1 Introduction

As we all know, the classical Fourier law is derived from the macroscopic experience. Though the parabolic heat conduction equation is widely applied in engineering practice, it implies that the heat signal has an infinite propagation speed in materials^[1-3], which is contradictory with the physical reality. The traditional Fourier law actually only reflects the equilibrium heat conduction process, which is not available to describe the thermodynamic process of the non-equilibrium state^[4]. With the development of science and technology, the ultra-short laser pulse heating and cooling technologies are applied more and more in engineering practice, in which the non-equilibrium heat conduction process has to be considered. That is to say, the heat relaxation time related to the mechanism of the non-equilibrium heat conduction must be introduced to the traditional Fourier law^[5-10]. Cattaneo^[11] and Vernotte^[12] developed Cattaneo and Vernotte equations by adding a partial time derivative of the heat flux multiplied by the thermal relaxation time of the material, respectively. Finally, the hyperbolic heat conduction equation can be obtained by combining the Cattaneo-Vernotte equation with the energy equation, which can predict the non-equilibrium heat conduction process, i.e., the wave-like behavior for the temperature and heat flux.

Though the Cattaneo-Vernotte model can efficiently overcome the unphysical infinite heat propagation speed for the non-equilibrium heat conduction process, some deviations from experimental results are observed by Narayanamurti and Dynes^[13] and Jiang et al.^[14]. In fact, the Cattaneo-Vernotte model is an anomalous diffusion one. The so-called anomalous diffusion is a non-equilibrium process. From a micro point of view, the random motion of particles is restricted by the structure of the irregular medium, and cannot be described by the standard statistical method of particles in the homogeneous medium. From a macro point of view, the statistical law for the motion of a large number of particles does not conform to the standard statistical distribution, and the central limit law is no longer available. The mean square displacement of the diffusion process in these complex systems does not follow the standard linear behavior, and takes the form of 〈 x²(t) 〉∝t^p, where p is the anomalous diffusion exponent. For p=1, it is the normal diffusion. For p≠1, it corresponds to the anomalous diffusion, in which p < 1 means sub-diffusion. It has been found that there is an intrinsic link of the fractional calculus, the power law, and the anomalous diffusion. The fractional order calculus has been proved to be efficient in modeling the intermediate anomalous behaviors observed in different physical phenomena^[16-17], and has been applied in complex viscoelastic and rheological media^[18-19], finance, control and bioengineering fields^[20-21].

In 1997, Compte and Metzler^[22] proposed three possible fractional Cattaneo models. Povstenko^[23] studied the time fractional Cattaneo heat conduction equation and established the corresponding thermal stress theory. Furthermore, the Cattaneo-type time fractional heat conduction equation was used to investigate temperature fields for laser heating by Qi et al.^[24]. Jiang and Qi^[25] studied the thermal wave model of bio-heat transfer with the modified Riemann-Liouville fractional derivative and demonstrated that the fractional models could provide a unified approach to examine the heat transfer in biological tissues. The transient temperature field based on the fractional heat conduction equation for laser heating was studied by Xu et al.^[26], in which the heat conduction mechanism in metal materials irradiated by the non-Gauss laser pulse was discussed. The anomalous heat conduction under the Neumann boundary condition in a semi-infinite medium was investigated by Xu et al.^[27]. Qi and Guo^[28] used the generalized Cattaneo model to study temperature variations in a finite slab with an initial sinusoidal distribution. Mishr and Rai^[29] obtained the fractional Cattaneo model by applying the fractional Taylor series formula to the single phase lagging heat conduction model, which was named as the fractional single phase lagging heat conduction model, and the effects of different parameters of laser on temperature variations were conducted.

