The aim of this article is to solve the Volterra-Fredholm integro-differential equations of fractional order numerically by using the shifted Jacobi polynomial collocation method. The Jacobi polynomial and collocation method properties are presented. This technique is used to convert the problem into the solution of linear algebraic equations. The fractional derivatives are considered in the Caputo sense. Numerical examples are given to show the accuracy and reliability of the proposed technique.

The scholastic view of public religion differed, and this difference was on two extremes. All economic schools agreed that public debt is a monetary liquidity that was unjustly deducted from the income and output cycle as a result of the imbalance in the economic balance and the departure from the conditions of balance between aggregate demand and aggregate supply. Debt is a waste of financial resources allocated to productive accumulation. Except for the Keynesian school, which considers public debt to be an addition to aggregate demand after the decline in the role of the private sector in investment as a result of pessimistic expectations that warn of signs of economic contraction. Public debt is linked to the ex

A total of 20 raw milk samples were used as the fouling agent for evaluating the bacteriological effectiveness of cleaning and sanitizing of domestic milking equipment by using ozonated water at 0.5 ppm comparing to the warm water at 55! for 5 minutes respectively. The mean values of total aerobic bacteria, Coliform and E.coli that present on the plastic and stainless-steel containers after using the raw milk as fouling agent were 3.4×10-6 , 6.7x10-5 and 5.8×10-3 cfu/cm2 respectively , after cleaning the stainless steel containers by the ozonated water the mean values of total aerobic bacterial counts, Coliforms and E.coli bacteria were reduced to 1.2×10-6, 4.7×10-5 and 3.3×10-3 CFU/cm2 respectively. while after cleaning by the warm wa

In this paper, a new technique is offered for solving three types of linear integral equations of the 2^nd kind including Volterra-Fredholm integral equations (LVFIE) (as a general case), Volterra integral equations (LVIE) and Fredholm integral equations (LFIE) (as special cases). The new technique depends on approximating the solution to a polynomial of degree  and therefore reducing the problem to a linear programming problem(LPP), which will be solved to find the approximate solution of LVFIE. Moreover, quadrature methods including trapezoidal rule (TR), Simpson 1/3 rule (SR), Boole rule (BR), and Romberg integration formula (RI) are used to approximate the integrals that exist in LVFIE. Also, a comparison between those

Stumpff functions are an infinite series that depends on the value of z. This value results from multiplying the reciprocal semi-major axis with a universal anomaly. The purpose from those functions is to calculate the variation of the universal parameter (variable) using Kepler's equation for different orbits. In this paper, each range for the reciprocal of the semi-major axis, universal anomaly, and z is calculated in order to study the behavior of Stumpff functions C(z) and S(z). The results showed that when z grew, Stumpff functions for hyperbola, parabola, and elliptical orbits were also growing. They intersected and had a tendency towards zero for both hyperbola and parabola orbits, but for elliptical orbits, Stumpff functions

In this paper, a new technique is offered for solving three types of linear integral equations of the 2nd kind including Volterra-Fredholm integral equations (LVFIE) (as a general case), Volterra integral equations (LVIE) and Fredholm integral equations (LFIE) (as special cases). The new technique depends on approximating the solution to a polynomial of degree  and therefore reducing the problem to a linear programming problem(LPP), which will be solved to find the approximate solution of LVFIE. Moreover, quadrature methods including trapezoidal rule (TR), Simpson 1/3 rule (SR), Boole rule (BR), and Romberg integration formula (RI) are used to approximate the integrals that exist in LVFIE. Also, a comparison between those methods i

In this study, hexadecyltrimethylammonium bromide (HDMAB) - bentonite was synthesized by placing alkylammonium cation onto bentonite. Adsorption of textile dye such as direct Yellow 50 on natural bentonite and HDMAB -bentonite was investigated. The effects of pH, contact time,dosage clay and temperature were investigated experimentally .The Langmuir and Freundlish isotherms equations were applied to the data and values of parameters of these isotherm equations were evaluated. The study indicated that using 0.2 g of HDMAB (hexadecyltrimethylammonium bromide) lead to increase the percentage removal(R%) from 78% for pure bentonite to 99 %. The optimum pH value for the adsorption experiments was found to be pH=3 and therefore all the experim

background: osteoporosis is a metabolic bone disease that affects women more than men, it is characterized by generalizes reduction of bone mineral density (BMD) leaving a fragile weak bone that is liable to fracture, gonial angle index (GAI) is one of the radio-morphometric indices, it has been controversial whether it is related to bone mineral density or ageing or none of them. The aim of study is to evaluate the role of cone beam computed tomography (CBCT) as a screening tool for diagnosis of osteoporosis and age effect in females using gonial angle index. Material and method: 60 females were divided into 3 groups according to age and (BMD) status into: Group1 (non-osteoporosis 20-30 years), Group2 (non-osteoporosis 50years and above),

Production logging is used to diagnose well production problems by evaluating the flow profile, entries of unwanted fluids and downhole flow regimes. Evaluating wells production performance can be easily induce from production logs through interpretation of production log data to provide velocity profile and contribution of each zone on total production. Production logging results supply information for reservoir modeling, provide data to optimize the productivity of existing wells and plan drilling and completion strategies for future wells. Production logging was carried out in a production oil well from Mishrif formation of West Qurna field, with the objective to determine the flow profile and fluid contributions from the perforations af