**1. Classical and Modern Heat Transfer Description**

Currently, deep development in technology has increased interest in heat transfer modeling. New thermal problems have been observed due to the continuous reduction of the size of electronic devices or miniaturization of integrated circuits. For instance, in such small electronic appliances, a significant rise of operational frequency and a rapid increase of generated internal heat density have been noticed which have a meaningful influence on temperature rise during the operation of the device.

It is worthwhile to highlight that it is extremely important to ensure the appropriate cooling conditions, however, this issue is very problematic in the case of nanosized electronic devices and can result in unstable and improper operation of the device. Moreover, most of the damages and malfunctions of the device are caused by unsuitable operation and thermal problems. Thus, thermal analysis is very important and is one of the most crucial steps in the design and planning of modern electronic appliances.

The heat transfer problems have been modeled using Fourier's theory [1]. This method is based on Fourier's law and the Fourier–Kirchhoff (FK) equation, and can be expressed as follows:

$$\begin{cases} \begin{aligned} q(\mathbf{x},t) &= -k \cdot \nabla T(\mathbf{x},t) \\ c\_v \frac{\partial T(\mathbf{x},t)}{\partial t} + \nabla \circ q(\mathbf{x},t) &= q\_V(\mathbf{x},t) \end{aligned} \quad \mathbf{x} \in R^n, n \in N, t \in R\_+ \cup \{0\} \end{cases} \tag{1}$$

In the above system of equations, the heat flux density vector is represented by *q*, thermal conductivity of analyzed material is expressed by *k*, and *T* describes the function of temperature rise. Moreover, the volumetric heat capacity and value of internally generated heat are represented by *cv* and *qV*, respectively.

The Fourier–Kirchhoff method has been successfully applied for two centuries. However, in modern nanosized structures, application of the FK method has some serious limitations [2–4]. First, the assumption of infinite speed propagation of the heat is postulated. Moreover, the instantaneous change of heat flux or temperature gradient should also be taken into account as a non-physical behavior of the structures. The phenomena mentioned have not been empirically proven in the case of nanosized electronic structures [5,6]. Thus, a new methodology, which significantly improves the FK method in the case of modern electronic structures, should be established.

In the mid-1990s, a new approach, called Dual-Phase-Lag (DPL), was introduced by Tzou as a more appropriate choice for modeling temperature changes in a nanosized electronic structure [7]. The DPL model based on the FK theory, however, as previously mentioned, includes improvements such as time lags [8]. These lags express the needed time change in the heat flux density, as well as the temperature gradient.

The lags mentioned above are represented by different values expressed by <sup>τ</sup>*q*, which is related to heat flux time lag, and τ*T* which represents the temperature time lag. Taking into consideration new time lags, the DPL model can be described by the following equations:

$$\begin{cases} \nabla \circ q(\mathbf{x},t) = -c\_v \frac{\partial T(\mathbf{x},t)}{\partial t} + q\_V(\mathbf{x},t) \\\ k\tau\_T \frac{\partial \nabla T(\mathbf{x},t)}{\partial t} + \tau\_q \frac{\partial q(\mathbf{x},t)}{\partial t} = -k \cdot \nabla T(\mathbf{x},t) - q(\mathbf{x},t) \end{cases} \quad \mathbf{x} \in R^n, n \in N, t \in R\_+ \cup \{0\} \tag{2}$$

The DPL model can be successfully applied instead of the classical FK model due to the fact that it is appropriate for parabolic partial differential equations, as well as for hyperbolic equations (see also [9,10]).

However, some disadvantages have also appeared. The DPL model is classified as a model with greater complexity than the FK model. Thus, to carry out the simulation, a longer computation time is needed, especially for complex electronic structures.

Considering the above limitation of the DPL model, the main aim of this paper is to implement a DPL-based method which reduces the time of simulation and can be as accurate as the original DPL model. One approach which can be applied is the Krylov subspace-based model order reduction method [11]. This approach significantly reduces the number of equations in the system describing an analyzed heat transfer problem.

The Krylov subspace method is used for the order reduction of large-scale systems, especially in a mechanical domain (e.g., J. White and J. G. Korvink papers, as well as [12,13]). However, the research presented in this paper is focused on aspects other than those of previously published papers. First of all, this manuscript includes the first application of the Krylov subspace-based method for the DPL heat transfer equation. Moreover, in previous research, the spectrum of mechanical systems generally contains frequencies from 0 to hundreds of Hz (sometimes up to 1–5 kHz). The model is validated typically for longer times, for example, 10 ms–1 s. However, our research includes significantly greater frequency values resulting in meaningfully shorter times and smaller DPL time lags, even hundreds of femtoseconds or a few nanoseconds. Therefore, the range of applicability of the described model order reduction methodology, presented in the manuscript, is different than other previous applications.

The structure of the paper is as follows: First, a short description of the investigated real test nanosized structure, its finite difference method approximation, and structure discretization are presented; then, the approximation scheme of the DPL model for the considered MEMS structure is proposed; after that, the Krylov subspace-based model order reduction technique is demonstrated; and finally, the simulation results obtained using both reduced and non-reduced versions of the DPL model are compared, analyzed, and discussed.
