in time
  104. Impact of permeability and fluid parameters in couple stress media on rotating eccentric spheres
  105. Review Article
  106. Transformer-based intelligent fault diagnosis methods of mechanical equipment: A survey
  107. Special Issue on Predicting pattern alterations in nature - Part II
  108. A comparative study of Bagley–Torvik equation under nonsingular kernel derivatives using Weeks method
  109. On the existence and numerical simulation of Cholera epidemic model
  110. Numerical solutions of generalized Atangana–Baleanu time-fractional FitzHugh–Nagumo equation using cubic B-spline functions
  111. Dynamic properties of the multimalware attacks in wireless sensor networks: Fractional derivative analysis of wireless sensor networks
  112. Prediction of COVID-19 spread with models in different patterns: A case study of Russia
  113. Study of chronic myeloid leukemia with T-cell under fractal-fractional order model
  114. Accumulation process in the environment for a generalized mass transport system
  115. Analysis of a generalized proportional fractional stochastic differential equation incorporating Carathéodory's approximation and applications
  116. Special Issue on Nanomaterial utilization and structural optimization - Part II
  117. Numerical study on flow and heat transfer performance of a spiral-wound heat exchanger for natural gas
  118. Study of ultrasonic influence on heat transfer and resistance performance of round tube with twisted belt
  119. Numerical study on bionic airfoil fins used in printed circuit plate heat exchanger
  120. Improving heat transfer efficiency via optimization and sensitivity assessment in hybrid nanofluid flow with variable magnetism using the Yamada–Ota model
  121. Special Issue on Nanofluids: Synthesis, Characterization, and Applications
  122. Exact solutions of a class of generalized nanofluidic models
  123. Stability enhancement of Al2O3, ZnO, and TiO2 binary nanofluids for heat transfer applications
  124. Thermal transport energy performance on tangent hyperbolic hybrid nanofluids and their implementation in concentrated solar aircraft wings
  125. Studying nonlinear vibration analysis of nanoelectro-mechanical resonators via analytical computational method
  126. Numerical analysis of non-linear radiative Casson fluids containing CNTs having length and radius over permeable moving plate
  127. Two-phase numerical simulation of thermal and solutal transport exploration of a non-Newtonian nanomaterial flow past a stretching surface with chemical reaction
  128. Natural convection and flow patterns of Cu–water nanofluids in hexagonal cavity: A novel thermal case study
  129. Solitonic solutions and study of nonlinear wave dynamics in a Murnaghan hyperelastic circular pipe
  130. Comparative study of couple stress fluid flow using OHAM and NIM
  131. Utilization of OHAM to investigate entropy generation with a temperature-dependent thermal conductivity model in hybrid nanofluid using the radiation phenomenon
  132. Slip effects on magnetized radiatively hybridized ferrofluid flow with acute magnetic force over shrinking/stretching surface
  133. Significance of 3D rectangular closed domain filled with charged particles and nanoparticles engaging finite element methodology
  134. Robustness and dynamical features of fractional difference spacecraft model with Mittag–Leffler stability
  135. Characterizing magnetohydrodynamic effects on developed nanofluid flow in an obstructed vertical duct under constant pressure gradient
  136. Study on dynamic and static tensile and puncture-resistant mechanical properties of impregnated STF multi-dimensional structure Kevlar fiber reinforced composites
  137. Thermosolutal Marangoni convective flow of MHD tangent hyperbolic hybrid nanofluids with elastic deformation and heat source
  138. Investigation of convective heat transport in a Carreau hybrid nanofluid between two stretchable rotatory disks
  139. Single-channel cooling system design by using perforated porous insert and modeling with POD for double conductive panel
  140. Special Issue on Fundamental Physics from Atoms to Cosmos - Part I
  141. Pulsed excitation of a quantum oscillator: A model accounting for damping
  142. Review of recent analytical advances in the spectroscopy of hydrogenic lines in plasmas
  143. Heavy mesons mass spectroscopy under a spin-dependent Cornell potential within the framework of the spinless Salpeter equation
  144. Coherent manipulation of bright and dark solitons of reflection and transmission pulses through sodium atomic medium
  145. Effect of the gravitational field strength on the rate of chemical reactions
  146. The kinetic relativity theory – hiding in plain sight
  147. Special Issue on Advanced Energy Materials - Part III
  148. Eco-friendly graphitic carbon nitride–poly(1H pyrrole) nanocomposite: A photocathode for green hydrogen production, paving the way for commercial applications
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