Attracting Sets Of Nonlinear Difference Equations With Time-Varying Delays

What follows is a discussion on the application of a digital computer to solve analytically a class of nonlinear ordinary differential equations. The asymptotic method of Krylov, Bogoliubov and Mitropolsky is employed to obtain analytical solutions. In addition, a procedure has been developed through which the initial amplitude and phase can be expressed as functions of initial displacement and velocity without the need of integration. Since the algorithms are analytical in nature, the truncated power series capability of the symbolic algebraic system ALTRAN has been used to generate analytical solutions automatically.

A three operator split-step method covering a larger set of non-linear partial differential equations

<p style='text-indent:20px;'>In this paper, we deal with the qualitative theory for a class of nonlinear differential equations with switching at variable times (SSVT), such as the existence and uniqueness of the solution, the continuous dependence and differentiability of the solution with respect to parameters and the stability. Firstly, we obtain the existence and uniqueness of a global solution by defining a reasonable solution (see Definition 2.1). Secondly, the continuous dependence and differentiability of the solution with respect to the initial state and the switching line are investigated. Finally, the global exponential stability of the system is discussed. Moreover, we give the necessary and sufficient conditions of SSVT just switching <inline-formula><tex-math id="M1">\begin{document}$ k\in \mathbb{N} $\end{document}</tex-math></inline-formula> times on bounded time intervals.</p>

A new technique is proposed to ensure global asymptotic stability for nonlinear switched time-varying systems with time-varying discontinuous delays. It uses an adaptation of Halanay's inequality to switched systems and a recent trajectory based technique. The result is applied to a family of linear time-varying systems with time-varying delays.

Nonlinear differential equations arise widely in applied mathematics, physics, and engineering, yet many conventional analytical and numerical methods remain limited in their ability to handle strong nonlinearities efficiently and accurately. This paper presents a novel computational framework based on the Modified Laplace–Adomian Polynomial Method (LAPM) for solving nonlinear differential equations. The proposed method integrates the Laplace transform with an enhanced form of the Adomian Decomposition Method, enabling complex nonlinear terms to be decomposed into rapidly convergent Adomian polynomials. This integration simplifies the solution procedure, reduces computational complexity, and preserves high accuracy. The performance of LAPM was validated using several benchmark nonlinear and linear differential equations, and the results demonstrated superior convergence speed, precision, and stability compared with traditional methods. The study concludes that the Modified Laplace–Adomian Polynomial Method is a reliable and efficient approach for solving a broad class of nonlinear differential equations. This work contributes to the advancement of computational methods by offering a robust alternative for the analysis of differential equation models encountered in mathematics, physics, and engineering.

A robust stabilization problem is investigated for a class of imperfectly known implicit systems with discrete and distributed time-varying delays. The delays are assumed to be unknown but bounded, with known bounds. The imperfections acting on the systems, which may be time-, state-, delayed state-, and/ or inputdependent, are modelled as additive nonlinear perturbations influencing a known set of implicit nonlinear functional differential equations of the neutral type. Sufficient conditions, which include delaydependent matrix inequalities and a delay-dependent stability criterion involving some bounding parameters for the uncertainty and delays in the system, are presented, and a class of robust feedback memoryless control inputs are designed to guarantee a prescribed asymptotic stability property for the class of implicit systems.

In this study, a collocation method based on Fibonacci polynomials is used for approximately solving a class of nonlinear differential equations with initial conditions. The problem is firstly reduced into a nonlinear algebraic system via collocation points, later the unknown coefficients of the approximate solution function are calculated. Also, some problems are presented to test the performance of the proposed method by using error functions. Additionally, the obtained numerical results are compared with exact solutions of the test problems and approximate ones obtained with other methods in literature.

We establish asymptotic representations for the solutions of one class of nonlinear differential equations of the second order with rapidly and regularly varying nonlinearities.

On a class of nonlinear differential equations of astrophysics

Solution of nonlinear partial differential equations from base equations

In this study, a single-degree-of-freedom structure with multiple nonlinear tuned mass dampers is investigated. A sinusoidal excitation is assumed to be applied to the structure. In order to obtain the results, a nonlinear mathematical model of the general system is obtained in the form of a set of nonlinear ordinary differential equations. Dry friction and cubic stiffness nonlinearities are considered for the nonlinear terms. To obtain steady-state frequency responses of the system, these nonlinear ordinary differential equations are transformed into a set of nonlinear algebraic equations by using Harmonic Balance Method. Fourier transformation is used in the calculation of the Fourier coefficients of the nonlinear forcing terms. Then, numerical solutions to the resulting nonlinear algebraic equations are obtained using Newton’s Method with Arc Length Continuation. The effect of the number of tuned mass dampers and their arrangement, i.e. in series or in parallel, on vibration reduction is observed by performing optimization on several cases and comparing them. In order to observe the separate and combined effects of the nonlinearities, and the number of tuned mass dampers used, several case studies are carried out using optimized case parameters with different combinations of the nonlinearities. The results of these case studies and the linear model are compared with each other to see the effect of the nonlinearities. Additionally, results of the system with optimum nonlinear tuned mass dampers are compared with that of the optimum linear tuned mass dampers and the effect of nonlinearities are discussed.

We establish asymptotic properties of some types of solutions of one class of essentially nonlinear nonautonomous differential equations of the second order and find necessary and sufficient conditions for the existence of these solutions.

New results on global exponential stability of recurrent neural networks with time-varying delays

We apply the Fourier‐least squares method (FLSM) which allows us to find approximate periodic solutions for a very general class of nonlinear differential equations modelling oscillatory phenomena. We illustrate the accuracy of the method by using several significant examples of nonlinear problems including the cubic Duffing oscillator, the Van der Pol oscillator, and the Jerk equations. The results are compared to those obtained by other methods.

Exponential stability of FCNNs with time-varying leakage delays and reaction–diffusion terms