It can be found that most researches on the fractional Cattaneo-Vernotte model have been mainly carried on the heat conduction problems of the semi-infinite space with the Neumann boundary condition^{[23, 26-27]} and those of the finite thickness with the Dirac boundary condition^[29]. However, the heat conduction of the finite thickness with the Neumann boundary condition has been less studied based on the fractional Cattaneo-Vernotte model. In this paper, therefore, the analytical solution for the time fractional Cattaneo heat conduction in a finite slab under a pulse heat flux is presented, and the heat conduction mechanism varying with the fractional order parameters and the anomalous diffusion phenomenon in the finite slab are analyzed.

2 Mathematical model 2.1 Fractional Cattaneo heat conduction equation

The Cattaneo model was proposed by Cattaneo^[11] and Vernotte^[12] in the following form:

(1)

where q is the heat flux, τ is the thermal relaxation time, k is the heat conductivity, T is the temperature, t is the time, and ∇ is the gradient operator. It is noted that the classical Cattaneo model could also be obtained from the Boltzmann equation^[30-31].

By combining Eq.(1) with the continuity equation,

(2)

the Cattaneo-Vernotte heat conduction equation is obtained as

(3)

where a =k/(ρc_V) is the thermal diffusivity. The Cattaneo-Vernotte heat conduction equation is studied both experimentally and theoretically in different fields of engineering. Different ranges of values for the relaxation time τ in the Cattaneo model have been given in different heat transfer processes and materials. The approximate ranges of heat relaxation time at the normal temperature include 10^-11 s~10^-14 s for metals, 10^-8 s~10^-10 s for gases, 10 s ~100 s for porous materials and biological tissues^[3].

The fractional Cattaneo model is given via the fractional Taylor series expansion^[32] as follows:

(4)

where 0≤ p≤ 1 is the fractional order of differentiation. is the time fractional derivative operator based on Caputo's definition^[32].

By combining Eq.(2) with Eq.(4), the time fractional Cattaneo heat conduction equation can be obtained as

(5)

It is noted that Eq.(5) can be reduced to the classical Fourier heat conduction equation as τ =0 and the Cattaneo-Vernotte heat conduction equation as p=1.

2.2 One-dimensional finite slab model

An isotropic slab with a thickness of L is considered, in which one-dimensional heat conduction and the constant thermal properties are assumed. The slab is initially in a state of thermal equilibrium T=T₀. As t>0, the left side surface at x=0 is irradiated by a short pulsed laser beam q=q₀ (U(t)-U(t-t₀)), where U(t) is a unit step Heaviside function, and q₀ is the absorbed laser intensity. Meanwhile, the right side surface at x=L is adiabatic. In this situation, the fractional Cattaneo heat conduction equation (5) can be reduced into its one-dimensional form,

(6)

The boundary conditions are given as

(7)

(8)

where f(t)=U(t)-U(t-t₀).

The initial condition is defined as

(9)

3 Analytical solutions for finite slab under pulse heat flux

In this section, the analytical solutions for the finite slab under pulse heat flux based on the time fractional Cattaneo heat conduction model, the classical Cattaneo-Vernotte one, and the Fourier one, are derived for exhibiting their different heat mechanisms.

For convenience, in the subsequent analysis, the following dimensionless quantities are introduced:

(10)

3.1 Solution of time fractional Cattaneo heat conduction

By substituting Eq.(10) into Eqs. (6)-(9), the corresponding dimensionless forms can be expressed as (for simplicity, the superscript * is omitted)

(11)

(12)

(13)

(14)

By taking the finite Cosine-Fourier transform of Eq.(11) and applying boundary conditions (12)-(13), it can be obtained as

(15)

where T_c (n, t)=∫₀¹T(x, t)cos (nπx)dx. The initial condition (14) is transformed into

(16)

The solution of Eq.(15) in the Laplace transformed domain is

(17)

where T (n, s)=∫₀^∞ T(n, t)exp (-st)dt, and F(s)=∫₀^∞f(t)exp (-st)dt.

When n≠0, Eq.(17) is expanded into two parts as follows:

(18)

where

(19)

(20)

Applying the convolution theorem of the Laplace transform, the inverse Laplace transform solution is written as

(21)

where

(22)

(23)

where φ =-τ^-pt^p, and E_{α, β} (z) is a double-parameter Mittag-Leffler function^[33].

