We study a damped Petrovsky equation with a Hartree type nonlocal source in a bounded domain. We establish a suitable energy framework, prove the local existence of weak solutions, and introduce the associated Nehari manifold and potential well structure. For weak solutions whose Nehari functional is continuous in time, we prove global continuation and uniform bounds for stable initial data below the potential well depth. The time derivative is square integrable on the positive half-line. For solutions satisfying the energy inequality on every time interval, we also obtain monotonicity and a finite limiting energy. We also prove a finite time blow up result under negative initial energy and suitable assumptions on the initial data and the parameters.
The notions of the classical two-scale convergence and periodic unfolding method, which both rely on an additional scale for the representation of periodic micro-oscillations, are in this paper traced back to the wider concept of general two-scale convergence. Moreover, a comparison is made between very weak two-scale convergence and periodic unfolding with respect to sequences bounded in Sobolev spaces, amplified by the scale of micro-oscillations. It turns out that, for amplified sequences, these approaches are not equivalent in an immediate sense.
The motivation for implementing this model arises from the need to analyze dependent count data that can take both positive and negative values, exhibit fluctuations around a zero mean, and demonstrate unimodal-bimodal aspects. To capture these features commonly observed in real-world applications, we propose a first-order integer-valued autoregressive process, INAR(1), with discrete double Lindley innovations. The model is constructed using a signed thinning mechanism, referred to as the relative binomial thinning operator, which extends the classical binomial thinning operator. The proposed model is appropriate for ℤ-valued time series with both positive and negative dependence. The main statistical properties of the process are derived. Parameter estimation is addressed using modified conditional least squares, Yule-Walker, and conditional maximum likelihood methods, whose performance is evaluated through simulation studies. The practical relevance of the model is demonstrated using data from the Saudi stock market. Model adequacy is assessed via standardized residuals and the cumulative periodogram, and predictive performance is evaluated using a hybrid framework combining Support Vector Regression (SVR) with the proposed model, in comparison with bootstrap and classical approaches.
We develop high-order numerical methods for solving second-order nonlinear Fredholm Integro-Differential Equations (FrIDEs). First, we establish the existence and uniqueness of solutions under easily verified conditions and propose a continuous iterative method. Subsequently, we construct a discrete iterative scheme by discretizing this continuous method using the trapezoidal quadrature formula augmented with correction terms. We rigorously prove that the proposed numerical method achieves eighth-order convergence. Extensive experiments on both linear and nonlinear equations demonstrate that our method attains eighth-order accuracy, surpassing existing methods whose accuracy does not exceed fifth order.
This article suggests a secure communication scheme with a new synchronization technique combining Adaptive Generalized Hybrid Projective Synchronization (AGHPS) and fixed-time stability for chaotic systems. The scheme is based on the structural flexibility of AGHPS but offers guaranteed and fast convergence under fixed-time control regardless of initial conditions. A comparative analysis with a recently published method is performed. The comparison shows a significant improvement in the synchronization time, indicating faster convergence under the same circumstances. In addition, higher performance is achieved by the new method in secure communication applications, especially encrypted image transmission and recovery. Simulations show unambiguous improvement in image recovery quality even under channel interference and parameter uncertainties, i.e., the scheme’s efficiency and reliability. This contribution therefore provides a computationally effective and secure synchronization framework with significant advantages over existing approaches and therefore a fine choice for real-time chaos-based communication systems.
