The internal heat source and reaction effects on the onset of thermosolutal convection in a local thermal non-equilibrium porous medium are examined, where the temperature of the fluid and the solid skeleton may differ. The linear instability and nonlinear stability theories of Darcy–Brinkman type with fixed boundary condition are carried out where the layer is heated and salted from below. The
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2
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Chebyshev tau technique is used to calculate the associated system of equations subject to the boundary conditions for both theories. Three different types of internal heat source function are considered, the first type increases across the layer, while the second decreases, and the third type heats and cools in a nonuniform way. The effect of different parameters on the Rayleigh number is depicted graphically. Moreover, the results detect that utilizing the internal heat source, reaction, and non-equilibrium have pronounced effects in determining the convection stability and instability thresholds.
In this paper, a least squares group finite element method for solving coupled Burgers' problem in 2-D is presented. A fully discrete formulation of least squares finite element method is analyzed, the backward-Euler scheme for the time variable is considered, the discretization with respect to space variable is applied as biquadratic quadrangular elements with nine nodes for each element. The continuity, ellipticity, stability condition and error estimate of least squares group finite element method are proved. The theoretical results show that the error estimate of this method is . The numerical results are compared with the exact solution and other available literature when the convection-dominated case to illustrate the efficiency of the proposed method that are solved through implementation in MATLAB R2018a.
In this paper, we apply weak Galerkin finite element method (WGFEM) and weak group finite element method (WGrFEM) for 2-D Burgers' problem by using the weak functions and their corresponding discrete weak derivatives in standard weak form. We consider the fully formulation using the backward-difference form for the time variable and we prove the stability and an error estimate for these methods. The numerical examples of our methods have been carried out through implementation in MATLAB programs and compared with the exact solutions and other available literature to illustrate the efficiency of the proposed methods, the WGFEM and the WGrFEM provide convergent approximations and handles the equation well in different cases, the results obtained of our methods are very acceptable and competent more than the results available in the other literature. Moreover, a WGrFEM is displayed to be better than the GFEM.
The study aimed to evaluate the content of the mathematics book for the third grade the new curriculum in the first and second part of the academic year 2018-2019 in light of the international standards issued by the National Council of Teachers of Mathematics (NCTM2000) to find strengths and weaknesses in it and make recommendations that contribute to the development of the content of the book. We relied on descriptive analytical methodology of the book and the tool was built to analyze content in four areas (number and operations, algebra, relations and functions, geometry and measurement, data analysis and probabilities).
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