The Wang-Ball polynomials operational matrices of the derivatives are used in this study to solve singular perturbed second-order differential equations (SPSODEs) with boundary conditions. Using the matrix of Wang-Ball polynomials, the main singular perturbation problem is converted into linear algebraic equation systems. The coefficients of the required approximate solution are obtained from the solution of this system. The residual correction approach was also used to improve an error, and the results were compared to other reported numerical methods. Several examples are used to illustrate both the reliability and usefulness of the Wang-Ball operational matrices. The Wang Ball approach has the ability to improve the outcomes by minimi

In this paper, Touchard polynomials (TPs) are presented for solving Linear Volterra integral equations of the second kind (LVIEs-2k) and the first kind (LVIEs-1k) besides, the singular kernel type of this equation. Illustrative examples show the efficiency of the presented method, and the approximate numerical (AN) solutions are compared with one another method in some examples. All calculations and graphs are performed by program MATLAB2018b.

A new method based on the Touchard polynomials (TPs) was presented for the numerical solution of the linear Fredholm integro-differential equation (FIDE) of the first order and second kind with condition. The derivative and integration of the (TPs) were simply obtained. The convergence analysis of the presented method was given and the applicability was proved by some numerical examples. The results obtained in this method are compared with other known results.

In this research, Haar wavelets method has been utilized to approximate a numerical solution for Linear state space systems. The solution technique is used Haar wavelet functions and Haar wavelet operational matrix with the operation to transform the state space system into a system of linear algebraic equations which can be resolved by MATLAB over an interval from 0 to . The exactness of the state variables can be enhanced by increasing the Haar wavelet resolution. The method has been applied for different examples and the simulation results have been illustrated in graphics and compared with the exact solution.

The study is devoted to both static and earthquake response analysis of retaining structures acted upon by lateral earth pressure. Two main approaches were implemented in the analysis, namely, the Mononobe-Okabe analytical method and the numerical Finite element procedure as provided in the ready software ABAQUS with explicit dynamic method. A basic case study considered in the present work is the bridge approach retaining walls as a part of AL-Jadiriya bridge intersection to obtain the effects of the backfill and the ground water on the retaining wall response including displacement of the retaining structure in addition to the behavior of the fill material. Parametric studies were carried out to evaluate the effects of several factors

This paper presents a novel idea as it investigates the rescue effect of the prey with fluctuation effect for the first time to propose a modified predator-prey model that forms a non-autonomous model. However, the approximation method is utilized to convert the non-autonomous model to an autonomous one by simplifying the mathematical analysis and following the dynamical behaviors. Some theoretical properties of the proposed autonomous model like the boundedness, stability, and Kolmogorov conditions are studied. This paper's analytical results demonstrate that the dynamic behaviors are globally stable and that the rescue effect improves the likelihood of coexistence compared to when there is no rescue impact. Furthermore, numerical simul

The inverse problem is important method in the design of electrostatic lenses which is used in this work, with new technique by suggesting an axial electrostatic potential distribution using polynomial functions of the third order. The paraxial-ray equation is solved to obtain the trajectory of particles that satisfy the suggested potential function.In this work design of immersion electrostatic lens operated under zero magnification condition. The electrode shape of sthe electrostatic lens was the dermined from the solution of laplace equation and plotted in two deimensions . The results showed low values of spherical and chromatic aberrations , which are considered as good criteria for good desigh.

In order to study the dynamic response of historical masonry structures, a scaled down brick masonry model constructed in civil engineering department at Baghdad University to simulate a part of a real case study, which is Alkifil historic minaret. Most of the previous researches about masonry structures try to understand the behavior of the masonry under seismic loading by experimental and numerical methods. In this paper, the masonry units (bricks) simulated in scale (S= 1/6) with the exact shape of the prototype bricks. Cementitious tile adhesive was selected to be the mortar for the modeling. The height of the model designed to be 1.5 m with a 0.5 m diameter. Detailed construction steps were presented in this paper. Experts buil