MULTIGRID SOLUTION OF THREE DIMENSIONAL BIHARMONIC EQUATIONS WITH DIRICHLET BOUNDARY CONDITIONS OF SECOND KIND

Ibrahim, S.A. Hoda,Hassan, Naglaa Ameen The Korean Society for Computational and Applied M 2012 Journal of applied mathematics & informatics Vol.30 No.1

In this paper, we solve the three-dimensional biharmonic equation with Dirichlet boundary conditions of second kind using the full multigrid (FMG) algorithm. We derive a finite difference approximations for the biharmonic equation on a 18 point compact stencil. The unknown solution and its second derivatives are carried as unknowns at grid points. In the multigrid methods, we use a fourth order interpolation to producing a new intermediate unknown functions values on a finer grid, and the full weighting restriction operators to calculating the residuals at coarse grid points. A set of test problems gives excellent results.

AN IMPROVED EXPONENTIAL REGULA FALSI METHODS WITH CUBIC CONVERGENCE FOR SOLVING NONLINEAR EQUATIONS

Hoda Ibrahim 한국전산응용수학회 2010 Journal of applied mathematics & informatics Vol.28 No.5

The aim of this paper is to propose a cubic convergent regula falsi iterative method for solving the nonlinear equation f(x) = 0, where f : [a,b] ⊂ □ → □ is a continuously di®erentiable. In [3,6] a quadratically convergent regula falsi iterative methods for solving this nonlinear equations is proposed. It is shown there that both the sequences of diameters and iterative points sequence converge to zero simultaneously. So The aim of this paper is to accelerate further the convergence of these methods from quadratic to cubic. This is done by replacing the parameter p in the iteration of [3,5,6] by a function p(x) defined suitably. The convergence analysis is carried out for the method. The method is tested on number of numerical examples and results obtained shows that our methods are better and more effective and comparable to well-known methods.

MULTIGRID SOLUTION OF THREE DIMENSIONAL BIHARMONIC EQUATIONS WITH DIRICHLET BOUNDARY CONDITIONS OF SECOND KIND

Hoda Ibrahim,Naglaa Ameen 한국전산응용수학회 2012 Journal of applied mathematics & informatics Vol.30 No.1

In this paper, we solve the three-dimensional biharmonic equation with Dirichlet boundary conditions of second kind using the full multigrid (FMG) algorithm. We derive a finite difference approximations for the biharmonic equation on a 18 point compact stencil. The unknown solution and its second derivatives are carried as unknowns at grid points. In the multigrid methods, we use a fourth order interpolation to producing a new intermediate unknown functions values on a finer grid, and the full weighting restriction operators to calculating the residuals at coarse grid points. A set of test problems gives excellent results.

AN IMPROVED EXPONENTIAL REGULA FALSI METHODS WITH CUBIC CONVERGENCE FOR SOLVING NONLINEAR EQUATIONS

Ibrahim, S.A. Hoda The Korean Society for Computational and Applied M 2010 Journal of applied mathematics & informatics Vol.28 No.5

The aim of this paper is to propose a cubic convergent regula falsi iterative method for solving the nonlinear equation f(x) = 0, where f : [a,b] $\subset$ R $\rightarrow$ R is a continuously differentiable. In [3,6] a quadratically convergent regula falsi iterative methods for solving this nonlinear equations is proposed. It is shown there that both the sequences of diameters and iterative points sequence converge to zero simultaneously. So The aim of this paper is to accelerate further the convergence of these methods from quadratic to cubic. This is done by replacing the parameter p in the iteration of [3,5,6] by a function p(x) defined suitably. The convergence analysis is carried out for the method. The method is tested on number of numerical examples and results obtained shows that our methods are better and more effective and comparable to well-known methods.

The Functional Role of Some Tomato Products on Lipid Profile and Liver Function in Adult Rats

Hoda Salama Ibrahim,Lamiaa Ali Ahmed,Maha Mohamed Essam El-din 한국식품영양과학회 2008 Journal of medicinal food Vol.11 No.3