When n=0, the solution of Eq.(17) is derived as

(24)

Therefore, the final solution of the time fractional Cattaneo heat conduction can be obtained with the finite cosine inverse Fourier transform as

(25)

3.2 Solution of classical Cattaneo-Vernotte heat conduction model

When p=1, Eq.(11) can be reduced to the classical Cattaneo-Vernotte heat conduction model. Thus, the dimensionless form of the classical Cattaneo-Vernotte heat conduction model is written as

(26)

Accordingly, the left boundary condition is modified as

(27)

The right boundary condition and the initial condition are the same as Subsection 3.1.

With the same solution methods as Subsection 3.1, the solution of the classical Cattaneo-Vernotte heat conduction model can be obtained as

(28)

where

(29)

(30)

(31)

3.3 Solution of Fourier heat conduction model

When τ =0, Eq.(11) can be reduced to the classical Fourier heat conduction model. Thus, the dimensionless form of the Fourier heat conduction model is expressed as

(32)

Similarly, the dimensionless left boundary condition is written as

(33)

The right boundary and the initial conditions are the same as Subsection 3.1.

With the same solution methods as Subsection 3.1, the solution of the classical Fourier heat conduction model can be obtained as

(34)

where

(35)

4 Results and discussion

In this section, the numerical results with different values of the fractional parameter p and the dimensionless relaxation time τ are given. The fractional parameter p represents the degree of deviation from the standard diffusion. The dimensionless relaxation time is taken as 0.01, 0.1, 1, 10 in Ref. [24], 1, 3 in Ref. [25], and 5.3 in Ref. [26]. Similar to previous studies, the dimensionless relaxation time is valued as 0~10.

Figure 1 gives the temperature distributions in the finite slab as p=0.8, τ =1, and t₀ =2. It can be observed that the temperature distribution range is enlarged gradually with the increase of time in the finite slab, which results from the finite heat propagation velocity included in the fractional Cattaneo-Vernotte model. In the stage of heating irradiating (t≤ t₀), the temperature of the heating surface is always larger than that of the interior of the slab. In the stage of cooling t>t₀, the temperature of the interior is larger than that of the heating surface of the slab. These phenomena are consistent with the second law of thermodynamics, proving that the present analytical solutions are reasonable.

Figure 2 shows the dimensionless temperature profiles for different values of the fractional order p when t=0.3 and τ =1. Meanwhile, the classical Fourier solution (as τ =0) is also shown in this figure. It is noted that the classical Cattaneo-Vernotte solution is the case as p=1. From Fig. 2, it can be found that the fractional Cattaneo solutions with 0 < p < 1 lie in the range between the classical Fourier and Cattaneo-Vernotte ones. The dimensionless temperature increases with the increase in the fractional order when the dimensionless position is less than 0.38. However, the dimensionless temperature is inverse when the dimensionless position is larger than 0.38. The area integral of the temperature distribution along the position denotes the absorption heat from the external heat source, and the temperature distributions of different fractional parameters at the certain time represent the different heat transfer mechanisms. It can be seen from Fig. 2 that the temperature distributions with different fractional order parameters are different. However, the area integrals of different temperature distributions must be equal because of the same amount of heat input, i.e., the present results comply with the energy conservation, which also proves that the present analytical solutions are reasonable. Obviously, the influence of the fractional order parameters on the temperature distribution is very important. From the thermal wave propagation point of view, the larger the parameter p, the slower the thermal wave. When p=0, the fractional Cattaneo-Vernotte thermal wave equation is transformed into the parabolic equation. However, it is shown that the corresponding solutions of p=0 are different from those corresponding to the Fourier heat conduction equation. It can also be inferred that the heat diffusion described by the fractional order heat wave equation behaves with the finite velocity, and the velocity is related with the fractional order parameter.

