A solid-state reaction technique was used to synthesize polycrystalline Na ₂ WO ₄ . Preliminary X-ray studies revealed that the compound has a cubic structure at room temperature. The formation of the compound has been confirmed by X-ray powder diffraction studies and Raman spectroscopy. Electrical and dielectric properties of the compound have been studied using complex impedance spectroscopy in the frequency range 209 Hz–1 MHz and temperature range 586–679 K. The impedance data were modellized by an equivalent circuit consisting of series of a combination of grains and grains boundary. We use complex electrical modulus M * at various temperatures to analyse dielectric data. The modulus plots are characterized by the presence of two relaxation peaks thermally activated. The morphologies and the average particle size of the resultant sodium tungstate sample were demonstrated by atomic force microscopy, scanning electron microscopy and transmission electron microscopy. The thicknesses and optical constants of the sample have been calculated using ellipsometric measurements in the range of 200–22 000 nm by means of new amorphous dispersion formula which is the objective of the present work. The results were obtained for Na ₂ WO ₄ particles from experimental (EXP) and measured (FIT) data showed an excellent agreement. In addition, the energy gap of the Na ₂ WO ₄ sample has been determined using ellipsometry and confirmed by spectrophotometry measurements.

In this work, numerical solution of nonlinear equations using Newton Raphson method (NRM) and a modified Newton-Raphson Method (MNRM) are utilized to solve and find the real roots of a nonlinear equation based on a single-diode PV cell. The proposed methods to solve nonlinear examples and obtain results with various values of a load resistance have been examined. The purpose of this paper is to obtain the results of solar cell parameters using two mathematical models with the comparison between them. The obtained results showed the proposed method (MNRM) is a powerful tool, sufficient way to solve this model with a least iterations.

The goal of this work is to find a numerical solution of nonlinear solar cell equation. This equation has been instructed using a single-diode model. The proposed method consists of solving the equation using two iterative methods with the initial value . Moreover, the Newton's and Two-step methods are used to determine the required the current, the voltage, and the power of the PV cell in the procedure of the present research. Different values of load resistance have introduced with these methods. The obtained results appeard that the proposed method is the most efficient compare with NRM and all the calculations are achieved using Matlab program.

This paper discussed the solution of an equivalent circuit of solar cell, where a single diode model is presented. The nonlinear equation of this model has suggested and analyzed an iterative algorithm, which work well for this equation with a suitable initial value for the iterative. The convergence of the proposed method is discussed. It is established that the algorithm has convergence of order six. The proposed algorithm is achieved with a various values of load resistance. Equation by means of equivalent circuit of a solar cell so all the determinations is achieved using Matlab in ambient temperature. The obtained results of this new method are given and the absolute errors is demonstrated.

The mathematical modeling of solar cell device is used to demonstrate the equivalent circuit of single-diode solar cell model operating under environmental conditions. In this work, bisection (BM) and secant (SM) Methods currently exists to demonstrate the non-linear electrical behavior of this device. The proposed method is tested in order to solve non-linear equation of PV cell with various values of load resistance are characterized. Comparisons between these numerical examples have been compared. The main idea of the present work is to calculate the PV parameters numerically using different methods with the comparison between them. The results showed that secant method is the best because it has least iteration to obtain the optimal values of the PV parameters.

In this paper, a general expression formula for Spline Scaling Functions (SSFs) operational matrix of derivative is constructed. Then it is used to study a new iterative parameterization direct technique for treating optimal control problems approximately. Optimal control problems describe several important phenomena in mathematical science. In the present technique, an operational matrix of derivatives for such functions is constructed for resolving the optimal control problem. Since the obtained operational matrix of derivative comprises, many zeros elements, it can bring about a numerical accurate result with high reliability of achieving the desired results. An attractive product operation matrix of SSFs with other properties is also included in this work. With solving some examples, the comparison with the actual solutions shows that our algorithm is acceptable.