In the paper, an oscillatory system with liquid dampers is considered, when the mass of the head is large enough. By means of expedient transformations, the equation of motion with fractional derivatives is reduced to an equation of fractional order containing a small parameter. The corresponding nonlocal boundary value problem is solved and the zero and first approximations of SOLUTIONs of the relative small parameter are constructed. The results are illustrated on the concrete example, where the SOLUTION differs from the analytical SOLUTION by 10−, 2 order.

Transversal vibrations of an axially moving string under boundary damping are investigated. Mathematically, it represents a homogenous linear partial differential equation subject to nonhomogeneous boundary conditions. The string is moving with a relatively (low) constant speed, which is considered to be positive. The string is kept fixed at the first end, while the other end is tied with the spring-dashpot system. The ASYMPTOTIC approximations for the SOLUTION of the equations are obtained by application of two time-scale perturbation technique and the characteristic coordinates method. The vertical displacement of the moving system under boundary damping is computed by using specific initial conditions. It is shown that how the introduced damping at the boundary may affect the vertical displacement of the axially moving system.

In the past decades, in the area of mathematical ecology, the dynamical properties occurring in the predator-prey models have been studied. Moreover, the stability and boundedness of the SOLUTION for population model such as cyclic, delayed and etc. have been studied. In the present paper, a nonlinear cyclic predator-prey system with sigmoidal type functional response is analyzed. Indeed, a model of four species predator-prey system has been investigated and the sufficient conditions for stability and boundedness of the SOLUTIONs of predator-prey system have been presented. For this purpose, the differential inequality theory is employed and finally, by constructing a suitable Lyapanov function the existence and uniqueness of ASYMPTOTICally periodic SOLUTION which is globally ASYMPTOTICally stable are proved.

Initial-boundary value problems including space-time fractional PDEs have been used to model a wide range of problems in physics and engineering fields. In this paper, a non-self adjoint initial boundary value problem containing a third order fractional differential equation is considered. First, a spectral problem for this problem is presented. Then the eigenvalues and eigenfunctions of the main spectral problem are calculated. In order to calculate the roots of their characteristic equation, the ASYMPTOTIC expansion of the roots is used. Finally, by suitable choice of these ASYMPTOTIC expansions, related eigenfunctions and Mittag-Lefler functions, the analytic and numerical SOLUTIONs to the main initial-boundary value problem are given.