**4. Conclusions**

This paper includes the investigation of the heat transfer problems at the nanoscale. A new approach to the heat transfer modeling in modern nanosized structures was considered. It combines the DPL model and the Grünwald–Letnikov fractional derivative. A combination of these mathematical tools allows for the preparation of a complex approach using the Finite Difference Method to temperature distribution determination at the nanoscale, with a high level of accuracy, as is confirmed by the measurement of a real test structure.

An important novelty described in this paper is the use of a DPL model with a fractional order space derivative of a temperature function based on the Grünwald–Letnikov derivative operator. This operator, as well as the proposed time–space discretization schema, is a bridge between experimentally confirmed DPL mesoscopic model with the ballistic heat transport model with dynamic temperature changes' intensification useful for quasi 1-D nanostructures and for radiative heat transport without phonon collisions. This solution allows for the consideration of such physical behaviors like time needed for a

heat flux or temperature gradient changes. Thus, a modeling of a heat diffusion can be investigated by making realistic assumptions, which was not possible in the case of the F–K model use and real relaxation thermal properties of material at mesoscopic scale. The research has shown that the proposed GL DPL model is more realistic than the commonly used Fourier–Kirchhoff model.

The manuscript describes also proposed an approximation scheme of a modern DPL heat transfer model based on the Finite Di fference Method approach prepared for the two-dimensional cross-section of the real test nanometric electronic structure manufactured at the Institute of Electron Technology in Warsaw. The investigation has shown that there is a possibility to e ffectively implement a prepared algorithm that allows for the determination of a temperature distribution inside real nanoscale electronic structures, based on proposed an approximation scheme.

Thermal simulation has provided results which coincide almost exactly with the real measurements. It means that prepared methodology is highly accurate and allows modeling of the heat transfer problems by using a modern approach based on the use of the Dual-Phase-Lag model. The considered thermal model is an appropriate methodology for heat-di ffusion modeling, especially at the nanoscale.

In the future, the reduction of the DPL model order reduction methodology will be considered in order to save simulation time, decrease a computational power requirements [47], and make the simulation process more e fficient.

**Author Contributions:** The algorithm for the FDM approximation scheme of DPL model for two-dimensional cross-section of investigated test structure, analysis of the Grünwald–Letnikov temperature derivative of a fractional order and the algorithm calculated its values, numerical simulations and evaluation of their results, and preparation of this manuscript were carried out by T.R. Preparation of the algorithm convergence analysis of numerical simulations, equivalent air conductance investigation, and photon tunneling, as well as the radiative heat transfer calculation, thermodynamic models aspects and time delayed PDEs analysis, were performed by M.Z.; M.Z. also supervised the research and made corrections to the manuscript. All authors have read and agreed to the published version of the manuscript.

**Funding:** The research presented in this paper was carried out and funded within the Polish National Science Centre project OPUS No. 2016/21/B/ST7/02247.

**Acknowledgments:** Authors would like to express their special thanks to M. Janicki and J. Topilko sharing papers [32,33].

**Conflicts of Interest:** The authors declare no conflicts of interest.
