**1. Introduction**

Heat transfer through thin films subjected to an ultrafast laser pulse is of vital importance in microtechnology applications and is a reason that the problems related to the fast heating of solids have become a very active research area. The problems of melting/resolidification processes modeling, which may be the result of heating with a laser beam, are also important from the technical point of view. So far, the method using the equation based on the second order model with two delay times for the phase changes modeling was not presented in literature. This was the most important motivation for the authors to undertake research in this area. In Section 5, the comparison of the results obtained with the similar solution on the basis of the first-order dual phase lag equation (DPLE) is presented.

The mathematical model of macroscale heat conduction is based on the parabolic Fourier equation. This equation was formulated under the assumption of instantaneous propagation of the thermal wave in the domain considered. It is obvious that this assumption is not correct, but for the problems concerning the analysis of macroscale heat conduction processes, the obtained results are fully satisfactory. Despite this, the attempts have been made to modify the Fourier equation to a form that better reproduces the real conditions of heat conduction in solids. Thus, about seventy years ago,

Cattaneo proposed a modification of the Fourier equation now called the Cattaneo–Vernotte equation. This is the hyperbolic partial differential equation (PDE) and contains the parameter τ*<sup>q</sup>* called the relaxation time (the lag time of the heat flux in relation to the temperature gradient) [1,2]. Especially important differences between the Fourier model and the real course of thermal processes appear in the case of microscale heat transfer problems. For example, the very high heating rates accompanying the heating of thin metal films with a laser beam mean that the inclusion of the finite value of thermal wave velocity must be taken into account. The deviations appear mainly when the mean free path of the heat carriers becomes comparable to the characteristic length of the domain considered and the time scale of interest becomes comparable to or smaller than the relaxation time of the heat carriers [3,4].

For the analysis of this type of process, the model with two delay times called a dual-phase lag model is presently applied. In addition to the relaxation time, the thermalization time is introduced. The relaxation time τ*<sup>q</sup>* takes into account the small-scale response in time, while the thermalization time τ*<sup>T</sup>* takes into account the small-scale response in space [3–6]. The dual phase lag equation (DPLE) results from the generalized form of the Fourier law

$$\mathbf{q(x,t+\tau\_q)} = -\lambda \,\nabla T(\mathbf{x,t+\tau\_T}) \tag{1}$$

where **q** is a heat flux vector; ∇*T* is a temperature gradient; λ is a thermal conductivity; and **x** and *t* denote the geometrical co-ordinates and time, respectively. Both sides of the last dependence are developed into a power series and, finally (depending on the number of components), the first- or second-order DPLE can be obtained.

The literature on equations with two delay times is very extensive (especially in the case of the first order equations), and here we quote only a few important articles. The first publications concerning the model with two delay times appeared in the early nineties of the last century. There may be mentioned, for example, the papers [7–9]. Currently, one can already find books devoted to this type of non-Fourier heat conduction model, for example, [10–13].

In this brief literature review, the selected papers on analytical and numerical solutions of first-order DPLE will be listed. First of all, the works containing the analytical solutions of first-order DPLE (usually 1D tasks) will be mentioned [14–18]. The paper [14] concerns the laser heating of ultra-thin metal film; in the papers [15,16], the bio-heat transfer problems are discussed; while in the paper [18], the multi-layered metal domain is considered.

A much larger number of articles concern the application of numerical methods in the tasks based on the models with two delay times. It should be pointed out that, first of all, different variants of finite difference method (FDM) are applied (see, for example, [19–24]). In the paper [19], the numerical model of heating of the double-layered thin film has been applied for the analysis of the thermal deformation process. In the paper [20], the 3D FDM numerical model of the thin metal film heating has been presented. In [21], the explicit scheme of the FDM has been used and the problem of biological tissue freezing process has been discussed. The stability problem of the algorithm of this type is analyzed in [22]. In [23], the problem based on DPLE has been solved using the alternating direction implicit FDM scheme. In the paper [24], the adaptation of typical boundary conditions for non-Fourier equations has been shown. The FDM numerical solutions of the inverse problems can also be found (e.g., [25]).

The number of works presenting solutions using the other numerical methods is significantly smaller. Here, one can mention the papers [26–32]. In particular, solutions based on the finite element method [26–28], the boundary element method [29], the control volume method [30], or the lattice Boltzmann method [31,32] are discussed in the above-mentioned papers.

Literature on the second-order DPLE is not as extensive as for the first-order equations. As an example, the papers [33–37] can be mentioned. The main subject of these papers (except [37]) is related to the construction of algorithms for numerical modeling of problems described by second-order DPLE (the different variants of FDM are used). Additionally, the transformed second-order equations are shown in [20,34,37] and the changed forms are more convenient at the stage of numerical modeling.

At the stage of melting/resolidification modeling, the concept presented in [38] (for the macroscale problems) is applied. The capacity of the internal heat source related to the phase change is proportional to the melting/solidification rate. To define this function (in particular, the volumetric fraction of liquid state *fL*(*T*)) in the form of a continuous one, the melting point corresponding to the temperature *Tm* is conventionally replaced by a certain interval [*Tm* − Δ*T*, *Tm* + Δ*T*], and then the function discussed can be described by a broken line. For this interval, the substitute thermal capacity is defined and the one domain approach can be used. It should be pointed out that the testing computations show a little impact of the interval Δ*T* width (within reasonable limits) on the results of numerical simulations.

At the beginning of the part of the article devoted to own research, the assumed form of the dualphase lag equation and the mathematical formulas determining the laser action and the evolution of phase change latent heat are presented. Both phenomena are taken into account by an introduction to the energy equation of the functions determining the efficiency of internal heat sources. Next, the numerical algorithm based on the implicit scheme of FDM is discussed. In the final part of the paper, the results of numerical computations concerning the heating/cooling process of the thin metal film made of chrome are shown. The conclusions resulting from the performed research are also formulated.
