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Nonlinear chains of atoms (NCA) are complex systems with rich dynamics, that influence various scientific disciplines. The lie symmetry approach is considered to analyze the NCA. The Lie symmetry method is a powerful mathematical tool for analyzing and solving differential equations with symmetries, facilitating the reduction of complexity and obtaining solutions. After getting the entire vector field by using the Lie scheme, we find the optimal system of symmetries. We have converted assumed PDE into nonlinear ODE by using the optimal system. The new auxiliary scheme is used to find the Travelling wave solutions, while graphical behaviour visually represents relationships and patterns in data or mathematical models. The multiplier method enables the identification of conservation laws, and fundamental principles in physics that assert certain quantities remain constant over time. [ABSTRACT FROM AUTHOR]

The computational techniques are a set of novel problem-solving methodologies that have attracted wider attention for their excellent performance. The handling strategies of real-world problems are artificial neural networks (ANN), evolutionary computing (EC), and many more. An estimated fifty thousand to ninety thousand new leishmaniasis cases occur annually, with only 25% to 45% reported to the World Health Organization (WHO). It remains one of the top parasitic diseases with outbreak and mortality potential. In 2020, more than ninety percent of new cases reported to World Health Organization (WHO) occurred in ten countries: Brazil, China, Ethiopia, Eritrea, India, Kenya, Somalia, South Sudan, Sudan, and Yemen. The transmission of visceral leishmaniasis is studied dynamically and numerically. The study included positivity, boundedness, equilibria, reproduction number, and local stability of the model in the dynamical analysis. Some detailed methods like Runge Kutta and Euler depend on time steps and violate the physical relevance of the disease. They produce negative and unbounded results, so in disease dynamics, such developments have no biological significance; in other words, these results are meaningless. But the implicit nonstandard finite difference method does not depend on time step, positive, bounded, dynamic and consistent. All the computational techniques and their results were compared using computer simulations. [ABSTRACT FROM AUTHOR]

On the basis of the analysis of thermoelastic motion, the current research develops a novel model of modified thermoelasticity. The rotating long hollow cylinders with fixed surfaces are considered in a generalized Moore–Gibson–Thompson thermoelastic model (MGTTE) framework, including the modified Ohm's law. The cylinders are made of a thermoelastic material that rotates at a uniform rotational speed and is elastic in the transverse direction. The set of equations for the MGT heat conduction in the new model is built under the influence of the electromagnetic field by including a delay time in the context of Green–Naghdi of the third kind (GN-III). The inner boundary of the hollow cylinder is not only restricted but also sensitive to heat loading. The outer surface, on the other hand, is also restricted but insulates the heat. The Laplace transform method is utilized to deal with the differential equations produced in the new domain and transfer the problem to the space domain. The Dubner and Abate method is used to compute dynamically and graphically depict the theoretical findings for an isotropic transverse material. After comparing the results of several thermoelastic theories, the implications for the electromagnetic field are discussed. [ABSTRACT FROM AUTHOR]

The main purpose of this research was to use the comparison approach with a first-order equation to derive criteria for non-oscillatory solutions of fourth-order nonlinear neutral differential equations with p Laplacian operators. We obtained new results for the behavior of solutions to these equations, and we showed their symmetric and non-oscillatory characteristics. These results complement some previously published articles. To find out the effectiveness of these results and validate the proposed work, two examples were discussed at the end of the paper. [ABSTRACT FROM AUTHOR]

In this work, we have investigated the fractional differential equation to describe the motion of a linear oscillator using fractional derivative operators with or without singular kernels. In order to be consistent with the physical systems the value of the fractional parameter that characterizes the existence of fractional structures in the system, lies within unit interval. The solutions of the non-integer order differential equation are obtained and expressed in terms of generalized G function depending upon the fractional parameter. The classical cases could be recovered by making the limit of fractional parameter approaches to unity. Moreover, we will analyse and compare the behaviour of the oscillator with different definitions of the fractional operators via graphical illustrations, phase portraits and Poincare maps. [ABSTRACT FROM AUTHOR]

