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आयतन 6, मुद्दा 4 (2017)

शोध आलेख

Combining Rational Approximation with Asymptotic Wiener–Hopf Factorization Algorithm for Matrix Functions: Implementation and Testing

Rougerie T and Kisil A

This paper discusses how an asymptotic Wiener–Hopf factorization can be implemented for a wide class of functions. Asymptotic Wiener–Hopf factorization was discussed [19] and the convergence for matrices sufficiently close to the identity matrix is shown. We demonstrate how the algorithm can be successfully implemented with the help of the rational approximations. The idea is to simplify the matrix first by rationally approximating and then perform the approximate factorisation. There is no compromise in accuracy since the factorisation is approximate anyway and rational approximations are very precise usually. There is also a mapping of real line discussed and implemented to make the rational approximation more optimal. The code is tested against some easy examples which are calculated by hand. The use of this code is illustrated with some more complicated matrix functions motivated by applications. The method has been implemented for 2 × 2 and 4 × 4 matrices but can easy be adapted for any size matrix. The code will be made available with the publication of this paper. We note that to date there are very few implemented Wiener–Hopf factorisation available due to instabilities, so this paper will make an important contribution to this area.

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New Adomian’s Polynomials Formulas for the Non-linear and Nonautonomous Ordinary Differential Equations

Zaouagui IN and Badredine T

In this paper, Adomian decomposition method has been adopted to resolve the non-linear and non-autonomous ordinary differential equations. It has been proved that this technique permits to give new expressions for the Adomian’s polynomials (??) and (??).

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Implementation of a Semi-implicit Time Integration Scheme in Non-Hydrostatic Euler Equations

Nam H and Choi SJ

A semi-implicit time integration scheme is implemented in a non-hydrostatic Euler problem on the cubed-sphere grid. The semi-implicit time integration scheme is a 3-stage additive RungeKutta method, which is an Implicit-Explicit (IMEX) multi-stage single-step scheme. The system treats acoustic and gravity waves implicitly and advection explicitly. In the implicit part, we compute the linear system by defining a proper linear operator. Then, the numerical results are presented to compare the semi-implicit time integration scheme with the explicit RungeKutta time integration scheme in non-hydrostatic Euler equations. In terms of accuracy, we demonstrate that the proposed method performs better than other schemes.

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The Unique Natural Number Set and Distributed Prime Numbers

Alabed TH and Bashir MB

The Natural number set has a new number set behind three main subset: odd, even and prime which are unique number set. It comes from a mathematical relationship between the two main types even and odd number set. It consists of prime and T-semi-prime numbers set. However, finding ways to understand how prime number was distributed was ambiguity and it is impossible, this new natural set show how the prime number was distributed and highlight focusing on T-semi-prime number as a new sub natural set.

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Axisymmetric Magneto Dynamic (MHD) Stability of a Compressible Fluid Cylinder

Barakat HM

The axisymmetric magneto dynamic (MHD) stability of a compressible fluid cylinder under the action of inertia, and electromagnetic forces is developed. A general eigenvalue relation is derived studied analytically and the results are confirmed numerically. In absence of the effect of the electromagnetic forces interior and exterior the fluid, so the model is only subjected to the capillary force. It is found that the model is unstable in the region 0<x<1. While it is stable in the region 1<x<∞. This means that model is just unstable in small domains of axisymmetric perturbation but it stable in all other domains. For very high intensity of magnetic field the model is completely stable for all values of wave lengths. The compressibility has a stabilizing tendency.

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Casting Simulation and Prediction of Shrinkage Cavities

Jabur AS and Kushnaw FM

This work aims to predict the shrinkage defects in Al-Si castings by determination the suitable parameters and techniques which can be applied in the casting simulation system. Also, it aims to specify the role of silicon content in amount, morphology, and distribution of these defects. The Numerical solution has been carried out using an explicit 3-D finite difference method for the given system of the casting and a mold. Additionally, an experimental casting of the studied samples was achieved. It was found that the shrinkage porosities increased with increasing the silicon content up to 7%, so at this peak, they spread in all cast regions and cannot be predicted. The low silicon alloys suffered from only the cavities defects that can be predicted by mapping the solidus time contours. Finally, it was concluded that the critical temperature gradient value of the porosities development in the eutectic (Al-12%Si) alloys was 1.3°C/cm.

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Analysis of Prey-Predator Model in Chemostat When the Predator Produces Inhibitor

Moniem AA

In this work, a prey-predator model in chemostat is considered when the predator produces inhibitor. This inhibitor is lethal to the prey by results in decrease of growth rate of the predator at some cost to its reproductive abilities. A Lyapunov function in the study of the global stability of a predator-free steady state is analysed. Local and global stability of other steady states, persistence analysis, as well as numerical simulations are also presented.

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