Search Results (121)

The aim of this paper is to study a nonlinear system of impulsive fractional differential equations and Caputo fractional derivatives with respect to another function (CFF). The main characteristics of these fractional derivatives are two-fold: first, the lower limit of CFF equals the impulsive time of the considered interval; second, the applied function in CFF is changeable at each interval without impulses. An auxiliary system of two linear scalar impulsive fractional differential equations with CFF is considered, and strict stability in a couple is defined. The behavior of its solutions is illustrated with several examples. Also, we use appropriate Lyapunov functions to obtain sufficient conditions for the strict stability of the studied system. These sufficient conditions depend significantly on the type of impulsive function. Full article

The Euler difference approach has become a prevalent tool in the research of integral order differential equations. Nevertheless, a review of the literature reveals a dearth of studies examining fractional order models using the exponential Euler difference approach. The present study employs an exponential Euler difference approach to examine the properties of nonlocal discrete-time oscillators with Mittag–Leffler kernels and piecewise features, with the aim of providing insights into a continuous-time nonlocal nonlinear system. By employing impulsive equations of variations in constants with different forms in conjunction with the Gronwall inequality, a controller that is capable of instantaneously responding and stabilizing the nonlocal discrete-time oscillator is devised. This controller is realized through an associated algorithm. As a case study, the primary outcome is applied to a problem of impulsive stabilization in nonlocal discrete-time Chua’s oscillator. This article presents a stabilizing algorithm for piecewise nonlocal discrete-time oscillators developed using a novel impulsive approach. Full article

A continuous point of a trajectory for an ordinary differential equation can be viewed as a special impulsive point; i.e., the pulsed proportional change rate and the instantaneous increment for the prey and predator populations can be taken as 0. By considering the variation multiple pulse intervention effects (i.e., several indefinite continuous points are regarded as impulsive points), an impulsive predator–prey model for characterizing chemical and biological control processes at different fixed times is first proposed. Our modeling approach can describe all possible realistic situations, and all of the traditional models are some special cases of our model. Due to the complexity of our modeling approach, it is essential to examine the dynamical properties of the periodic solutions using new methods. For example, we investigate the permanence of the system by constructing two uniform lower impulsive comparison systems, indicating the mathematical (or biological) essence of the permanence of our system; furthermore, the existence and global attractiveness of the pest-present periodic solution is analyzed by constructing an impulsive comparison system for a norm $V\left(t\right)$, which has not been addressed to date. Based on the implicit function theorem, the bifurcation of the pest-present periodic solution of the system is investigated under certain conditions, which is more rigorous than the corresponding traditional proving method. In addition, by employing the variational method, the eigenvalues of the Jacobian matrix at the fixed point corresponding to the pest-free periodic solution are determined, resulting in a sufficient condition for its local stability, and the threshold condition for the global attractiveness of the pest-free periodic solution is provided in terms of an indicator $R_a$. Finally, the sensitivity of indicator $R_a$ and bifurcations with respect to several key parameters are determined through numerical simulations, and then the switch-like transitions among two coexisting attractors show that varying dosages of insecticide applications and the numbers of natural enemies released are crucial. Full article

This study develops a high-dimensional impulsive differential equation model to analyze Huanglongbing (HLB) transmission dynamics, incorporating seasonal fluctuations in vector psyllid populations and multi-pronged control measures: (1) periodic removal of infected/dead citrus trees to eliminate pathogen reservoirs and (2) non-uniform pesticide applications timed to disrupt psyllid life cycles. The model analytically derives the basic reproduction number ($R_0$) and proves the existence of a unique disease-free periodic solution. Theoretical analysis reveals a threshold-dependent stability: when $R_0<1$, the disease-free solution is globally asymptotically stable, ensuring pathogen extinction; when $R_0>1$, the system becomes uniformly persistent, indicating endemic HLB. Numerical simulations validate these findings and demonstrate that integrated interventions, combining psyllid population control and removal of infected plants, can significantly suppress HLB spread. The results provide a mathematical framework for optimizing intervention timing and intensity, offering actionable strategies for citrus growers. Full article

This paper presents a Finite Element Method (FEM) framework for solving the screened Poisson equation with a Dirac delta function as the forcing term. The singularity introduced by the delta function poses challenges for standard numerical methods, particularly in higher dimensions. To address this, we employ integrated Legendre basis functions, which yield sparse and structured system matrices characterized by a Banded-Block-Banded-Arrowhead ($B^3$-Arrowhead) form. In one dimension, the resulting linear system can be solved directly. In two and three dimensions, the equation can be efficiently solved using a generalized Alternating Direction Implicit (ADI) method combined with reverse Cholesky factorization. Numerical results in 1D, 2D, and 3D confirm that the method accurately captures the localized impulse response and reproduces the expected Green’s function behavior. The proposed approach offers a robust and scalable solution framework for partial differential equations with singular source terms and has potential applications in physics, engineering, and computational science. Full article