This study was carried out to investigate the functional role of lycopene obtained from powder prepared from fresh tomato, tomato paste, and ketchup that contained equal amounts of lycopene based on levels of intake on body weight gain (BWG), feed intake, feed efficiency ratio (FER), lipid profiles, atherogenic index, and liver enzymes of hyperlipidemic rats. Forty-eight male albino rats were divided into two main groups: the first group (n = 6 rats) was kept on the basal diet as a normal control, while the second group (n = 42 rats) was fed a hyperlipidemic diet for 5 weeks to induce hyperlipidemia. The latter group was divided into seven subgroups: the first subgroup was the positive control group, while the others were supplemented with one of the tomato products at one of two levels (10 or 20 mg of lycopene/kg of diet). BWG, feed intake, and FER were calculated, and blood samples were collected to determine total lipids, total cholesterol, triglycerides, lipoprotein fractions, atherogenic index, and liver function in sera. Relative organ weights were also calculated. Results revealed that administration of various tomato products produced a significant reduction in feed intake except for the hyperlipidemic group that supplemented with the lower lycopene level from tomato paste. In addition, BWG and FER were not influenced by addition of tomato products at any level of intake. Hyperlipidemic rats supplemented with tomato powder, tomato paste, or ketchup showed significant improvement in almost all the parameters studied compared to the positive control group. Results showed that the higher lycopene level from tomato paste produced significant improvement in all lipid parameters, followed by 10 mg of lycopene/kg from tomato paste, which caused significant elevation in high-density lipoprotein cholesterol comparable to that of the negative control group. The lowest atherogenic index was achieved by addition of the lower lycopene level from tomato paste followed by the higher lycopene level from the same source. So, because of the positive effect of tomato products on the tested parameters an increase in consumption of tomato and its products in the diet is recommended. Nutrition education programs should be encouraged to inform the public of the importance of tomato and its products, especially tomato paste and ketchup, in decreasing the risk of hyperlipidemia.

Effect of different storage media on elemental analysis and microhardness of cervical cavity margins restored with a bioactive material

Ismail Hoda Saleh,Morrow Brian Ray,Ali Ashraf Ibrahim,Mehesen Rabab Elsayed Elaraby,Mahmoud Salah Hasab,Garcia-Godoy Franklin 대한치과보존학회 2024 Restorative Dentistry & Endodontics Vol.49 No.1

Objectives This study aimed to investigate the elemental analysis and microhardness of a bioactive material (Activa) and marginal tooth structure after storage in different media. Materials and Methods Fifteen teeth received cervical restorations with occlusal enamel and gingival dentin margins using the tested material bonded with a universal adhesive, 5 of them on the 4 axial surfaces and the other 10 on only the 2 proximal surfaces. The first 5 teeth were sectioned into 4 restorations each, then stored in 4 different media; deionized water, Dulbecco's phosphate buffered saline (DPBS), Tris buffer, and saliva. The storage period for deionized water was 24 hours while it was 3 months for the other media. Each part was analyzed by scanning electron microscopy-energy dispersive spectroscopy (SEM-EDS) analysis for different substrates/distances and the wt% of calcium, phosphorus, silica, and fluoride were calculated. The other 10 teeth were sectioned across the restoration, stored in either Tris buffer or saliva for 24 hours or 3 months, and were evaluated for microhardness of different substrates/areas. Data were analyzed using analysis of variance and Tukey’s post hoc test. Results Enamel and dentin interfaces in the DPBS group exhibited a significant increase in calcium and phosphorus wt%. Both silica and fluoride significantly increased in tooth structure up to a distance of 75 μm in the 3-month-media groups than the immediate group. Storage media did not affect the microhardness values. Conclusions SEM-EDS analysis suggests an ion movement between Activa and tooth structure through a universal adhesive while stored in DPBS. Objectives This study aimed to investigate the elemental analysis and microhardness of a bioactive material (Activa) and marginal tooth structure after storage in different media. Materials and Methods Fifteen teeth received cervical restorations with occlusal enamel and gingival dentin margins using the tested material bonded with a universal adhesive, 5 of them on the 4 axial surfaces and the other 10 on only the 2 proximal surfaces. The first 5 teeth were sectioned into 4 restorations each, then stored in 4 different media; deionized water, Dulbecco's phosphate buffered saline (DPBS), Tris buffer, and saliva. The storage period for deionized water was 24 hours while it was 3 months for the other media. Each part was analyzed by scanning electron microscopy-energy dispersive spectroscopy (SEM-EDS) analysis for different substrates/distances and the wt% of calcium, phosphorus, silica, and fluoride were calculated. The other 10 teeth were sectioned across the restoration, stored in either Tris buffer or saliva for 24 hours or 3 months, and were evaluated for microhardness of different substrates/areas. Data were analyzed using analysis of variance and Tukey’s post hoc test. Results Enamel and dentin interfaces in the DPBS group exhibited a significant increase in calcium and phosphorus wt%. Both silica and fluoride significantly increased in tooth structure up to a distance of 75 μm in the 3-month-media groups than the immediate group. Storage media did not affect the microhardness values. Conclusions SEM-EDS analysis suggests an ion movement between Activa and tooth structure through a universal adhesive while stored in DPBS.