NUMERICAL SOLUTIONS OF SINGULAR NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS USING SAID-BALL POLYNOMIALS

In this article, the collocation method based on Said-Ball polynomials have been used to solve the singular nonlinear ordinary differential equations of various orders numerically. An operational matrix forms of these ordinary differential equations are obtained from Said-Ball polynomial with variated relations of solution and different derivatives. The presented method reduces the given problem to a system of nonlinear algebraic equations, which removed the singularity of ordinary differential equations. Resulting system is solved using Newton\u27s iteration method to get the coefficients of Said-Ball polynomials. We obtained approximate solutions of the problem under study. Numerical results have been obtained and compared with exact and other works. The presented method gives impressive solutions, that show the accuracy and reliability of the proposed method

An improved positivity preserving odd degree-n Said-Ball boundary curves on rectangular grid using partial differential equation

This paper discusses the sufficient conditions for positivity preserving odd degree-n Said-Ball boundary curves defined on a rectangular grid.We derive a sufficient condition on boundary curves of rectangular Said-Ball patches where the lower bound ordinates are adjusted independently.To construct the boundary curves for each rectangular patch, the Said-Ball polynomial solution of fourth order PDE will be considered where its coefficients can be calculated using edge Said-Ball ordinates which fulfill the positivity preserving conditions.Graphical examples are presented using well-known test functions

A Numerical scheme to Solve Boundary Value Problems Involving Singular Perturbation

نستخدم المصفوفات العملياتية لمشتقات وانج-بول متعددة الحدود في هذه الدراسة لحل المعادلات التفاضلية الشاذه المضطربة من الدرجة الثانية (WPSODEs) ذات الشروط الحدية. باستخدام مصفوفة كثيرات حدود وانج-بول، يمكن تحويل مشكلة الاضطراب الرئيسية الشاذ إلى أنظمة معادلات جبرية خطية. كما يمكن الحصول على معاملات الحل التقريبي المطلوبة عن طريق حل نظام المعادلات المذكور. وتم استخدام أسلوب الخطاء المتبقي أيضًا لتحسين الخطأ، كما تمت مقارنة النتائج بالطرق المنشورة في عدد من المقالات العلمية. استُخدِمت العديد من الأمثلة لتوضيح موثوقية وفائدة مصفوفات وانج بول العملياتية. طريقة وانج بول لديها القدرة على تحسين النتائج عن طريق تقليل درجة الخطأ بين الحلول التقريبية والدقيقة. أظهرت سلسلة وانج-بول فائدتها في حل أي نموذج واقعي كمعادلات تفاضلية من الدرجة الأولى أو الثانيةThe Wang-Ball polynomials operational matrices of the derivatives are used in this study to solve singular perturbed second-order differential equations (SPSODEs) with boundary conditions. Using the matrix of Wang-Ball polynomials, the main singular perturbation problem is converted into linear algebraic equation systems. The coefficients of the required approximate solution are obtained from the solution of this system. The residual correction approach was also used to improve an error, and the results were compared to other reported numerical methods. Several examples are used to illustrate both the reliability and usefulness of the Wang-Ball operational matrices. The Wang Ball approach has the ability to improve the outcomes by minimizing the degree of error between approximate and exact solutions. The Wang-Ball series has shown its usefulness in solving any real-life scenario model as first- or second-order differential equations (DEs)

Improved Operational Matrices of DP-Ball Polynomials for Solving Singular Second Order Linear Dirichlet-type Boundary Value Problems

Solving Dirichlet-type boundary value problems (BVPs) using a novel numerical approach is presented in this study. The operational matrices of DP-Ball Polynomials are used to solve the linear second-order BVPs. The modification of the operational matrix eliminates the BVP\u27s singularity. Consequently, guaranteeing a solution is reached. In this article, three different examples were taken into consideration in order to demonstrate the applicability of the method. Based on the findings, it seems that the methodology may be used effectively to provide accurate solutions

