## 1 Introduction
[PERSON] [6] employed a novel isogeometric Bezier finite element approach to solve the vibration problem of functionally gradient piezoelectric plates, all the numerical results obtained were precise enough since the proposed approach inherited all the advantages of the accurate geometry of isogeometric analysis and traditional FEM. Within the framework of the isogeometric approach, [PERSON] [7] adopted the Carrera's unified formulation as the approximate field solution, which yielded accurate numerical results for free vibration and buckling problems of cross-ply laminated plates. [PERSON] [8] developed the coupled motion equations for concrete slabs reinforced by agglomerated SiO2 nanoparticles through using the third order shear deformation theory. The authors provided new, precise approximate results for forced vibration of concrete slabs by combining the Newmark method and the harmonic differential quadrature approach. Results indicated that linear frequencies of such type structures increased when the volume percent of SiO2 nanoparticles grew up to 0.37. [PERSON] explored new, precise numerical solutions of plate [9; 10; 11] and shell [12] problems by adopting the advanced discrete singular convolution approach which was treated as a reliable and applicable tool for its excellent accuracy and efficiency. Compared with the numerical/approximate methods, the progress made by analytical solutions is relatively few owing to the mathematical difficulty in solving the governing PDEs under the given boundaries. Semi-inverse approaches on the basis of Fourier series theory [13; 14; 15] are treated as the most popular analytical ways to solve plate problems, in which the preselected solutions are mainly in the form of single or double Sine series and the majority of boundaries investigated are mainly restricted to simple classical boundaries, such as the fully simply supported boundaries and Levy-type boundaries. For example, [PERSON] [16] employed a modified Fourier series method to acquire new exact analytical free vibration solutions for rectangular plates with general elastic boundary supports. Based on nonlocal elasticity theory and the generalized heat conduction model with phase delays, [PERSON] [17] provided Sinusoidal form solutions for the thermomechanical problem of rotating size-dependent nanobeams with clamped-clamped boundary conditions. Except for the classical single/double Sine series solutions for plate problems, various types of trial functions such as Quasi-Green's function [18], beam function [19], power series [20], displacement potential functions [21], etc. allow researchers to pursue accurate results for problems of plates with some certain boundaries. Based on the generalized thermoelasticity theory, [PERSON] [22] obtained the analytical solution for the temperature, displacement components, and stresses of thermoelastic material plates by utilizing an eigenvalue approach. By using the same method, the author [23] also provided new accurate analytical solutions for the free vibration problem of thermoelastic hollow sphere, in which the dispersion relations for various types of possible vibration modes of the hollow sphere were well derived. [PERSON] [24] developed the extended Kantorovich approach to handle the buckling problem of laminated composite plates. [PERSON] [25] proposed a new Spectral stiffness approach for buckling analysis of Winkler foundation plates with general boundaries. By using an extended separation-of-variable approach, [PERSON] [26, 27] obtained close-form solutions of thin plates with several non-Levy boundaries, the given solution procedure enables one to solve eigenvalue equations simultaneously. [PERSON] [28] developed a modified Superposition-Galerkin approach to investigate vibration behaviors of thin plates, in which the difficulties in determining vast families of roots of traditional superposition method can be avoided. [PERSON] [29] derived an exact in-plane vibration solution for Mindlin plates through utilizing the Helmholtz decomposition.