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

Higher order numerical methods for singular perturbation problems

Philosophiae Doctor - PhDIn recent years, there has been a great interest towards the higher order numerical methods for singularly perturbed problems. As compared to their lower order counterparts, they provide better accuracy with fewer mesh points. Construction and/or implementation of direct higher order methods is usually very complicated. Thus a natural choice is to use some convergence acceleration techniques, e.g., Richardson extrapolation, defect correction, etc. In this thesis, we will consider various classes of problems described by singularly perturbed ordinary and partial differential equations. For these problems, we design some novel numerical methods and attempt to increase their accuracy as well as the order of convergence. We also do the same for existing numerical methods in some instances. We find that, even though the Richardson extrapolation technique always improves the accuracy, it does not perform equally well when applied to different methods for certain classes of problems. Moreover, while in some cases it improves the order of convergence, in other cases it does not. These issues are discussed in this thesis for linear and nonlinear singularly perturbed ODEs as well as PDEs. Extrapolation techniques are analyzed thoroughly in all the cases, whereas the limitations of the defect correction approach for certain problems is indicated at the end of the thesis.South Afric

Application of the b-spline collocation method to a geometrically non-linear beam problem

Engineers are researching solutions to resolve many of today\u27s technical challenges. Numerical techniques are used to solve the mathematical models that arise in engineering problems. A numerical technique that is increasingly being used to solve mathematical models in engineering research is called the B-spline Collocation Method. The B-spline Collocation Method has a few distinct advantages over the Finite Element and Finite Difference Methods. The main advantage is that the B-spline Collocation Method efficiently provides a piecewise-continuous, closed form solution. Another advantage is that the B-spline Collocation Method procedure is very simple and easy to apply to many problems involving partial differential equations. The current research involves developing, and extensively documenting, a comprehensive, step-by-step procedure for applying the B-spline Collocation Method to the solution of Boundary Value problems. In addition, the current research involves applying the B-spline Collocation Method to solve the mathematical model that arises in the deflection of a geometrically nonlinear, cantilevered beam. The solution is then compared to a known solution found in the literature

Spline collocation method for singular perturbation problem

We introduce piecewise interpolating polynomials as an approximation for the driving terms in the numerical solution of the singularly perturbed differential equation. In this way we obtain the difference scheme which is second order accurate in uniform norm. We verify the convergence rate of presented scheme by numerical experiments

Cubic B-spline collocation method for coupled system of ordinary differential equations with various boundary conditions

This paper is concerned with collocation approach using cubic B-spline to solve coupled system of boundary value problems with various boundary conditions. The collocation equations are methodically derived using cubic B splines, for problems with Dirichlet data and an iterative method with assured convergence is described to solve the resulting system of algebraic equations. Problems with Cauchy or mixed boundary condition have been converted into series of Dirichlet problems using the bisection method. Nonlinear problem is linearized using quasilinearization to be handled by our method. Fourth order equation is converted into a coupled second order equations and solved by the proposed method . Several illustrative examples are presented with their error norms and order of convergence.Publisher's Versio

On Ɛ-uniform convergence of exponentially fitted methods

A class of methods constructed to numerically approximate solution of two-point singularly perturbed boundary value problems of the form $varepsilon u\u27\u27 + b u\u27 + c u = f$ use exponentials to mimic exponential behavior of the solution in the boundary layer(s). We refer to them as exponentially fitted methods. Such methods are usually exact on polynomials of certain degree and some exponential functions. Shortly, they are exact on exponential sums. It is often possible that consistency of the method follows from the convergence of interpolating function standing behind the method. Because of that, we consider interpolation error for exponential sums. A main result of the paper is an error bound for interpolation by exponential sum to the solution of singularly perturbed problem that does not depend on perturbation parameter $varepsilon$ when $varepsilon$ is small with the respect to mesh width. Numerical experiment implies that the use of dense mesh in the boundary layer for small meshwidth results with $varepsilon$-uniform convergence

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 problems using Haar wavelet collocation method

Abstract: In this paper, a collocation method based on Haar wavelets is proposed for the numerical solutions of singularly perturbed boundary value problems. The properties of the Haar wavelet expansions together with operational matrix of integration are utilized to convert the problems into systems of algebraic equations with unknown coefficients. To demonstrate the effectiveness and efficiency of the method various benchmark problems are implemented and the comparisons are given with other methods existing in the recent literature. The demonstrated results confirm that the proposed method is considerably efficient, accurate, simple, and computationally attractive

On Ɛ-uniform convergence of exponentially fitted methods

A class of methods constructed to numerically approximate solution of two-point singularly perturbed boundary value problems of the form $varepsilon u\u27\u27 + b u\u27 + c u = f$ use exponentials to mimic exponential behavior of the solution in the boundary layer(s). We refer to them as exponentially fitted methods. Such methods are usually exact on polynomials of certain degree and some exponential functions. Shortly, they are exact on exponential sums. It is often possible that consistency of the method follows from the convergence of interpolating function standing behind the method. Because of that, we consider interpolation error for exponential sums. A main result of the paper is an error bound for interpolation by exponential sum to the solution of singularly perturbed problem that does not depend on perturbation parameter $varepsilon$ when $varepsilon$ is small with the respect to mesh width. Numerical experiment implies that the use of dense mesh in the boundary layer for small meshwidth results with $varepsilon$-uniform convergence

Spline collocation method for singular perturbation problem

We introduce piecewise interpolating polynomials as an approximation for the driving terms in the numerical solution of the singularly perturbed differential equation. In this way we obtain the difference scheme which is second order accurate in uniform norm. We verify the convergence rate of presented scheme by numerical experiments