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.
We generalize an optimal fourth-order scalar method to solve systems of nonlinear equations. The proposed generalized scheme is optimal for solving systems of nonlinear equations. The fourth-order convergence of the proposed method is proved. The computational efficiency of the proposed iteration scheme is analysed and compared with several methods for nonlinear systems. To evaluate the numerical performance of the presented scheme, extensive numerical experiments are conducted and the results are compared with some existing fourth-order methods from the literature. The stability of the scheme is rigorously analysed through convergence planes, confirming its advantage over other fourth-order iteration techniques for systems of nonlinear equations.