Figures 3(a) and 3(b) show the dimensionless temperature profiles for different values of the fractional order p when t=1.5 and t=2.5, respectively. From Figs. 3(a) and 3(b), it can be observed that the heat flux has reached the right boundary. Figure 3(a) shows that the temperature distribution is degraded smoothly from the left boundary to the right one for the fractional Cattaneo model and the Fourier model, while the temperature distribution for the classical Cattaneo-Vernotte model is unsmooth due to the heat flux reflection at the right boundary. Figure 3(b) shows that the temperature distribution changes with the fractional order parameters after the pulse heat flux of the surface x=0 is stopped. It can be found from Figs. 3(a) and 3(b) that the classical Cattaneo-Vernotte model exhibits the unphysical phenomena of the heat propagation from the low temperature to the high one in the range of x>0.5.

Figure 4 presents the dimensionless temperature response at x=0 for different models. It can be found that the temperature response based on the classical Cattaneo-Vernotte model firstly jumps into a higher temperature at the onset of heating, then it reaches a maximum temperature at the end of heating (i.e., t=t₀ =2). After several oscillations due to the heat flux reflection, the temperature response for the Cattaneo-Vernotte mode becomes a stable value. For the fractional order and the Fourier model, the temperatures both firstly reach their maximum values at different temperature change rates from zero and then converge into the stable value. At the beginning of the pulse irradiating, the temperature rise rate for the fractional order model is between those for the Fourier model with the smallest value and the Cattaneo-Vernotte model with the largest value. At the end of the pulse irradiating, the similar temperature change rates can be observed. Furthermore, it can be found that the fractional order parameters have significant effects on the temperature response, i.e., the peak temperature values at the end of the pulse heating are decreased with the increase of the fractional order parameters.

Figure 5 presents the dimensionless temperature response at x=1 for different models. x=1 corresponds to the adiabatic surface, and it is also the thermal wave reflection surface. It can be clearly found that the initial response time for different models is different, that is to say, different models represent different heat propagation velocities. Meanwhile, the initial response time for different fractional order parameters is also different. From Fig. 5, the peak temperature values at the end of the pulse heating are increased with the increase of the fractional order parameters.

For studying the influence of the relaxation time τ on the temperature distribution, p is taken as 0.6. Figure 6 shows the change of the dimensionless temperature distribution in the finite slab at t=0.5. It is clearly observed that the slower the heat flux propagates, the higher the relaxation time is. Obviously, the relaxation time has some effects on the heat propagation velocity.

5 Conclusions

In this paper, the fractional Cattaneo model is derived for modeling the heat transfer in a finite slab irradiated by a short pulse laser. The finite Fourier transform and the Laplace one are used to obtain the analytical solutions for the fractional Cattaneo, the classical Cattaneo-Vernotte, and the Fourier models. The wave behavior of the heat propagation in the finite slab is reproduced based on the fractional Cattaneo model. The dependence of the temperature distribution on the fractional order is studied at different time. From the thermal wave propagation point of view, the larger the parameter p, the slower the thermal wave. The influence of the fractional order parameter on the temperature response is also investigated at different positions. The peak temperature values at the end of the pulse heating are decreased with the increase of the fractional order parameter at the heating surface, while it is inverse at the adiabatic surface. The effect of the relaxation time on the temperature distribution is also studied. The slower the heat flux propagates, the higher the relaxation time is. By comparing the fractional order Cattaneo model with the classical Cattaneo-Vernotte and Fourier models, finally, the heat flux predicted using the fractional Cattaneo model always transports from the high temperature to the low one, while the Cattaneo-Vernotte model shows that the heat flux sometimes transports from the low temperature to the high one. At any position, the fractional Cattaneo model gives the continuous temperature variation, while the classical Cattaneo-Vernotte model predicts the leaping change of temperature.

Appendix A

The formulated definition of the Mittag-Leffler double parameters function is

(A1)

Its Laplace transforms are

(A2)

(A3)

where L and R represent the Laplace transform operator and the real part operator, respectively.