This paper shows the special functions are a mathematical tool to solving nonlinear equations. The gamma function is used as an example to show the one-step solution process for a special nonlinear oscillator. Comparison with the exact solution and the approximate solution obtained by the homotopy perturbation method reveals the gamma function method is extremely simple and remarkably accurate.

A way to obtain approximate periodic solutions to nonlinear oscillators arising in a micro-electro-mechanical system (MEMS) is presented for the case of zero initial conditions and magnetostatic excitation. The frequency-amplitude relationship is derived by adopting He’s frequency formulation. The obtained analytical results are illustrated graphically. The proposed simple approach gives the fast insight into the dynamics of the singular oscillator and can be useful for design of MEMS devices.

Wang’s fractal variational principle for a fractal nano/microelectromechanical oscillator is studied(K.L. Wang. Mathematical Methods in the Applied Sciences, 2020, DOI: 10.1002/mma.6726), and its Euler-Lagrange equation is obtained. A new variational principle is obtained by the semi-inverse method.

A document by Naveed Anjum. Click on the document to view its contents.

The nano/microelectromechanical systems (N/MEMS) have been caught much attention in the past few decades for their attractive properties such as small size, high reliability, batch fabrication, and low power consumption. The dynamic oscillatory behavior of these systems is very complex due to strong nonlinearities in these systems. The basic aim of this manuscript is to investigate the nonlinear vibration property of N/MEMS oscillators by the homotopy perturbation method coupled with Laplace transform (also called as He-Laplace method in literature). A generalized N/MEMS oscillator is systematically studied, and a fairly accurate analytic solution is obtained. Three special cases for electrically actuated MEMS, multi-walled Carbon nanotubes-based MEMS, and MEMS subjected to van der Waals attraction are considered for comparison, and a good agreement with exact solutions is observed.

This paper suggests a simple analytical method for Volterra-Fredholm integral equations, the solution process is similar to that by variational-based analytical method, e.g., Ritz method. The examples show the method is straightforward and effective.

Fangzhu, which has been lost for thousands of years, is an ancient device for water collection from air, its mechanism is unknown yet. Here we elucidate its possible surface-geometric and related physical properties by the oldest the Yin-Yang contradiction. In view of modern nanotechnology, we reveal that Fangzhu’s water-harvesting ability is obtained through a hydrophilic-hydrophobic hierarchy of the surface, mimicking spider web’s water collection, lotus or desert beetle’s water intake. The convex-concave hierarchy of Fangzhu’s textured surface enables it to have low wettability(high geometric potential) to attract water molecules from air through the nano-scale convex surface and transfer the attracted water along the concave surface to the collector. A mathematical model is established to reveal three main factors affecting its effectiveness, i.e., the air velocity, the surface temperature and surface structure. The lost technology can play an extremely important role in modern architecture, ocean engineering, transportation and others to catch water from air for everyday use.