This paper studies the boundedness and convergence properties of the sequences generated by strict and weak contractions in metric spaces, as well as their fixed points, in the event that finite jumps can take place from the left to the right limits of the successive values of the generated sequences. An application is devoted to the stabilization and the asymptotic stabilization of impulsive linear time-varying dynamic systems of the n-th order. The impulses are formalized based on the theory of Dirac distributions. Several results are stated and proved, namely, (a) for the case when the time derivative of the differential system is impulsive at isolated time instants; (b) for the case when the matrix function of dynamics is almost everywhere differentiable with impulsive effects at isolated time instants; and (c) for the case of combinations of the two above effects, which can either jointly take place at the same time instants or at distinct time instants. In the first case, finite discontinuities of the first order in the solution are generated; that is, equivalently, finite jumps take place between the corresponding left and right limits of the solution at the impulsive time instants. The second case generates, equivalently, finite jumps in the first derivative of the solution with respect to time from their left to their right limits at the corresponding impulsive time instants. Finally, the third case exhibits both of the above effects in a combined way. Full article

Differential equations are frequently used to mathematically describe many problems in real life, but they are always subject to intrinsic phenomena that are neglected and could influence how the model behaves. In some cases like ecosystems, electrical circuits, or even economic models, the model may suddenly change due to outside influences. Occasionally, such changes start off impulsively and continue to exist for specific amounts of time. Non-instantaneous impulses are used in the creation of the models for this kind of scenario. In this paper, a new class of non-instantaneous impulsive $\psi$-Caputo fractional stochastic differential equations under integral boundary conditions driven by the Rosenblatt process was examined. Semigroup theory, stochastic theory, the Banach fixed-point theorem, and fractional calculus were applied to investigating the existence of piecewise continuous mild solutions for the systems under consideration. The impulsive Gronwall’s inequality was employed to establish the unique stability conditions for the system under consideration. Furthermore, we examined the controllability results of the proposed system. Finally, some examples were provided to demonstrate the validity of the presented work. Full article

This paper investigates a general class of variable-kernel discrete delay differential equations (DDDEs) with integral boundary conditions and impulsive effects, analyzed using Caputo piecewise derivatives. We establish results for the existence and uniqueness of solutions, as well as their stability. The existence of at least one solution is proven using Schaefer’s fixed-point theorem, while uniqueness is established via Banach’s fixed-point theorem. Stability is examined through the lens of Ulam–Hyers (U-H) stability. Finally, we illustrate the application of our theoretical findings with a numerical example. Full article

The built physical and social environments are critical drivers of child neural and cognitive development. This study aimed to identify the factor structure and correlates of 29 environmental, education, and socioeconomic indicators of neighborhood resources as measured by the Child Opportunity Index 2.0 (COI 2.0) in a sample of youths aged 9–10 enrolled in the Adolescent Brain Cognitive Development (ABCD) Study. This study used the baseline data of the ABCD Study (n = 9767, ages 9–10). We used structural equation modeling to investigate the factor structure of neighborhood variables (e.g., indicators of neighborhood quality including access to early child education, health insurance, walkability). We externally validated these factors with measures of psychopathology, impulsivity, and behavioral activation and inhibition. Exploratory factor analyses identified four factors: Neighborhood Enrichment, Socioeconomic Attainment, Child Education, and Poverty Level. Socioeconomic Attainment and Child Education were associated with overall reduced impulsivity and the behavioral activation system, whereas increased Poverty Level was associated with increased externalizing symptoms, an increased behavioral activation system, and increased aspects of impulsivity. Distinct dimensions of neighborhood opportunity were differentially associated with aspects of psychopathology, impulsivity, and behavioral approach, suggesting that neighborhood opportunity may have a unique impact on neurodevelopment and cognition. This study can help to inform future public health efforts and policy about improving built and natural environmental structures that may aid in supporting emotional development and downstream behaviors. Full article

This paper explores the mild solutions of partial impulsive fractional integro-differential systems of order $1<\alpha <2$ in a Banach space. We derive the solution of the system under the assumption that the homogeneous part of the system admits an $\alpha$-resolvent operator. Krasnoselskii’s fixed point theorem is used for the existence of solution, while uniqueness is ensured using Banach’s fixed point theorem. The stability of the system is analyzed through the framework of Hyers–Ulam stability using Lipschitz conditions. Finally, examples are presented to illustrate the applicability of the theoretical results. Full article

