This research includes the synthesis of new series of heterocyclic compounds. Reaction of 2-nitro benzylidene)thiosemicarbazide(1) with ethyl chloro acetate gave (2-nitro benzylidene amino)-2-thioxomidazolidine-4-one(2) ,treatment(2) with methyl iodide to give(3)which was reacted with hydrazine to give 2-hydrazinyl-1-[(2-nitrobenzylidene)amino]- 1H-imidazol-5(4H)-one, andreation of compound(2) with aromatic aldehydes to give 5arylidene -3-({2-nitro benzylidene}amino)2-thioxo-3,5-dihydro-4H-imidazole-4-one(5a,5b), which was reacted with ethyl aceto acetate to give 4-aryl-1-[2-nitrobenzylidene, amino -6oxo-2-thioxo octa hydro-1H-benzo[d]imidazole-5-carboxylate and followed synthesis of βlactamederivtives(9a,9b) by treatment derivatives(

This paper is an attempt to help the manager of a manufactory to

plan for the next year by a scientific approach, to maximize the profit and Ø¢ provide optimal Ø¢ monthly quantities of Ø¢ production, Ø¢ inventory,

work-force, prices and sales. The computer programming helps us to execute that huge number of calculations.

  In this paper, we introduce and discuss an algorithm for the numerical solution of two- dimensional fractional dispersion equation.  The algorithm for the numerical solution of this equation is based on explicit finite difference approximation. Consistency, conditional stability, and convergence of this numerical method are described. Finally, numerical example is presented to show the dispersion behavior according to the order of the fractional derivative and we demonstrate that our explicit finite difference approximation is a computationally efficient method for solving two-dimensional fractional dispersion equation

     An Alternating Directions Implicit method is presented to solve the homogeneous heat diffusion equation when the governing equation is a bi-harmonic equation (X) based on Alternative Direction Implicit (ADI). Numerical results are compared with other results obtained by other numerical (explicit and implicit) methods. We apply these methods it two examples (X): the first one, we apply explicit when the temperature .

Necessary and sufficient conditions for the operator equation I AXAX n*, to have a real positive definite solution X are given. Based on these conditions, some properties of the operator A as well as relation between the solutions X andAare given.