# A Comparative Study of Differential Quadrature Methods for METE Nanobeam Vibrations
## 1 Introduction  
Magneto-electro-thermo-elastic (METE) composites, comprising piezoelectric and piezomagnetic phases, exhibit remarkable energy conversion capabilities, transforming electric, thermal, elastic, and magnetic energy [1, 2]. Their versatility has led to widespread adoption in various fields, including vibration control, actuation, sensing, medical devices, health monitoring systems, and energy harvesting [3, 4]. This section analyzes the available literature on the analysis of magneto-electro-elastic systems. [PERSON] et al. [5] investigated the static behavior of 3D, doubly curved, functionally graded magneto-electro-elastic shells under combined mechanical, electrical, and magnetic loads. [PERSON] et al. [6] proposed analytical and semi-analytical solutions for anisotropic, functionally graded magneto-electro-elastic beams under arbitrary stresses, based on sinusoidal series expansions. [PERSON] [7] investigated the vibration characteristics of transversely isotropic magneto-electro-elastic rectangular plates, considering various vibration conditions (free, deterministic, and random) within fluid environments. [PERSON] et al. [8] developed a nonlocal, geometrically nonlinear beam model for magneto-electro-thermo-elasticnanobeams subjected to external electric voltage, magnetic potential, and temperature changes. Finally, [PERSON] et al. [9] analyzed the free vibration of embedded magneto-electro-elastic cylindrical nanoshells using [PERSON]'s shell theory. These studies collectively provide a foundation for understanding the diverse mechanical and vibrational behaviors of magneto-electro-elastic materials across different structural configurations. Prior research has examined the response of magneto-electro-elastic materials under various loading conditions. [PERSON] et al. [5] investigated the static behavior of three-dimensional (3D), doubly curved, functionally graded magneto-electro-elastic shells subjected to combined mechanical, electrical, and magnetic loads. [PERSON] et al. [6] developed analytical and semi-analytical solutions for anisotropic, functionally graded magneto-electro-elastic beams under arbitrary loading. These studies provide a foundation for understanding the complex interactions between mechanical, electrical, and magnetic fields within these materials. [PERSON] [7] studied the vibration characteristics of transversely isotropic magneto-electro-elastic rectangular plates, considering free, deterministic, and random vibrations in fluid environments. Furthermore, studies have explored the behavior of magneto-electro-elastic nanostructures. [PERSON] et al. [8] developed a nonlocal, geometrically nonlinear beam model to analyze the behavior of magneto-electro-thermo-elastic nanobeams under various external stimuli. [PERSON] et al. [9] investigated the free vibration properties of embedded magneto-electro-elastic cylindrical nanoshells, considering size-dependent effects. The field has seen substantial growth, particularly in the investigation of magneto-electro-thermo-elastic (METE) nanomaterials and nanostructures. Studies have focused on materials such as BiFeO\({}_{3}\), BiTiO\({}_{3}\)-CoFe\({}_{2}\)O\({}_{4}\), NiFe\({}_{2}\)O\({}_{4}\)-PZT, and various nanowire and nanobeam configurations [10; 11]. The recent incorporation of PTZ-5H-COFe\({}_{2}\)O\({}_{4}\) further expands the potential applications of these materials. The investigation of nanoscale phenomena frequently employs nonlocal elasticity theory [12; 13]. While experimental characterization at the nanoscale level presents significant challenges, and molecular dynamics simulations incur high computational costs, continuum models remain crucial for nanostructure research [14; 15; 16]. However, compelling evidence suggests that the nonlocal effect, arising from the inherent small-length scales, significantly influences the mechanical properties of nanostructures [15; 16; 17].