The sensitivity of Na+, K+ ATPase as an indicator of blood diseases

Background: Blood-related hereditary diseases are widespread in Eastern and SouthWestern regions of Saudi Arabia until recently. In this study, we used Na+, K+ATPase as an enzymatic indicator for the diagnosis of the diseases.Materials and methods: Individuals with different blood diseases (iron deficiency (n=13), anemia (n=14), thalassemia (n=16) and sickle cell anemia (n=12) were studied for Na+, K+-ATPase activity in the plasma membrane of red blood cell and compared with those of the healthy ones (n=20) of the same age and gender living in Jeddah, Saudi Arabia.Results: There was a significant elevation in the specific activity of Na+, K+ATPase in individuals with anemia compared with those of control (0.0094 + 0.001 nmol / mg protein/min versus 0.0061 0.001). On the other hand, there was a significant reduction in enzyme activity in thalassemia (0.0028 0.002 nmol / mg protein/min) and sickle cell anemia cases (0.0042 0.001 nmol / mg protein/min) compared to the control group. The cut off value for Na+, K+ATPase activity is 0.005 μmol Pi/minshowing 94% sensitivity and 93% specificity for the differentiation of blood abnormality.Conclusion: It can be recommended that the activity of Na+, K+-ATPase can be used for the diagnosis of individuals with blood diseases/disorders.Keywords: Na+, K+-ATPase, red blood cell, plasma membrane, iron deficiency anemia, thalassemia, sickle cell anemia, indicato

The sensitivity of Na+, K+ ATPase as an indicator of blood diseases.

Background: Blood-related hereditary diseases are widespread in Eastern and SouthWestern regions of Saudi Arabia until recently. In this study, we used Na+, K+ATPase as an enzymatic indicator for the diagnosis of the diseases. Materials and methods: Individuals with different blood diseases (iron deficiency (n=13), anemia (n=14), thalassemia (n=16) and sickle cell anemia (n=12) were studied for Na+, K+-ATPase activity in the plasma membrane of red blood cell and compared with those of the healthy ones (n=20) of the same age and gender living in Jeddah, Saudi Arabia. Results: There was a significant elevation in the specific activity of Na+, K+ATPase in individuals with anemia compared with those of control (0.0094 + 0.001 nmol / mg protein/min versus 0.0061 \ub10.001). On the other hand, there was a significant reduction in enzyme activity in thalassemia (0.0028 \ub1 0.002 nmol / mg protein/min) and sickle cell anemia cases (0.0042 \ub1 0.001 nmol / mg protein/min) compared to the control group. The cut off value for Na+, K+ATPase activity is 0.005 \u3bcmol Pi/minshowing 94% sensitivity and 93% specificity for the differentiation of blood abnormality. Conclusion: It can be recommended that the activity of Na+, K+-ATPase can be used for the diagnosis of individuals with blood diseases/disorders

Surface interpolation using partial differentiation equation with positivity preserving cubic Said-Ball curves boundary condition

This paper proposes the sufficient conditions for positivity preserving cubic boundary curves defined on rectangular grid using polynomial solution of fourth order linear PDEs in order to improve the positivity preserving of the interpolating surface. We derive a sufficient condition on boundary curves for each each of bicubic rectangular Bezier patches where the lower bounds of edge Bezier ordinates are adjusted independently.By using two well-known test functions, our result shows that the proposed method is well performed in terms of preserving the positivity of boundary curves and improves the positivity preserving of overall interpolating surfaces

Cubic Bézier-Like Triangular Patches for Rainfall Scattered Data Interpolation and Visualization

Scattered data interpolation plays an important role in computer graphics and scientific visualization. This method can be used to interpolate any regular or irregular data sets. For instance, rainfall distribution is an example of an irregular data that usually appear in statistics. In this study, cubic Bézier-Like triangular patches defined on triangular domain is used to interpolate rainfall data sets. From graphical results, we obtain a very smooth surface. Therefore, the proposed scheme can be used to interpolate and provide a good visualization to the given irregular data sets