This paper formalizes the analytic expressions and some properties of the evolution operator that generates the state-trajectory of dynamical systems combining delay-free dynamics with a set of discrete, or point, constant (and not necessarily commensurate) delays, where the parameterizations of both the delay-free and the delayed parts can undergo impulsive changes. Also, particular evolution operators are defined explicitly for the non-impulsive and impulsive time-varying delay-free case, and also for the case of impulsive delayed time-varying systems. In the impulsive cases, in general, the evolution operators are non-unique. The delays are assumed to be a finite number of constant delays that are not necessarily commensurate, that is, all of them being integer multiples of a minimum delay. On the other hand, the impulsive actions through time are assumed to be state-dependent and to take place at certain isolated time instants on the matrix functions that define the delay-free and the delayed dynamics. Some variants are also proposed for the cases when the impulsive actions are state-independent or state- and dynamics-independent. The intervals in-between consecutive impulses can be, in general, time-varying while subject to a minimum threshold. The boundedness of the state-trajectory solutions, which imply the system’s global stability, is investigated in the most general case for any given piecewise-continuous bounded function of initial conditions defined on the initial maximum delay interval. Such a solution boundedness property can be achieved, even if the delay-free dynamics is unstable, by an appropriate distribution of the impulsive actions. An illustrative first-order example is developed in detail to illustrate the impulsive stabilization results. Full article

This article studies the performance of the copper wire communication channel in the existence of the Middleton impulsive noise model. Differential chaos shift keying (DCSK) scheme is implemented with a noise reduction (NR) technique to mitigate the impulsive noise effect and improve system performance. The proposed communication system is simulated using Monte Carlo simulation using MATLAB 2023 and the result is compared with the derived theoretical performance. The use of NR technique in accordance with the proposed model has improved the performance and promoted its use with G.fast Communication. Full article

This paper investigates the asymptotic eventual stability (AE-S) for nonlinear impulsive Caputo fractional differential equations (ICFDEs) with fixed impulse moments, employing auxiliary Lyapunov functions (ALF) which are specifically constructed as analogues of vector Lyapunov functions (VLF). A novel derivative tailored for VLF is introduced, offering a more robust framework than existing approaches based on scalar Lyapunov functions (SLF). Adequate conditions for AE-S involving ICFDEs are provided. We also used the predictor corrector method to implement a numerical solution for a given impulsive Caputo fractional differential equation. These findings extend and improve upon existing results, providing significant advancements in the stability analysis of systems with memory effects and impulsive dynamics. The study holds practical relevance for modeling and analyzing real-world systems, including control processes, biological systems, and economic dynamics where fractional-order behavior and impulses play a crucial role. Full article

Purpose: To evaluate the ability of acoustic radiation force impulse (ARFI) elastography in differentiating benign from malignant etiologies of splenomegaly based on differences in splenic stiffness. Materials and Methods: Between September 2020 and November 2022, we evaluated 40 patients with splenomegaly—defined by a splenic long axis greater than 13 cm and/or a short axis greater than 6 cm, without visible focal or infiltrative mass lesions—using abdominal ultrasound at our university hospital. Each patient also underwent a standardized ARFI elastographic assessment of the enlarged spleen, with data collected prospectively. We then retrospectively analyzed the cases with confirmed etiologies of splenomegaly from their final medical reports. Mean ARFI velocities (MAV) were compared across patients with splenomegaly due to malignant infiltration (MIS) from hematological malignancy, congestive splenomegaly (CS) due to portal or splenic vein congestion/occlusion, and immune-related splenomegaly (IRS) associated with systemic infectious or autoimmune diseases. Results: Among the 40 patients with splenomegaly, 21 (52.5%) were diagnosed with malignant infiltrative splenomegaly (MIS), 11 (27.5%) with congestive splenomegaly (CS), and 8 (20%) with immune-related splenomegaly (IRS). The mean ARFI velocities (MAV) for the MIS, CS, and IRS groups were 3.25 ± 0.68 m/s, 3.52 ± 0.47 m/s, and 2.84 ± 0.92 m/s, respectively. No significant differences were observed in splenic stiffness (MAV) among these groups. Conclusions: Differentiating between benign and malignant etiologies of splenomegaly based on stiffness differences observed in ARFI elastography is not feasible. Larger prospective studies are necessary to validate these findings. Full article

In this paper, we study the existence and uniqueness of second-order impulsive delay differential systems. Firstly, we define cosine-type and sine-type delay matrix functions, which are used to derive the solutions of the impulsive delay differential systems. Secondly, based on the Schauder and Banach fixed-point theorems, we establish sufficient conditions that guarantee the existence and uniqueness of solutions to nonlinear impulsive delay differential systems. Finally, several examples are given to illustrate our theoretical results. Full article