# A Comparative Study of Differential Quadrature Methods for METE Nanobeam Vibrations
## 1 Introduction
This necessitates the incorporation of nonlocal considerations into traditional structural theories to accurately capture the size-dependent behavior observed in these systems [18]. Numerous recent publications have focused on exploring the static and dynamic characteristics of magneto-electro-thermo-Elastic (METE) nanobeams. These works introduce exact and semi-analytical techniques for addressing linear free and forced vibrations, as well as buckling phenomena in nanobeams [19; 20; 21]. Additionally, alternative numerical approaches such as finite element analysis [22], meshless methods [23], higher-order B-spline finite strip modeling [24], and Rayleigh-Ritz methods [25] have been investigated for tackling these issues. However, all these methodologies demand a significant number of grid points and substantial computational resources to achieve a satisfactory level of precision. The Polynomial-Based Differential Quadrature Method (PDQM) has been proven to be able to produce correct results with fewer grid points [26; 27; 28; 29; 30], in contrast to many numerical methods [31; 32; 33]. More reliable substitutes are provided by the Discrete Singular-Convolution Differential Quadrature Method (DSCQM) [34] and the Sinc Differential Quadrature Method (SDQM) [35]. The selection of shape functions, such as the Regularized Shannon Kernel (RSK), Delta Lagrange Kernel (DLK), and cardinal sine function, which enhance solution convergence and stability, is crucial to the efficacy of these techniques [36; 37; 38; 39; 40]. Using the Sinc Differential Quadrature Method (SDQM) and Discrete Singular-Convolution Differential Quadrature Method (DSCQM), this study examines the vibration properties of magneto-electro-thermo-elastic (METE) nanobeams sitting on nonlinear elastic substrates. Although there are several numerical methods [31; 32; 33], the PDQM has been proven to be able to provide precise results using fewer grid points [26; 27; 28; 29; 30]. Better accuracy and stability are provided by the SDQM [35] and DSCQM [34] than by the conventional Polynomial-Based DQM. To improve the convergence and stability of the numerical solutions, these techniques rely on the selection of suitable shape functions, such as the Regularized Shannon Kernel (RSK), Delta Lagrange Kernel (DLK), and cardinal sine function [36; 37; 38; 39; 40]. By creating innovative numerical algorithms and comparing them to proven analytical and numerical results, this study pioneers the application of the SDQM and DSCQM to this particular situation. The impact of various parameters on the natural frequencies and mode shapes of the METE nanobeams are analyzed through a thorough parametric investigation. Axial loads, external magnetic potentials, temperature fluctuations, boundary conditions, material qualities, nonlocal parameters, axial loads, nonlinear and linear elastic foundations, and aspect ratios are some of these variables. The structure of this paper is as follows: the mathematical formulation is explained in depth in Section 2, the numerical methods are described in Section 3, the results and discussion are presented in Sections 4 and 5, and the main conclusions and future research areas are discussed in Section 6.