In this paper, the process for finding an approximate solution of nonlinear three-dimensional (3D) Volterra type integral operator equation (N3D-VIOE) in R³ is introduced. The modelling of the majorant function (MF) with the modified Newton method (MNM) is employed to convert N3D-VIOE to the linear 3D Volterra type integral operator equation (L3D-VIOE). The method of trapezoidal rule (TR) and collocation points are utilized to determine the approximate solution of L3D-VIOE by dealing with the linear form of the algebraic system. The existence of the approximate solution and its uniqueness are proved, and illustrative examples are provided to show the accuracy and efficiency of the model.

Mathematical Subject Classificat

This article suggests and explores a three-species food chain model that includes fear effects, refuges depending on predators, and cannibalism at the second level. The Holling type II functional response determines food consumption between stages of the food chain. This study examined the long-term behavior and impacts of the suggested model's essential elements. The model's solution properties were studied. The existence and stability of every probable equilibrium point were examined. The persistence needs of the system have been determined. It was discovered what conditions could lead to local bifurcation at equilibrium points. Appropriate Lyapunov functions are utilized to investigate the overall dynamics of the system. To support the a

The aim of this paper is to construct cyclic subgroups of the projective general linear group over  from the companion matrix, and then form caps of various degrees in . Geometric properties of these caps as secant distributions and index distributions are given and determined if they are complete. Also, partitioned of  into disjoint lines is discussed.

In this paper, the dynamical behavior of a three-dimensional fractional-order prey-predator model is investigated with Holling type III functional response and constant rate harvesting. It is assumed that the middle predator species consumes only the prey species, and the top predator species consumes only the middle predator species. We also prove the boundedness, the non-negativity, the uniqueness, and the existence of the solutions of the proposed model. Then, all possible equilibria are determined, and the dynamical behaviors of the proposed model around the equilibrium points are investigated. Finally, numerical simulations results are presented to confirm the theoretical results and to give a better understanding of the dynami

the first part of the research involves investigate the aspect of the radiation superposed on the one bright soliton pulse propagated on ideal single mode

Proxy-based sliding mode control PSMC is an improved version of PID control that combines the features of PID and sliding mode control SMC with continuously dynamic behaviour. However, the stability of the control architecture maybe not well addressed. Consequently, this work is focused on modification of the original version of the proxy-based sliding mode control PSMC by adding an adaptive approximation compensator AAC term for vibration control of an Euler-Bernoulli beam. The role of the AAC term is to compensate for unmodelled dynamics and make the stability proof more easily. The stability of the proposed control algorithm is systematically proved using Lyapunov theory. Multi-modal equation of motion is derived using the Galerkin metho

In the Iraqi educational context, the academic presentation of learning and teaching Spanish faces several challenges due to the lack of updated teach-ing resources, absence of new teaching and learning methodologies, and in-adequate infrastructure to accommodate students in their study locations, among other issues. Given this comprehensive landscape, this article pro-poses a documentary analysis investigating the potential advantages of im-plementing self-regulation as a strategy for teaching and learning Spanish in the Iraqi educational setting. Accessing clarification of some fundamental concepts, self-regulation can be understood as the control each individual can exert over their thoughts, actions, emotions, and strategies,

RÉSUMÉ                                                    

            LeMauger Bleu est une méthode emblématique de l’enseignement traditionnel, qui présente une littérature française riche et patrimoniale, et dont l’objectif est de faciliter l’accès au texte par le vocabulaire et la grammaire et d’étendre la culture générale et littéraire de l’apprenant. La démarche (traditionnelle) est toujours la même : le texte, le vocabulaire et la grammaire, la traduct

The essential objective of this study presents a analyzes of the combination of reality and fantasy in three dramas of Spanish author Alejandro Casona. "Trees die standing" (premiered in Buenos Aires in 1949), “Prohibited suicide in spring” (premiered in Mexico in 1937) and "The boat without fisherman" (premiered in Buenos Aires in 1945).

  The first part is an introduction about the biography of Spanish author Alejandro Casona that will allow us to understand some peculiarities of his work in relation to the social historical context of his time.

 In the second part of this study we have analyzed elements of reality and fantasy in the th