Nonlinear vibrations of double-layer nanoplate system are studied by numerical simulation. The governing equations are derived on a base of the nonlocal elasticity theory, von Kármán nonlinear theory, and Kirchhoff's hypotheses. Both layers are assumed to be with the same material properties, while interaction between them is taken by the Winkler model. Discretization of the system of governing differential equations is performed by the Bubnov-Galerkin method with the two-mode approximation model, and the obtained system of ordinary differential equations is analysed by the Runge–Kutta method. The study of a linearized system allows the construction of regions of instability. The application of nonlinear dynamics methods made it possible to determine the nature of oscillations when stability is lost, and when small-scale effects are manifested. • Nonlinear vibrations of double-layer graphene system under in-plane load are studied; • the nonlocal governing system based on von Kármán nonlinear relations is applied; • the proposed approach uses the Bubnov-Galerkin method with the two-mode model; • the Floquet theory is used to define instability zones; • it is concluded that small-scale effects on the stability zones are not significant; • small-scale effects can influence the nature of vibrations after loss of stability. [ABSTRACT FROM AUTHOR]

The externally excited and damped vibration of the double pendulum in the vertical plane are considered. The pendulum can collide many times during the motion with a motionless obstacle having the rough surface. The double pendulum colliding with this object has been modeled as a piecewise smooth system constrained by the unilateral constraint. In the relatively long time between the collisions, the differential equations govern the motion of the pendulum. When the contact with the obstacle appears, the pendulum exhibits a discontinuous behavior. An important element of the solving algorithm is aimed on the continuous tracking of the position of the pendulum in order to detect the collision with the unilateral constraints and to determine the state vector of the pendulum at the impact time instant. A single collision is described by the Euler's laws of motion in the integral form. The equations are supplemented by the Poisson's hypothesis and Coulomb's law of friction. The friction law is formulated for the instantaneous values of the reaction forces. The values of their impulses depend on the existence of a slip between the contacting bodies. Therefore, the Coulomb law cannot be generalized for the linear impulses of the forces in a simple way. We have applied the Routh method in order to solve the problem. The method has a simple geometrical interpretation in the impulse space. The angular velocities of both pendulum parts as well as the reaction forces at the joints of the system, which change in stepwise manner, have been presented in the paper. [ABSTRACT FROM AUTHOR]

A boundary value problem associated with filtering governed by differential equations with time dependent coefficients and exhibiting a weak nonlinearity is solved. Namely, the boundary value problem is split into two independent boundary problems, i.e., that of Goursat for the concentration of a sorbate and that absorbed by a sorbent. Then each of the formulated problems is solved separately applying the Riemann transformation. [ABSTRACT FROM AUTHOR]

Complex vibration of flexible elastic shells subjected to transversal and sign-changeable local load in the frame of nonlinear classical theory is studied. A transition from partial to ordinary differential equations is carried out using the higher-order Bubnov-Galerkin approach. Numerical analysis is performed applying theoretical background of nonlinear dynamics and qualitative theory of differential equations. Mainly the so-called Sharkovskiy periodicity is studied. [ABSTRACT FROM AUTHOR]

Abstract: By using Tichonov theorem for singularly perturbed differential equations, we study the relationship between dynamics of discontinuous differential equations and their continuous approximations along periodic solutions. [Copyright &y& Elsevier]

A self-excited friction pair is modeled by a mechanical system of two degrees-of-freedom with a normal force varying during the block displacement. Two different friction coefficients are used, and a normal force pressing the sliding body to a belt depending on an angle bar motion of the analyzed system is applied. In addition, the numerical analysis is supported by the investigation of a real laboratory object modeling the feedback reinforcement of friction forces acting on the block. Both numerical and experimental results are compared showing good agreement. [ABSTRACT FROM AUTHOR]

