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A Numerical Algorithm for the Resolution of Scalar and Matrix Algebraic Equations Using Runge-Kutta Method
Issue:
Volume 3, Issue 3, June 2014
Pages:
68-74
Received:
3 May 2014
Accepted:
17 May 2014
Published:
30 May 2014
Abstract: The Runge-Kutta method is an interesting and precise method for the resolution of ordinary differential equations. Fortunately, when supposing the differentiation by any variable that the equation to solve is not variable of, and after iterations, the solution of this equation stretches to the algebraic roots of this equation. This feature of this algorithm, indeed, allows to solve precisely any scalar or matrix equation. The numerical algorithm proposed herein is an iterative procedure of the fourth-order Runge-Kutta method with an adopted precision tolerance of convergence. Also, a method to determine all the roots of the polynomial equations is presented. Some scalar and matrix algebraic equations are resolved using this proposed algorithm, and show how this algorithm featuring with an excellent precision, a good speed and a simplicity for programming to solve equations and deduct the roots.
Abstract: The Runge-Kutta method is an interesting and precise method for the resolution of ordinary differential equations. Fortunately, when supposing the differentiation by any variable that the equation to solve is not variable of, and after iterations, the solution of this equation stretches to the algebraic roots of this equation. This feature of this ...
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Modelling the Effects of Variable Viscosity in Unsteady Flow of Nanofluids in a Pipe with Permeable Wall and Convective Cooling
Sara Khamis,
Oluwole Daniel Makinde,
Yaw Nkansah-Gyekye
Issue:
Volume 3, Issue 3, June 2014
Pages:
75-84
Received:
13 May 2014
Accepted:
27 May 2014
Published:
30 May 2014
Abstract: In this paper, the combined effects of variable viscosity, Brownian motion, thermophoresis and convective cooling on unsteady flow of nanofluids in a pipe with permeable wall are investigated. It is assumed that the pipe surface exchange heat with the ambient following the Newton’s law of cooling. Using a semi discretization finite difference method coupled with Runge-Kutta Fehlberg integration scheme, the nonlinear governing equations of momentum and energy balance, and the equation for nanoparticles concentration are tackled numerically. Useful results for the velocity, temperature, nanoparticles concentration profiles, skin friction and Nusselt number are obtained graphically and discussed quantitatively.
Abstract: In this paper, the combined effects of variable viscosity, Brownian motion, thermophoresis and convective cooling on unsteady flow of nanofluids in a pipe with permeable wall are investigated. It is assumed that the pipe surface exchange heat with the ambient following the Newton’s law of cooling. Using a semi discretization finite difference meth...
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Single Machine Scheduling Problems with Delivery Times under Simple Linear Deterioration
Issue:
Volume 3, Issue 3, June 2014
Pages:
85-89
Received:
30 April 2014
Accepted:
20 May 2014
Published:
10 June 2014
Abstract: We consider several single machine scheduling problems in which the processing time of a job is a simple linear increasing function of its starting time and each job has a delivery time. The objectives are to minimize the functions about delivery completion times. For the former three problems, we propose polynomial-time algorithms to solve them. For the last problem, we prove that it is NP-hard when all jobs have release dates.
Abstract: We consider several single machine scheduling problems in which the processing time of a job is a simple linear increasing function of its starting time and each job has a delivery time. The objectives are to minimize the functions about delivery completion times. For the former three problems, we propose polynomial-time algorithms to solve them. F...
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Boundary Value Problems on Triangular Domains and MKSOR Methods
I. K. Youssef,
Sh. A. Meligy
Issue:
Volume 3, Issue 3, June 2014
Pages:
90-99
Received:
29 May 2014
Accepted:
16 June 2014
Published:
30 June 2014
Abstract: The performance of six variants of the successive overrelaxation methods (SOR) are considered for an algebraic system arising from a finite difference treatment of an elliptic equation of Partial Differential Equations (PDEs) on a triangular region. The consistency of the finite difference representation of the system is achieved. In the finite difference method one obtains an algebraic system corresponding to the boundary value problem (BVP). The block structure of the algebraic system corresponding to four different labeling (the natural, the red- black and green (RBG), the electronic and the spiral) of the grid points is considered. Also, algebraic systems obtained from BVP with mixed derivatives are well established. Determination of the optimal relaxation parameters on the bases of the graphical representation of the spectral radius of the iteration matrices for the SOR, the Modified Successive over relaxation (MSOR) and their new variants KSOR, MKSOR, MKSOR1 and MKSOR2 are considered. Application of the treatment to two numerical examples is considered.
Abstract: The performance of six variants of the successive overrelaxation methods (SOR) are considered for an algebraic system arising from a finite difference treatment of an elliptic equation of Partial Differential Equations (PDEs) on a triangular region. The consistency of the finite difference representation of the system is achieved. In the finite dif...
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Analysis of Turbulent Hydromagnetic Flow with Radiative Heat over a Moving Vertical Plate in a Rotating System
Dawit H. Gebre,
O. D. Makinde,
M. Kinyanjui
Issue:
Volume 3, Issue 3, June 2014
Pages:
100-109
Received:
12 June 2014
Accepted:
24 June 2014
Published:
30 June 2014
Abstract: In this paper, the combined effects of magnetic fields, buoyancy force, thermal radiation, viscous and Ohmic heating on turbulent hydromagnetic flow of an incompressible electrically conducting fluid over a moving vertical plate in a rotating system is investigated numerically. The governing equations are reduced to non-linear ordinary differential equations using the time-averaged approach known as Reynolds-averaged Navier–Stokes equations (or RANS equations) and tackled by employing an efficient Runge-Kutta Fehlberg integration technique coupled with shooting scheme. Graphical results showing the effects of various thermophysical parameters on the velocity, temperature, local skin friction and local Nusselt number are presented and discussed quantitatively.
Abstract: In this paper, the combined effects of magnetic fields, buoyancy force, thermal radiation, viscous and Ohmic heating on turbulent hydromagnetic flow of an incompressible electrically conducting fluid over a moving vertical plate in a rotating system is investigated numerically. The governing equations are reduced to non-linear ordinary differential...
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Conversion of Energy Equation for Turbulent Motion and its Applications
Issue:
Volume 3, Issue 3, June 2014
Pages:
110-116
Received:
26 June 2014
Accepted:
4 July 2014
Published:
20 July 2014
Abstract: Turbulent energy has developed revolutionary technology in the form of a portfolio of devices for the mixing, separation and the homogenization of liquids with liquids, liquids with gasses and gasses with gasses. The mixing technology may be applied to a wide variety including chemicals, pharmaceuticals, cosmetics, foods, agricultural, water treatment with purification and hybrid fuels. The paper reports the transformation of energy equation for turbulent flow in terms of correlation tensors of second order, where the correlation tensors are the functions of space coordinates, distance between two points and time. To reveal the relation of turbulent energy between two points, one point has been taken as the origin of the coordinate system. Correlation between pressure fluctuations and velocity fluctuations at the two points of flow field is applied to the turbulent energy equation. The applications of turbulent energy are discussed for the source of oceanic turbulence by means of Richardson number. A multiplication factor in terms of kinetic energy and potential energy is considered for finding the correlation between the multiplication factor and critical flux Richardson number and to signify the relative efficiency of mixing by Kelvin-Helmholtz billows and the critical flux Richardson number.
Abstract: Turbulent energy has developed revolutionary technology in the form of a portfolio of devices for the mixing, separation and the homogenization of liquids with liquids, liquids with gasses and gasses with gasses. The mixing technology may be applied to a wide variety including chemicals, pharmaceuticals, cosmetics, foods, agricultural, water treatm...
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