In this paper, we built a mathematical model for convection and thermal radiation heat transfer of fluid flowing through a vertical channel with porous medium under effects of horizontal magnetic field (MF) at the channel. This model represents a 2-dimensional system of non-linear partial differential equations. Then, we solved this system numerically by finite difference methods using Alternating Direction Implicit (ADI) Scheme in two phases (steady state and unsteady state). Moreover, we found the distribution and behaviour of the heat temperature inside the channel and studied the effects of Brinkman number, Reynolds number, and Boltzmann number on the heat temperature behaviour. We solved the system by building a computer program using MATLAB.

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In this paper we give some properties of the zero commut idempotent matrices ,and give some properties of non-singular matrices. Some basic properties of idempotent matrices.

This paper deals with a mathematical model of a fluid flowing between two parallel plates in a porous medium under the influence of electromagnetic forces (EMF). The continuity, momentum, and energy equations were utilized to describe the flow. These equations were stated in their nondimensional forms and then processed numerically using the method of lines. Dimensionless velocity and temperature profiles were also investigated due to the impacts of assumed parameters in the relevant problem. Moreover, we investigated the effects of Reynolds number , Hartmann number M, magnetic Reynolds number , Prandtl number , Brinkman number , and Bouger number , beside those of new physical quantities (N , ). We solved this system by creating a computer program using MATLAB.

In this paper, the method of estimating the Generalized Additive Models (GAM) was highlighted, and a proposed robust weighted composition was found by combining the robust M method with the smoothing splines to estimate the Robust Generalized Additive Model and its notation is (RGAM). This estimator is used to deal with the effect of the presence of outliers in the data that do not fit into the overall data pattern by relying on some of the weight functions of the robust M method. Wavelet Shrinkage technique is used as well, which has been proposed as a smoothing of data using several types of wavelet filters in calculating the discrete wavelet transformation and relying on it in estimating the wavelet generalized additive model symbolized by (WGAM). In the case of using the simulation method, when data is contaminated with distributions ((t) Dis., Exp. Dis.) And with contamination rates (5%, 15%, 35%) and with sample sizes (50,150,300) it is noted that the smoothing method is with the Bisequare weight (BRGAM). It had a better performance compared to the rest of the methods for the simulated scenarios covered. The GCV criterion showed a marked advantage over other criteria, especially when estimating the model in the proposed robust M (RGAM) model. Some statistical criteria have been adopted. These criteria of the Generalized Additive Model (GAM) is used to compare estimation methods, the proposed methods were tested on simulation experiments as well as on real data collected from Ibn Sina Learning Hospital on cases of short stature, and the RGAM method gave the best results compared to the ordinary GAM and WGAM methods, and that by obtaining the smallest GCV value, this is because it is responsible for selecting the most suitable smoothing parameter for the smoothing spline estimator.