System Of Non-homogeneous Differential Equations Research Articles

Transient behavior of a plate partially immersed in the fluid subjected to impact loadings: Theoretical analysis and experimental measurements

This study aimed to investigate the transient behavior of a rectangular plate partially in contact with fluid subjected to a dynamic external force. To this purpose, a theoretical model was developed to analyze vibration characteristics and transient wave propagation. Based on superposition method, the dry mode shapes and natural frequencies of the plate under vacuum could be obtained. The dry mode shapes were treated as the fundamental function to construct the wet mode that describes the vibration behavior of the fluid-plate coupled system. The velocity potential and fluid pressure within a finite tank due to the plate deflection were derived using an equation governing the incompressible fluid. Based on the relationship between dry mode shape and wet mode shape, the fluid-plate coupled system's wet mode shape and resonant frequency could be determined from the frequency response function. Applying the normal mode method, the transient displacement of plate and fluid pressure can be obtained by solving a system of non-homogeneous differential equations. The theoretical predictions were verified by finite element method (FEM) and experimental measurements. Experiments were conducted using piezoelectric film sensors (polyvinylidene fluoride, PVDF) to measure the force history induced by a steel ball impact to quantitatively analyze the transient response. The comparison results proved that the theoretical predictions and experiments were in good agreement, including the transient responses of the displacement and in-plane strain of a plate partially submerged in the fluid. The results indicate that changes in water depth can induce resonance frequency shifts and wet mode shape distortions, which also illustrate that the vibrational properties of wet modes affect transient behavior. The proposed transient solution demonstrates an analytical approach that connects the physical significance of the dynamic behavior of the fluid-plate coupled system in time and frequency domains; it provides a connection between the transient behaviors and vibration characteristics.

Mathematical Modelling of Malware Intrusion in Computer Networks

Abstract Malware attacks cause great harms in the contemporary information systems and that requires analysis of computer networks reaction in case of malware impact. The focus of the present study is on the analysis of the computer network’s states and reactions in case of malware attacks defined by the susceptibility, exposition, infection and recoverability of computer nodes. Two scenarios are considered – equilibrium without secure software and not equilibrium with secure software in the computer network. The behavior of the computer network under a malware attack is described by a system of nonhomogeneous differential equations. The system of the nonhomogeneous differential equations is solved, and analytical expressions are derived to analyze network characteristics in case of susceptibility, exposition, infection and recoverability of computer nodes during malware attack. The analytical expressions derived are illustrated with results of numerical experiments. The conception developed in this work can be applied to control, prevent and protect computer networks from malware intrusions.

Etiology of Multiple Sclerosis

Multiple Sclerosis (MS) is characterized by degradation of the essential myelin sheath around the nerves in the white matter of the central nervous system (CNS). I argue that this is caused by insufficient production of myelin, as myelin-forming oligodendrocytes become damaged by excess steroid receptors type 1 in the cytoplasm, which clump and release zinc due to the relative lack of their ligands. This may gradually lead to hyperphosphorylation of tau protein. Accordingly, the etiological basis for MS is thought to be at least three synergistic systems with the components, steroid receptors type 1 and their ligands: vitamin D, estrogen and testosterone. To prove this hypothesis with its many variables, it has been necessary to use a system of non-homogeneous differential equations.

A System of Differential-Operator Equations of Different Orders in Hilbert Spaces

We consider an abstract Cauchy problem for a system of nonhomogeneous abstract differential equations in Hilbert spaces. The “main” equation is of the second order and “boundary” equations are of the first order. Existence of a solution is proved. Application to mixed (initial boundary-value) problems for one-dimensional second order hyperbolic equations and for fourth order PDEs with the time derivative in boundary conditions has been shown.