Discontinuous Nonlinearities Research Articles

Second-order strongly nonlinear impulsive coupled systems

This paper presents sufficient conditions for the solvability of a second-order coupled system, composed of two differential equations involving different laplacians applied to fully discontinuous nonlinearities, two-point boundary conditions, and generalized impulsive effects. Applying the lower and upper solutions technique and Schauder’s fixed point theorem, it is obtained an existence and localization theorem, based on local monotone properties on the nonlinearities and the impulsive functions.

Dynamics of N-elastically longitudinal coupled rotating pendulums with smooth and discontinuous nonlinearities: Generation of chaotic bursting with many orbits as a solution

The dynamics of a mechanical network, consisting of a certain number (N) of cells, in which each unit cell consists of the irrational coupling of a simple pendulum and inclined mass-spring oscillator, is investigated. The single cell was recently introduced by Ning Han and Qingjie Cao, and it has strong irrational nonlinearity, exhibiting smooth and discontinuous characteristics due to the geometry configuration. The obtained network has a large degree of freedom and consequently has richer dynamics as compared to a single one. The set of discrete equations of this network is found, while the first equation is driven by an external sinusoidal force; the variables here are the angles between the vertical direction and the directions of mass in each unit cell. By using the non-driven version of this set of equations, the equilibrium points are found as well as their stability using the Jacobian matrix. Then, by using the multiple time scale method, the solutions of the set of the network equations are found for weak amplitude oscillations of the system, while the large amplitude solutions are found numerically, showing the ability of the network to transform the shape of some solutions at the input to other forms as the signals evolve in the network. One notices the generation of simple chaotic signals as well as the train of impulse signals. What is important here is the generation of the chaotic bursting with many orbits by the system, which are the simultaneous oscillations around several fixed points. For the case of numerous cells, the system can be used as a waveguide; this is why the set of the equation of the network is reduced to the extended complex Ginzburg Landau (ECGL) equation, with complex nonlinear dispersion and nonlocality terms, using the Tanuiti's reduction perturbation methods. The dissipative dark compacton and peakon as solutions for the ECGL equation are then found. The input-output matching of the network is next investigated via the transfer function of the system near the equilibrium points. The inverse Fourier transform of the transfer function multiplied by the Fourier transform of the input sinusoidal force gives the solution of the network equation for low amplitude oscillations. Finally, the supratransmission condition is studied, leading to the fact that the input sinusoidal signal with frequency nearly inside the forbidden gap zone gradually transforms into the train of chaotic bursting.

Existence and concentration of solutions for a fractional Schrödinger–Poisson system with discontinuous nonlinearity

Existence and concentration of solutions for a fractional Schrödinger–Poisson system with discontinuous nonlinearity

A New Second-order Maximum-principle-preserving Finite-volume Method for Flow Problems Involving Discontinuous Coefficients

A new development of Finite Volumes (FV, for short) and its theoretical analysis are the purpose of this work. Recall that FV are known as powerful tools to address equations of conservation laws (mass, energy, momentum,...). Over the last two decades investigators have succeeded in putting in place a mathematical framework for the theoretical analysis of FV. A perfect illustration of this progress is the design and mathematical analysis of Discrete Duality Finite Volumes (DDFV, for short). We propose now a new class of DDFV for 2nd order elliptic equations involving discontinuous diffusion coefficients or nonlinearities. A one-dimensional linear elliptic equation is addressed here for illustrating the ideas behind our numerical strategy. The algebraic structure of the discrete system we have got is different from that of standard DDFV. The main novelty is that the so-called diamond mesh elements are confined in homogeneous zones for flow problems governed by piecewise constant coefficients. This is got from our judicious definition of the primal mesh. The gain is that there is no need to compute homogenized coefficients to be allocated to the so-called diamond cells as required to conventional DDFV. Notice that poor homogenized permeability allocated to diamond elements leads to poor approximations of fluxes across grid-block interfaces. Moreover for 1-D flow problems in a porous medium involving permeability discontinuities (piecewise constant permeability for instance) the proposed FV scheme leads to a symmetric positive-definite discrete system that meets the discrete maximum principle; we have shown its second order convergence under relevant assumptions.

