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Numerical simulations are carried out to assess the quality of the circular and square apodize apertures in observing extrasolar planets. The logarithmic scale of the normalized point spread function of these apertures showed sharp decline in the radial frequency components reaching to 10-36 and 10-34 respectively and demonstrating promising results. This decline is associated with an increase in the full width of the point spread function. A trade off must be done between this full width and the radial frequency components to overcome the problem of imaging extrasolar planets.

This paper is dealing with non-polynomial spline functions "generalized spline" to find the approximate solution of linear Volterra integro-differential equations of the second kind and extension of this work to solve system of linear Volterra integro-differential equations. The performance of generalized spline functions are illustrated in test examples

Volterra – Fredholm integral equations (VFIEs) have a massive interest from researchers recently. The current study suggests a collocation method for the mixed Volterra - Fredholm integral equations (MVFIEs)."A point interpolation collocation method is considered by combining the radial and polynomial basis functions using collocation points". The main purpose of the radial and polynomial basis functions is to overcome the singularity that could associate with the collocation methods. The obtained interpolation function passes through all Scattered Point in a domain and therefore, the Delta function property is the shape of the functions. The exact solution of selective solutions was compared with the results obtained

In this study, a new technique is considered for solving linear fractional Volterra-Fredholm integro-differential equations (LFVFIDE's) with fractional derivative qualified in the Caputo sense. The method is established in three types of Lagrange polynomials (LP’s), Original Lagrange polynomial (OLP), Barycentric Lagrange polynomial (BLP), and Modified Lagrange polynomial (MLP). General Algorithm is suggested and examples are included to get the best effectiveness, and implementation of these types. Also, as special case fractional differential equation is taken to evaluate the validity of the proposed method. Finally, a comparison between the proposed method and other methods are taken to present the effectiveness of the proposal meth

This paper is concerned with the numerical solutions of the vorticity transport equation (VTE) in two-dimensional space with homogenous Dirichlet boundary conditions. Namely, for this problem, the Crank-Nicolson finite difference equation is derived.  In addition, the consistency and stability of the Crank-Nicolson method are studied. Moreover, a numerical experiment is considered to study the convergence of the Crank-Nicolson scheme and to visualize the discrete graphs for the vorticity and stream functions. The analytical result shows that the proposed scheme is consistent, whereas the numerical results show that the solutions are stable with small space-steps and at any time levels.

Most of the world is seeking attention to the agricultural sector, which occupies a distinguished economic center, including Iraq, it is no wonder that the focus is on this sector, especially in developing countries. Descriptive analysis of the sample tables of wheat and barley crops in Iraq showed that adherence to the principles and rules of total quality, including modern irrigation methods (irrigated) has a significant impact in increasing productivity, reducing costs and improving quality compared to traditional irrigation methods, so it became necessary to take agricultural economic units All procedures and means that will help in the application of the rules of total quality to promote the reality of wheat and barley cultivation in I