Search

In this paper, we investigate the Lipschitz stability of a perturbed impulsive differential system concerning the unperturbed system. We employ the variation of parameters or the constant of variation for impulsive differential systems with an initial time difference. [ABSTRACT FROM AUTHOR]

Abstract This paper presents a new approach for finding exact solutions to certain classes of nonlinear partial differential equations (NLPDEs) by combining the variation of parameters method with classical techniques such as the method of characteristics. Our primary focus is on NLPDEs of the form $$u_{tt}+a(x,t)u_{xt}+b(t)u_{t}=\alpha (x,t)+ G(u)(u_{t}+a(x,t)u_{x})e^{-\int b(t)dt}$$ u tt + a ( x , t ) u xt + b ( t ) u t = α ( x , t ) + G ( u ) ( u t + a ( x , t ) u x ) e - ∫ b ( t ) d t and $$u_{t}^{m}(u_{tt}+a(x,t)u_{xt})+b(t)u_{t}^{m+1}=e^{-(m+1)\int b(t)dt}(u_{t}+a(x,t)u_{x}) F(u,u_{t}e^{\int b(t)dt}).$$ u t m ( u tt + a ( x , t ) u xt ) + b ( t ) u t m + 1 = e - ( m + 1 ) ∫ b ( t ) d t ( u t + a ( x , t ) u x ) F ( u , u t e ∫ b ( t ) d t ) . We provide numerical validation through several examples to ensure accuracy and reliability. Our approach enhances the applicability of analytical solution methods for a broader range of NLPDEs.

A review of the basic Hansen's ideal frame algorithms for accurate numerical integration of perturbed elliptic motion is carried out. The fundamental approaches rely on the use of nonsingular variables and differ in the ways in which the ellipse in the orbital plane is determined. It is well known that the accuracy of the propagation of the orbit geometry is notably increased when using time-regularization techniques to transform the independent variable. However, this is at the expense of adding a differential equation to compute the time, which gathers the Lyapunov-type instabilities that are removed from the coordinates. The asynchronism resulting from errors in the numerical integration of the time may be palliated with the use of time elements, to which end a constant and a linear nonsingular time element are presented, which are new to our knowledge. [ABSTRACT FROM AUTHOR]

Summary: In this article, the global and exponential synchronization of Lur'e networks with hybrid time‐varying delays is investigated. For sake of saving control costs, a novel impulsive adaptive controller is presented, where the adaptive feedback control input is utilized to compensate the control effects of the impulsive signal. Meanwhile, suitable control gains are obtained according to the designing on adaptive updating laws. In view of the definition of average impulsive interval, the generalized comparison principle and the formula of parameter variation, sufficient conditions for achieving global and exponential synchronization of Lur'e dynamical networks are eventually derived. Two relevant numerical simulations are provided to illustrate the effectiveness of theoretical analysis and the control scheme. [ABSTRACT FROM AUTHOR]

A review of the basic Hansen’s ideal frame algorithms for accurate numerical integration of perturbed elliptic motion is carried out. The fundamental approaches rely on the use of nonsingular variables and differ in the ways in which the ellipse in the orbital plane is determined. It is well known that the accuracy of the propagation of the orbit geometry is notably increased when using time-regularization techniques to transform the independent variable. However, this is at the expense of adding a differential equation to compute the time, which gathers the Lyapunov-type instabilities that are removed from the coordinates. The asynchronism resulting from errors in the numerical integration of the time may be palliated with the use of time elements, to which end a constant and a linear nonsingular time element are presented, which are new to our knowledge.

This paper investigates the synchronization of inertial reaction-diffusion Cohen-Grossberg-type neural networks. Compared with the existing works concerning reaction-diffusion neural networks, the main innovation of this paper is that two design strategies of feedback synchronization controllers are proposed based on the types of time delays. For the systems with bounded differentiable delays, the sufficient conditions for synchronization are derived under the framework of Lyapunov method. If the time delay of the addressed system is unbounded or non-differentiable, it can also realize synchronization by employing the method of variation of parameters and some analytical techniques. Moreover, the proposed methods are applicable to various boundary conditions. The correctness of the obtained criteria is verified by three numerical examples. [ABSTRACT FROM AUTHOR]

In this study, multiplicative conformable fractional differential equations are presented. Wronskian concept, linear dependence-independence concepts are defined on multiplicative conformable fractional calculus and some theorems and results are given among them. Finally, some examples are solved by giving some methods for finding general solutions of multiplicative conformable fractional differential equations. [ABSTRACT FROM AUTHOR]

The theoretical substantiations of an indirect analytical method for assessing the wear of a dry friction damper structure are presented. The wear is simulated by variations in the parameters of the load–damper dynamic system, leading to changes in the kinematic characteristics of the motion of the system, based on which the wear effect has been estimated. Analytical solutions are obtained for the differential equations of the motion of the system and for their variations as functions of the parameters characterizing wear. This makes it possible to determine the dynamic response of the damper and to assess its dissipative properties. [ABSTRACT FROM AUTHOR]

Summary: This article is devoted to studying the problem of globally and exponentially adaptive cluster synchronization for a class of Lur'e dynamical networks with multiple time‐varying delays and hybrid couplings. A novel impulsive adaptive pinning feedback control protocol is proposed with fully considering the cluster‐tree topology structures of the networks and only imposed on the Lur'e systems in current cluster which exist directed paths with those Lur'e systems in the other clusters. In view of the concept of average impulsive interval, the comparison principle, the extended parameter variation methods, and the reductio ad absurdum, sufficient conditions are acquired for achieving the cluster synchronization of the derivative coupled Lur'e networks with considering different functions of the impulsive effects, respectively. Furthermore, according to the designed adaptive updating law, suitable feedback control strengths are obtained, which largely save the control costs. Finally, the effectiveness of the control schemes and the theoretical results have been proved by executing two numerical simulation examples. [ABSTRACT FROM AUTHOR]

This paper presents a new approach for finding exact solutions to certain classes of nonlinear partial differential equations (NLPDEs) by combining the variation of parameters method with classical techniques such as the method of characteristics. Our primary focus is on NLPDEs of the form u tt + a ( x , t ) u xt + b ( t ) u t = α ( x , t ) + G ( u ) ( u t + a ( x , t ) u x ) e - ∫ b ( t ) d t and u t m ( u tt + a ( x , t ) u xt ) + b ( t ) u t m + 1 = e - ( m + 1 ) ∫ b ( t ) d t ( u t + a ( x , t ) u x ) F ( u , u t e ∫ b ( t ) d t ) . We provide numerical validation through several examples to ensure accuracy and reliability. Our approach enhances the applicability of analytical solution methods for a broader range of NLPDEs., (© 2024. The Author(s).)