Semi-Regular Solutions of Integral Equations with Discontinuous Nonlinearities

Semi-Regular Solutions of Integral Equations with Discontinuous Nonlinearities

Existence of Weak Solutions for a Class of $(p,q)$-biharmonic Equations with Critical Exponent and Discontinuous Nonlinearity

Existence of Weak Solutions for a Class of $(p,q)$-biharmonic Equations with Critical Exponent and Discontinuous Nonlinearity

Impulsive coupled systems with regular and singular ϕ-Laplacians and generalized jump conditions

This work contains sufficient conditions for the solvability of a third-order coupled system with two differential equations involving different Laplacians, fully discontinuous nonlinearities, two-point boundary conditions, and two sets of impulsive effects. The first existing result is obtained from Schauder’s fixed point theorem, and the second one provides also the localization of a solution via the lower and upper solutions technique.We point out that it is the first time that impulsive coupled systems with strongly nonlinear fully differential equations and generalized impulse effects are considered simultaneously. Moreover, the singular case is applied to a special relativity model in classical electrodynamics.

Existence of Solutions to the Non-Self-Adjoint Sturm–Liouville Problem with Discontinuous Nonlinearity

Existence of Solutions to the Non-Self-Adjoint Sturm–Liouville Problem with Discontinuous Nonlinearity

Physics enhanced sparse identification of dynamical systems with discontinuous nonlinearities

Physics enhanced sparse identification of dynamical systems with discontinuous nonlinearities

Biharmonic equation with discontinuous nonlinearities

We study the biharmonic equation with discontinuous nonlinearity and homogeneous Dirichlet type boundary conditions $$\displaylines{ \Delta^2u=H(u-a)q(u) \quad \hbox{in }\Omega,\cr u=0 \quad \hbox{on }\partial\Omega,\cr \frac{\partial u}{\partial n}=0 \quad \hbox{on }\partial\Omega, }$$ where \(\Delta\) is the Laplace operator, \(a> 0\), \(H\) denotes the Heaviside function, \(q\) is a continuous function, and \(\Omega\) is a domain in \(R^N \) with \(N\geq 3\). Adapting the method introduced by Ambrosetti and Badiale (The Dual Variational Principle), which is a modification of Clarke and Ekeland's Dual Action Principle, we prove the existence of nontrivial solutions. This method provides a differentiable functional whose critical points yield solutions despite the discontinuity of \(H(s-a)q(s)\) at \(s=a\). Considering \(\Omega\) of class \(\mathcal{C}^{4,\gamma}\) for some \(\gamma\in(0,1)\), and the function \(q\) constrained under certain conditions, we show the existence of two non-trivial solutions. Furthermore, we prove that the free boundary set \(\Omega_a=\{x\in\Omega:u(x)=a\}\) for the solution obtained through the minimizer has measure zero. For more information see https://ejde.math.txstate.edu/Volumes/2024/15/abstr.html

Finite-Time Stability of Fractional-Order Discontinuous Nonlinear Systems With State-Dependent Delayed Impulses

Finite-Time Stability of Fractional-Order Discontinuous Nonlinear Systems With State-Dependent Delayed Impulses

The index theory for multivalued dynamical systems with applications to reaction-diffusion equations with discontinuous nonlinearity

The index theory for multivalued dynamical systems with applications to reaction-diffusion equations with discontinuous nonlinearity

Полуправильные решения интегральных уравнений с разрывными нелинейностями

В пространствах Лебега изучаются интегральные уравнения с разрывными нелинейностями. Вариационным методом, базирующемся на понятии квазипотенциального оператора, установлена теорема существования полуправильных решений. Для уравнений с параметром получена теорема существования нетривиальных полуправильных решений при достаточно больших значениях параметра. Приводится пример прикладной задачи, для которой выполняются условия доказанных теорем. Библиография: 22 названия.