In this work a triple pendulum with damping, external forcing and impacts is investigated. We integrate the obtained system of governing equations between two successive impacts, and the discontinuity points are detected (by halving time step until the required precision is obtained). For each impact time, the state of the system is transformed using the extended restitution coefficient rule. The theory of Aizerman and Gantmakher is used to calculate the fundamental solution matrices in the analyzed system exhibiting discontinuities. Fundamental matrices are used during calculation of Lyapunov exponents, in stability analysis of periodic solution (Floquet multipliers) and in shooting method of finding periodic orbits. Then some examples for three coupled identical rods with horizontal barrier are shown. [ABSTRACT FROM AUTHOR]

The objective of this study is to establish new sufficient criteria for oscillation of solutions of even-order delay Emden-Fowler differential equations with neutral term r ı y ı + m ı y g ı n − 1 γ ′ + ∑ i = 1 j q i ı y γ μ i ı = 0 . We use Riccati transformation and the comparison with first-order differential inequalities to obtain theses criteria. Moreover, the presented oscillation conditions essentially simplify and extend known criteria in the literature. To show the importance of our results, we provide some examples. Symmetry plays an essential role in determining the correct methods for solutions to differential equations. [ABSTRACT FROM AUTHOR]

In this paper, new oscillatory properties for fourth-order delay differential equations with p-Laplacian-like operators are established, using the Riccati transformation and comparison method. Moreover, our results are an extension and complement to previous results in the literature. We provide some examples to examine the applicability of our results. [ABSTRACT FROM AUTHOR]

In this work, we address an interesting problem in studying the oscillatory behavior of solutions of fourth-order neutral delay differential equations with a non-canonical operator. We obtained new criteria that improve upon previous results in the literature, concerning more than one aspect. Some examples are presented to illustrate the importance of the new results. [ABSTRACT FROM AUTHOR]

In this paper, we apply the concept of fractional calculus to study three-dimensional Lotka-Volterra differential equations. We incorporate the Caputo-Fabrizio fractional derivative into this model and investigate the existence of a solution. We discuss the uniqueness of the solution and determine under what conditions the model offers a unique solution. We prove the stability of the nonlinear model and analyse the properties, considering the non-singular kernel of the Caputo-Fabrizio operator. We compare the stability conditions of this system with respect to the Caputo-Fabrizio operator and the Caputo fractional derivative. In addition, we derive a new numerical method based on the Adams-Bashforth scheme. We show that the type of differential operators and the value of orders significantly influence the stability of the Lotka-Volterra system and numerical results demonstrate that different fractional operator derivatives of the nonlinear population model lead to different dynamical behaviors. [ABSTRACT FROM AUTHOR]

By establishing new conditions for the non-existence of so-called Kneser solutions, we can generate sufficient conditions to ensure that all solutions of odd-order equations are oscillatory. Our results improve and expand the previous results in the literature. [ABSTRACT FROM AUTHOR]

Based on the comparison with first-order delay equations, we establish a new oscillation criterion for a class of even-order neutral differential equations. Our new criterion improves a number of existing ones. An illustrative example is provided. [ABSTRACT FROM AUTHOR]

Dynamical regular response of an oscillator with two serially connected springs with nonlinear characteristics of cubic type and governed by a set of differential-algebraic equations (DAEs) is studied. The classical approach of the multiple scales method (MSM) in time domain has been employed and appropriately modified to solve the governing DAEs of two systems, i.e. with one- and two degrees-of-freedom. The approximate analytical solutions have been verified by numerical simulations. [ABSTRACT FROM AUTHOR]

This study aims at offering a new criterion for the oscillation of the solutions of Emden–Fowler second-order neutral differential equations. The approach used presents a non-traditional Riccati substitution and takes into account all the delay functions, which had been neglected in previous analyses. The new results essentially improve and simplify some well-known results reported in the literature. Furthermore, the new criterion provides precise results in the ordinary case (e.g. Euler-type equations), whereas the existing results fail in this case. [ABSTRACT FROM AUTHOR]