**Tomasz Raszkowski \* and Mariusz Zubert**

Department of Microelectronics and Computer Science, Lodz University of Technology, 90-924 Lodz, Poland; mariuszz@dmcs.p.lodz.pl

**\*** Correspondence: traszk@dmcs.pl

Received: 25 March 2020; Accepted: 13 May 2020; Published: 15 May 2020

**Abstract:** This paper presents an analysis of the time complexity of algorithms prepared for solving heat transfer problems at nanoscale. The first algorithm uses the classic Dual-Phase-Lag model, whereas the second algorithm employs a reduced version of the model obtained using a Krylov subspace method. This manuscript includes a description of the finite difference method approximation prepared for analysis of the real microelectromechanical system (MEMS) structure manufactured by the Polish Institute of Electron Technology. In addition, an approximation scheme of the model, as well as the Krylov subspace-based model order reduction technique are also described. The paper considers simulation results obtained using both investigated versions of the Dual-Phase-Lag model. Moreover, the relative error generated by the reduced model, as well as the computational complexity of both algorithms, and a convergence of the proposed approach are analyzed. Finally, all analyses are discussed in detail.

**Keywords:** Dual-Phase-Lag heat transfer model; Krylov subspace-based model order reduction; algorithm efficiency analysis; relative error analysis; algorithm convergence analysis; computational complexity analysis; thermal simulation algorithm; finite difference method scheme; Grünwald– Letnikov fractional derivative
