**4. Conclusions**

This paper includes an analysis of the quality and time complexity of two algorithms dedicated to solving heat transfer problems at nanoscale. The first one uses the modern DPL model which is a significant improvement as compared with one of the most common approaches based on the FK model. The second one also employs the DPL model, however in its reduced version. The DPL model order reduction is prepared based on the Krylov subspace method.

The analyses have shown that both the reduced and non-reduced DPL models produce high quality results that coincide with real measurements of the test structure. Moreover, the results obtained using the reduced DPL model are very similar to the results yielded based on the non-reduced DPL approach. It is also worthwhile highlighting that the relative error of approximation of temperature distribution determination inside the test structure, which was obtained using the reduced DPL model, was at a very low level. Furthermore, this error decreased over the time, which suggested a convergence of the proposed approach.

In addition, the reduced DPL model prepares a solution for a heat transfer problem significantly faster than the classic version of this heat transfer model. The time complexity of the non-reduced approach is O(*n2.544*), whereas in the case of the reduced model, the complexity is O(*n*) only. Considering all the mentioned facts, it can be stated that the proposed approach obtained the high quality solution of the temperature distribution at nanoscale in a significantly shorter time than the classic approach, which is especially important to the future of designing and investigating advanced nanosized electronic structures.

**Author Contributions:** The algorithm for the FDM approximation scheme of reduced and non-reduced DPL model for 2D cross-section of analyzed test structure, investigation of the fractional Grünwald-Letnikov temperature derivative, application of Krylov subspace-based model order reduction technique in the case of DPL model, numerical simulations and evaluation of their results, relative error as well as computational complexity analysis of prepared algorithms and preparation of this manuscript have been carried out by T.R., M.Z. investigated algorithm convergence, the analysis of simplification error for internal heat generation source, 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 within the Polish National Science Centre project OPUS No. 2016/21/B/ST7/02247.

**Acknowledgments:** The authors would like to express their special thanks to M. Janicki and J. Topilko for sharing papers 14.

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

### **Appendix A. Analysis of Internal Heat Generation Source in DPL Model**

The DPL equation contains term *qv*(*t*) + τ*q* ∂*qv*(*t*) ∂*t* , which is required for the accurate modeling of heat generation sources:

$$c\_v \tau\_q \frac{\partial^2 T}{\partial t^2} + c\_v \frac{\partial T}{\partial t} = k \Delta T + k \tau\_T \frac{\partial T}{\partial t} + \left( q\_v + \tau\_q \frac{\partial q\_v}{\partial t} \right) \tag{A1}$$

Hence, the Taylor series expansions mapping of the heat generation in the DPL Equation (A1) into a time-shifted function 23 is considered as presented in Equation (A3). The higher order terms of the Taylor series expansions are neglected.

$$f(\mathbf{x},t) + \tau \frac{\partial f(\mathbf{x},t+\tau)}{\partial t} \to f(\mathbf{x},t+\tau) \tag{A2}$$

$$q\_v(t) + \tau\_q \frac{\partial q\_v(t)}{\partial t} \approx q\_v(t + \tau\_q) \tag{A3}$$

After that, the accurate heat generation model *qv*(*t*) + τ*q* ∂*qv*(*t*) ∂*t* can be interpreted as a time-shifted function *qv t* + τ*q* . It is su fficient to use a time-shifted delayed heat generation function *qv t* + τ*q* instead of term *qv*(*t*) + τ*q* ∂*qv*(*t*) ∂*t* in the case of absence of electro-thermal couplings ( ... → *q*v → *T* → *q*v ... ), as well as known excitation function *q*v(*t*), with the simulation time domain correction.

The advanced analysis based on the generalized functions theory 46 allows the estimation of relative error introduced by the simplified expression *qv*(*t*) instead of the *qv*(*t*) + τ*q* ∂*qv*(*t*) ∂*t* expression. Let us consider a more accurate model of internal heat source with the Heaviside excitation function *qv*(*t*) = *H*(*t*) :

*Energies* **2020**, *13*, 2520

$$H(t) = \begin{cases} 1 & \text{for} \quad t > 0 \\ 0 & \text{for} \quad t < 0 \end{cases}, \text{ and} \\ H'(t) = \delta(t) \tag{A4}$$

where δ(*t*) is the Dirac measure [46].

Suppose also ϕ*(t)* is a border measured test function with compact support [46], then detailed probed calculated for internal heat source generalized function is described by the following Equation:

$$\begin{split} A &= \left\langle q\_{\mathbb{P}}(t) + \tau\_{q}\frac{\partial q\_{\mathbb{P}}(t)}{\partial t}, q\right\rangle = \left\langle q\_{\mathbb{P}\prime}\,\rho\right\rangle + \tau\_{q}\left\langle \frac{\partial q\_{\mathbb{P}}}{\partial t}, q\right\rangle = \left\langle q\_{\mathbb{P}\prime}\,\rho\right\rangle - \tau\_{q}\left\langle q\_{\mathbb{P}\prime}\frac{\partial \boldsymbol{\psi}}{\partial t}\right\rangle = \\ &= \left\langle q\_{\mathbb{P}\prime}\,\rho\right\rangle + \left\langle q\_{\mathbb{P}\prime} - \tau\_{q}\frac{\partial \boldsymbol{\psi}}{\partial t}\right\rangle = \left\langle q\_{\mathbb{P}\prime}\,\rho - \tau\_{q}\frac{\partial \boldsymbol{\psi}}{\partial t}\right\rangle \end{split} \tag{A5}$$

where

$$
\langle f, \boldsymbol{\varrho} \rangle = \int\_{-\infty}^{+\infty} f(t) \, \boldsymbol{\varrho}(t) dt \tag{A6}
$$

Let us apply the same procedure for a simplified internal heat generation model for the DPL equation:

$$B = \langle q\_v(t), \varphi \rangle \tag{A7}$$

Then, the relative error can be estimated by the following Equation:

$$err = \frac{B - A}{A} = \frac{-\tau\_q \wp(0)}{\int\_0^{+\infty} \wp(t)dt + \tau\_q \wp(0)} = \frac{-1}{\frac{1}{\tau\_q} \int\_0^{+\infty} \frac{\wp(t)}{\wp(0)} dt + 1} \approx \frac{-1}{\frac{1}{\tau\_q} \int\_0^b \frac{\wp(t)}{\wp(0)} dt + 1} \tag{A8}$$

where *b* is determined by the support of ϕ*(t)*. One of the most popular test function Equation (A9) has been used for the error evaluation *err*

$$\varphi(t) = \begin{cases} \ \exp\left(\frac{-a^2}{a^2 - t^2}\right) & \text{for} \quad |t| < a\\ 0 & \text{for} \quad |t| \ge a \end{cases} \tag{A9}$$

where *a* is an arbitrarily selected constant. The errors calculated for *a* = <sup>τ</sup>q and several simulation times *b* ∈ {<sup>τ</sup>q, τT, 3<sup>τ</sup>T, 10τT} are presented in Table A1. It can be seen that the simplified heat transfer model *qv*(*t*) = *H*(*t*) can be used, with relative error about −1.91%, for *a* = <sup>τ</sup>q*,* and simulation time about *t* ≥ 3<sup>τ</sup>T, *b* = 3<sup>τ</sup>T. The relative error err is neglected for *t* ≥ 10<sup>τ</sup>T and *b* = 10<sup>τ</sup>T.

**Table A1.** Value of relative error calculated for different time values a, b, and the test function Equation (A9).


It is worth emphasizing that calculations presented in the above table have been carried out for temperature and heat flux time lags for platinum.
