Vibrations and dynamic chaos should be controlled in structures and machines. The wing of the airplane should be free from vibrations or it should be kept minimum. To do so, two main strategies are used. They are passive and active control methods. In this paper we present a mathematical study of passive and active control in some non-linear differential equations describing the vibration of the wing. Firstly, non-linear differential equation representing the wing system subjected to multi-excitation force is considered and solved using the method of multiple scale perturbation. Secondly, a tuned mass absorber (TMA) is applied to the system at simultaneous primary resonance. Thirdly, the same system is considered with 1:2 internal resonance active control absorber. The approximate solution is derived up to the fourth order approximation, the stability of the system is investigated applying both frequency response equations and phase plane methods. Previous work regarding the wing vibration dealt only with a linear system describing its vibration. Some recommendations are given by the end of the work.

In this paper, the two-layer viscous shallow-water equations are derived from the threedimensional Navier-Stokes equations under the hydrostatic assumption. It is noted that the combination of upper and lower equations in the two-layer model produces the classical one-layer equations if the density of each layer is the same. The two-layer equations are approximated by a finite element method which follows our numerical scheme established for the one-layer model in 1978. Finally, it is numerically demonstrated that the interfacial instability generated when the densities are the same can be eliminated by providing a sufficient density difference.

A method for numerical solution of boundary value problems with ordinary differential equation based on the method of Green's function incorporated with the double exponential transformation is presented. The method proposed does not require solving a system of linear equations and gives an approximate solution of very high accuracy with a small number of function evaluations. The error of the method is $O\left(\exp\left(-C_1N/\log(C_2N)\right)\right)$ where $N$ is a parameter representing the number of function evaluations and $C_1$ and $C_2$ are some positive constants. Numerical examples also prove the high efficiency of the method. An alternative method via an integral equation is presented which can be used when the Green's function corresponding to the given equation is not available.

Railway track irregularities need to be kept at a satisfactory level by taking appropriate maintenance activities. This paper aims at obtaining an optimal maintenance schedule for improving the railway track irregularities using all-integer linear programming (AILP) optimization model analyses. Firstly, we try to predict a change of surface irregularities by investigating the transition process through degradation and restoration model analyses. Then we develop an AILP model for obtaining an optimal schedule of multiple tie tamper (MTT) operation. The model takes both maintenance costs and the level of surface irregularities that reflects riding quality and safety into account, then finally gives an optimal tamping schedule of MTT for the whole year. Then we apply the results of this model to solve the optimal MTT's maintenance scheduling problem for the actual railway network system and show that it is effective and useful enough by comparing our model results with actual existing data.