An Efficient Method for Calculating Hysteretic Dry Friction Response of Dynamic Systems Subjected to Combined Harmonic and Random Excitation.

Many examples of dynamic analysis require modelling of dry friction. Often this is represented by a so-called Jenkins element or multiple Jenkins elements in parallel, which is sometimes termed a parallel-series Iwan element. This study considers the case where a system that includes these representations of dry friction are loaded dynamically with a combination of deterministic harmonic and random excitation. This paper presents a new efficient method for predicting the response of systems subject to combined deterministic and random excitation. The method is based on equivalent linearisation and involves averaging across both an ensemble of random responses as well as over a harmonic excitation period. The key novelty in the approach is the use of an auxiliary harmonic term to facilitate an analytical representation of the nonlinear force. This overcomes the challenge of analysing a discontinuous hysteretic nonlinearity in the presence of random excitation. Analytical solutions are presented and it is shown that the proposed approach can predict the approximate response at a significantly reduced computational cost.

Analysis of a multiphase free boundary problem

In this paper, we investigate a free boundary problem relevant in several applications, such as tumor growth models. Our problem is expressed as an elliptic equation involving discontinuous nonlinearities in a specified domain with a moving boundary. We establish the existence and uniqueness of solutions and provide a qualitative analysis of the free boundaries generated by the nonlinear term (inner boundaries). Furthermore, we analyze the dynamics of the outer region boundary. The final result demonstrates that under certain conditions, our problem is solvable in the neighborhood of a radial solution.

A weak solution to a Kirkoff type problem with discontinuous nonlinearities

This paper is devoted to the study of a class of Kirchhoff-type problems with discontinuous nonlinearities with Neumann boundary data. Here,
 by employing the topological degree methods for the abstract Hammerstein equation, we establish the existence of at least one solution.

Approximation to the Sturm–Liouville Problem with a Discontinuous Nonlinearity

We consider a continuous approximation to the Sturm–Liouville problem with a nonlinearity discontinuous in the phase variable. The approximating problem is obtained from the original one by small perturbations of the spectral parameter and by approximating the nonlinearity by Carathéodory functions. The variational method is used to prove the theorem on the proximity of solutions of the approximating and original problems. The resulting theorem is applied to the one-dimensional Gol’dshtik and Lavrent’ev models of separated flows

Identification of Hammerstein-Wiener model with discontinuous input nonlinearity

Identification of Hammerstein-Wiener model with discontinuous input nonlinearity

A time and ensemble equivalent linearization method for nonlinear systems under combined harmonic and random excitation

An Equivalent Linearization technique, termed an Equivalent Linearization Time and Ensemble Expectation (EL-TEE) approach, is used to develop an alternative method for estimating the response of a nonlinear oscillator to a combination of deterministic harmonic and random white noise excitation. The approach is based on applying equivalent linearization and averaging over the time period of one harmonic excitation cycle. This gives a set of coupled nonlinear equations that can be solved for the response averaged over time and across the ensemble. The primary advantages of the proposed method are its computational speed, ability to return physically meaningful linearization matrices and that it can be applied to a wide variety of nonlinearities. The method is applied to three example test systems: the well-known single degree of freedom Duffing oscillator; a single degree of freedom system with a displacement constraint imposing a discontinuous nonlinearity; and a multi degree of freedom oscillator with a localized polynomial nonlinearity that has also been examined experimentally. It is shown that the response predicted matches well with Monte Carlo results from direct time integration at a fraction of the computational cost, and the method is capable of reproducing key results observed experimentally.

Approximation to the Sturm–Liouville Problem with a Discontinuous Nonlinearity

Approximation to the Sturm–Liouville Problem with a Discontinuous Nonlinearity