Review of the Methods of Transition from Partial to Ordinary Differential Equations: From Macro- to Nano-structural Dynamics

Abstract This review/research paper deals with the reduction of nonlinear partial differential equations governing dynamic behavior structural mechanical members emphasis put on theoretical aspects applied methods and signal processing. Owing to rapid development technology, materials science in particular micro/nano systems, there is a need not only revise approaches mathematical modeling vibrations, but also choose/propose novel (extended) theoretically based hence, motivating numerical algorithms, get authentic, reliable, validated accurate solutions complex models derived (nonlinear PDEs). The review introduces reader traditional broad spectrum Fourier-type methods, Galerkin-type Kantorovich–Vlasov variational iteration as well Vaindiner Agranovskii–Baglai–Smirnov. While some them are known by computational engineering-oriented community, attention paid important (from our point view) widely used classical approaches. In addition, considerations supported most popular frequently employed algorithms direct schemes finite element method (FEM) difference (FDM) validate results obtained. spite general aspect paper, mentioned so far quantified compared respect applications branch mechanics, i.e. vibrational nanostructures, which includes own research presented throughout paper. Namely, considerable effort has been devoted investigate features Germain–Lagrange nanoplate (including physical nonlinearity inhomogeneity materials). Modified obtained using Kirchhoff’s hypothesis relations modified couple stress theory Hamilton’s principle. A comparative analysis carried out identify effective for solving physics taking an example equation nanoplate. experiments reducing problem PDEs ODEs Fourier’s ideas (separation variables), Bubnov–Galerkin static problems Faedo–Galerkin quantified. An exact solution nanoplates served quantify efficiency various including method, iterations Vaindiner’s (the last three include theorems regarding their convergence). have combinations FDM second order accuracy FEM triangular quadrangular elements. studied ordinary show high feasibility solve numerous systems