Oscillatory Behavior of Third-Order Quasi-Linear Neutral Differential Equations

In this paper, we consider a class of quasilinear third-order differential equations with a delay argument. We establish some conditions of such certain third-order quasi-linear neutral differential equation as oscillatory or almost oscillatory. Those criteria improve, complement and simplify a number of existing results in the literature. Some examples are given to illustrate the importance of our results.

Digital simulation cylindrical milling process with end mills

Any cutting process is accompanied by vibrations that can lead to a chatter of machining surface and even to a loss of stability. Mathematical modeling helps to solve such problems of the choice of cutting mode that provide vibration-free machining. The developed mathematical model takes into account the closed of the cutting process and machining "on the trail", which is implemented in its structural diagram representing the process, feedbacks and a function of a delay argument. The dynamic model of a technological machining system is represented by a single-mass model with two degrees of freedom. The developed block diagram of the system together with numerical algorithms for solving the geometric intersection of the cutter tooth with the workpiece and the numerical method of integrating differential equations form the basis of the created application program for digital modeling of the process. The simulation results indicate an adequate response of the program as a whole, which allows us to take the proposed methodology as the basis for further improvement of the virtual research process and the prediction of its real properties. Key words: Cylindrical milling, end mill, digital simulation.

A New Multi-Step Method for Solving Delay Differential Equations using Lagrange Interpolation

This paper presents 2-step p-th order (p = 2,3,4) multi-step methods that are based on the combination of both polynomial and exponential functions for the solution of Delay Differential Equations (DDEs). Furthermore, the delay argument is approximated using the Lagrange interpolation. The local truncation errors and stability polynomials for each order are derived. The Local Grid Search Algorithm (LGSA) is used to determine the stability regions of the method. Moreover, applicability and suitability of the method have been demonstrated by some numerical examples of DDEs with constant delay, time dependent and state dependent delays. The numerical results are compared with the theoretical solution as well as the existing Rational Multi-step Method2 (RMM2).

Even-order differential equation with continuous delay: nonexistence criteria of Kneser solutions

In this paper, we study even-order DEs where we deduce new conditions for nonexistence Kneser solutions for this type of DEs. Based on the nonexistence criteria of Kneser solutions, we establish the criteria for oscillation that take into account the effect of the delay argument, where to our knowledge all the previous results neglected the effect of the delay argument, so our results improve the previous results. The effectiveness of our new criteria is illustrated by examples.

Oscillation results for a certain class of fourth-order nonlinear delay differential equations

In this work, we study the oscillation of the fourth order neutral differential equations with delay argument. By means of generalized Riccati transformation technique, we obtain new oscillation criteria for oscillation of this equation. An example is given to clarify the main results in this paper.

The stability analysis of the market price using Lambert function method

In this article we are going to analyze market price stability with different market intensity coefficient and delay argument values. Market price is described as a scalar differential equation with a delay argument. In order to find solutions for the transcendental equation we will use method based on Lambert function. We will present examples of the applications of the method.

Criteria for the Nonexistence of Kneser Solutions of DDEs and Their Applications in Oscillation Theory

The objective of this study was to improve existing oscillation criteria for delay differential equations (DDEs) of the fourth order by establishing new criteria for the nonexistence of so-called Kneser solutions. The new criteria are characterized by taking into account the effect of delay argument. All previous relevant results have neglected the effect of the delay argument, so our results substantially improve the well-known results reported in the literature. The effectiveness of our new criteria is illustrated via an example.

Some New Oscillation Results for Fourth-Order Neutral Differential Equations with Delay Argument

The aim of this paper is to study the oscillatory properties of 4th-order neutral differential equations. We obtain some oscillation criteria for the equation by the theory of comparison. The obtained results improve well-known oscillation results in the literate. Symmetry plays an important role in determining the right way to study these equation. An example to illustrate the results is given.

The Problem of Two Bodies with a Finite Velocity of Gravity

From the moment of Newton’s discoveryof the law of universal gravitation, ordinary differentialequations were used to study the motion of bodies,since it was assumed that the velocity of gravitationis infinite. However, in reality the velocity of gravityis finite, which is consistent with the theory of relativityof Einstein, which postulated that the velocity ofgravity matches the velocity of light, and the studiesconducted by S. Kopeikin and E. Fomalont on the fundamentallimit of the velocity of gravity. Due to thedelay of the gravitational field for studying the motionof bodies, the mathematical apparatus based on differentialequations with a delay argument is the mostacceptable one. These equations are used to constructand study the mathematical model of the motion of twobodies. It is shown that the motion of these bodies withidentical masses (with finite velocity of gravity!) is notcarried out in accordance with Kepler’s laws.