This work describes two efficient and useful methods for solving fractional pantograph delay equations (FPDEs) with initial and boundary conditions. These two methods depend mainly on orthogonal polynomials, which are the method of the operational matrix of fractional derivative that depends on Bernstein polynomials and the operational matrix of the fractional derivative with Shifted Legendre polynomials. The basic procedure of this method is to convert the pantograph delay equation to a system of linear equations and by using, the operational matrices we get rid of the integration and differentiation operations, which makes solving the problem easier. The concept of Caputo has been used to describe fractional derivatives. Finally, some

Because the Coronavirus epidemic spread in Iraq, the COVID-19 epidemic of people quarantined due to infection is our application in this work. The numerical simulation methods used in this research are more suitable than other analytical and numerical methods because they solve random systems. Since the Covid-19 epidemic system has random variables coefficients, these methods are used. Suitable numerical simulation methods have been applied to solve the COVID-19 epidemic model in Iraq. The analytical results of the Variation iteration method (VIM) are executed to compare the results. One numerical method which is the Finite difference method (FD) has been used to solve the Coronavirus model and for comparison purposes. The numerical simulat

In this article, the numerical and approximate solutions for the nonlinear differential equation systems, represented by the epidemic SIR model, are determined. The effective iterative methods, namely the Daftardar-Jafari method (DJM), Temimi-Ansari method (TAM), and the Banach contraction method (BCM), are used to obtain the approximate solutions. The results showed many advantages over other iterative methods, such as Adomian decomposition method (ADM) and the variation iteration method (VIM) which were applied to the non-linear terms of the Adomian polynomial and the Lagrange multiplier, respectively. Furthermore, numerical solutions were obtained by using the fourth-orde Runge-Kutta (RK4), where the maximum remaining errors showed th

     The method of operational matrices is based on the Bernoulli and Shifted Legendre polynomials which is used to solve the Falkner-Skan equation. The nonlinear differential equation converting to a system of nonlinear equations is solved using Mathematica^®12, and the approximate solutions are obtained. The efficiency of these methods was studied by calculating the maximum error remainder ( ), and it was found that their efficiency increases as  increases. Moreover, the obtained approximate solutions are compared with the numerical solution obtained by the fourth-order Runge-Kutta method (RK4), which gives  a good agreement.

The current research seeks to Analyze third intermediate chemistry book following sustainable development standards for academic year (2016-2017). To do this, a list of sustainable development standards that should be included in chemistry book was designed based on the previous studies. The first version of the list consisted of (50) sub-case divided into three standards (social, economic, and environmental) which was exposed to group of experts in teaching chemistry and teaching methods. The list has modified to (43) sub-case. The researcher followed the implicit and explicit meaning in his analysis: one for recording and repetition and the other for frequency. The result showed that third intermediate chemistry book has achieved (20)

 AbstractThis study aimed to demonstrate the impact of human development on economic growth in Jordan during the period (1980-2014), Where some  diagnoses tests were applied, the results of these tests concluded that the standard models used in the study were free of statistical problems, and hence ordinary least squares (OLS) standard has been used as a tool for analysis to get efficient and unbiased estimates to parameters according to the theory of Gauss Markov.

The results showed that there is a strong and positive impact of human development represented by the Human Development Index (HDI) on economic growth in Jordan represented by the average of real productivity of the Jordanian worker (

2 - 5,5 cm mosquito fishes (Gambusia affinis) were collected from brooks of Baghdad university . This study aimed to identify the histological structure and embryonic development of the gas bladder . The results revealed that the gas bladder of the adult fish consists of single chamber, paraphysoclistous , and its wall consist of three layers. The primordium of the gas bladder appears in 2,5 mm fish embryo as an evagination from the right side of endoderm of foreguts wall. This primordium consists of two thecae, inner, is simple columnar epithelium and outer originates from undifferentiated mesoderm. Simultaneously with the appearance of gas gland in the ventral side of the sac in 4 mm embryo, morphological and histological chan