Nonlinear partial differential equations with higher order dispersion terms play an important role in dynamics research. In this paper, the fifth order KdV equation with high order dispersion term is studied and discussed. Firstly, the bilinear form of the fifth order KdV equation with high order dispersion term is derived by Hirota bilinear form. Then, the combined test function of the positive quartic function, quadratic function, exponential function and the interaction solution of the hyperbolic function of the fifth order KdV equation with variable coefficients is constructed, and the resonance multi-soliton test function of the equation is constructed by using the linear superposition principle.By means of mathematical symbol calculation, the interaction solution between high-order Lump solution and periodic cross kink solution of the fifth order KdV equation with variable coefficients and its resonance multi-solitons are solved.And by observing its corresponding graph analysis of its physical phenomenon.

A nonlinear oscillator arising in a micro-electro-mechanical system (MEMS) is difficult to be solved analytically due to the zero conditions. so the main objective of this work is to analyze the mathematical model of this system, and its approximate analytical solution is solved via the coupling Variational Iteration method and Laplace transform(LVIM). This method provides an efficient way to obtain the approximate nonlinear frequency and approximate solutions of MEMS. Moreover, LVIM also approximates the pull-in threshold in terms of model parameters. Finally, the results are compared with the exact one and a good result is obtained.