On supraconvergence phenomenon for second order centered finite differences on non-uniform grids

In the present study we consider an example of a boundary value problem for a simple second order ordinary differential equation, which may exhibit a boundary layer phenomenon. We show that usual central finite differences, which are second order accurate on a uniform grid, can be substantially upgraded to the fourth order by a suitable choice of the underlying non-uniform grid. This example is quite pedagogical and may give some ideas for more complex problems.Comment: 26 pages, 2 figures, 2 tables, 37 references. Other author's papers can be downloaded at http://www.denys-dutykh.com

A Numerical Slow Manifold Approach to Model Reduction for Optimal Control of Multiple Time Scale ODE

Time scale separation is a natural property of many control systems that can be ex- ploited, theoretically and numerically. We present a numerical scheme to solve optimal control problems with considerable time scale separation that is based on a model reduction approach that does not need the system to be explicitly stated in singularly perturbed form. We present examples that highlight the advantages and disadvantages of the method

A Maximum Entropy Method for Solving the Boundary Value Problem of Second Order Ordinary Differential Equations

We propose a new method to solve the boundary value problem for a class of second order linear ordinary differential equations, which has a non-negative solution. The method applies the maximum entropy principle to approximating the solution numerically. The theoretical analysis and numerical examples show that our method is convergent

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)

Numerical solution of singularly perturbed convection–diffusion problem using parameter uniform B-spline collocation method

AbstractThis paper is concerned with a numerical scheme to solve a singularly perturbed convection–diffusion problem. The solution of this problem exhibits the boundary layer on the right-hand side of the domain due to the presence of singular perturbation parameter ɛ. The scheme involves B-spline collocation method and appropriate piecewise-uniform Shishkin mesh. Bounds are established for the derivative of the analytical solution. Moreover, the present method is boundary layer resolving as well as second-order uniformly convergent in the maximum norm. A comprehensive analysis has been given to prove the uniform convergence with respect to singular perturbation parameter. Several numerical examples are also given to demonstrate the efficiency of B-spline collocation method and to validate the theoretical aspects