scholarly journals A comparison between MMAE and SCEM for solving singularly perturbed linear boundary layer problems

Filomat ◽  
2019 ◽  
Vol 33 (7) ◽  
pp. 2135-2148
Author(s):  
Süleyman Cengizci
Keyword(s):  
Boundary Layer ◽  
Asymptotic Expansions ◽  
Present Method ◽  
Numerical Experiments ◽  
Expansion Method ◽  
Singularly Perturbed ◽  
Linear Boundary ◽  
Point Boundary ◽  
Matching Process ◽  
Boundary Layer Problems

In this study, we propose an efficient method so-called Successive Complementary Expansion Method (SCEM), that is based on generalized asymptotic expansions, for approximating to the solutions of singularly perturbed two-point boundary value problems. In this easy-applicable method, in contrast to the well-known method the Method of Matched Asymptotic Expansions (MMAE), any matching process is not required to obtain uniformly valid approximations. The key point: A uniformly valid approximation is adopted first, and complementary functions are obtained imposing the corresponding boundary conditions. An illustrative and two numerical experiments are provided to show the implementation and numerical properties of the present method. Furthermore, MMAE results are also obtained in order to compare the numerical robustnesses of the methods.

An Approximate Solution of the Laminar Boundary Layer on a Flat Plate with Uniform Suction

1955 ◽  
Vol 59 (538) ◽  
pp. 697-698
Author(s):  
S. J. Peerless ◽  
D. B. Spalding
Keyword(s):  
Boundary Layer ◽  
Boundary Conditions ◽  
Present Method ◽  
Integral Method ◽  
Boundary Layer Thickness ◽  
Numerical Solutions ◽  
Approximate Methods ◽  
Boundary Layer Problems ◽  
Momentum Integral ◽  
Similar Solutions

Boundary layer problems may be divided into two classes: (a) those for which similar solutions can be found, i.e. where the boundary conditions are such that similar profiles differing only in scale factor exist at different sections; and (b) those where the boundary conditions do not effect similarity, so that the development of the boundary layer must be calculated in stages. The latter class are known as “continuation problems,” and very few numerical solutions have been obtained because of the labour involved.Approximate methods of solving continuation problems are known, using the Karman momentum integral method (e.g. Ref. 1) or variants. Some of these methods make use of velocity profiles calculated for “similar” boundary layers. This note presents a new approximate method which uses “similar” profiles but avoids using the momentum integral. Instead of characterising the boundary layer thickness by the “momentum thickness,” which needs to be calculated yet is of less direct interest, the wall shear stress is used; this stress usually has to be calculated in any case and the present method is therefore comparatively simple.

scholarly journals The Solution of a Class of Singularly Perturbed Two-Point Boundary Value Problems by the Iterative Reproducing Kernel Method

2012 ◽  
Vol 2012 ◽  
pp. 1-7 ◽  
Author(s):  
Zhiyuan Li ◽  
YuLan Wang ◽  
Fugui Tan ◽  
Xiaohui Wan ◽  
Tingfang Nie
Keyword(s):  
Singular Perturbation ◽  
Boundary Layers ◽  
Present Method ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Singularly Perturbed ◽  
Point Boundary ◽  
Reproducing Kernel Method ◽  
Numerical Examples ◽  
Perturbation Problems

In (Wang et al., 2011), we give an iterative reproducing kernel method (IRKM). The main contribution of this paper is to use an IRKM (Wang et al., 2011), in singular perturbation problems with boundary layers. Two numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by the method indicate that the method is simple and effective.

scholarly journals MULTIPLE SOLUTIONS OF A GENERALIZED SINGULAR PERTURBED BRATU PROBLEM

2012 ◽  
Vol 22 (04) ◽  
pp. 1250095 ◽  
Author(s):  
TAOUFIK BAKRI ◽  
YURI A. KUZNETSOV ◽  
FERDINAND VERHULST ◽  
EUSEBIUS DOEDEL
Keyword(s):  
Boundary Value Problems ◽  
Asymptotic Expansions ◽  
Multiple Solutions ◽  
Boundary Value ◽  
Slow Manifold ◽  
Singularly Perturbed ◽  
Point Boundary ◽  
Numerical Bifurcation ◽  
Manifold Theory ◽  
Bratu Problem

Nonlinear two-point boundary value problems (BVPs) may have none or more than one solution. For the singularly perturbed two-point BVP εu″ + 2u′ + f(u) = 0, 0 < x < 1, u(0) = 0, u(1) = 0, a condition is given to have one and only one solution; also cases of more solutions have been analyzed. After attention to the form and validity of the corresponding asymptotic expansions, partially based on slow manifold theory, we reconsider the BVP within the framework of small and large values of the parameter. In the case of a special nonlinearity, numerical bifurcation patterns are studied that improve our understanding of the multivaluedness of the solutions.

scholarly journals Numerical Treatment of Singularly Perturbed Two-Point Boundary Value Problems by Using Differential Transformation Method

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Nurettin Doğan ◽  
Vedat Suat Ertürk ◽  
Ömer Akın
Keyword(s):  
Boundary Value Problems ◽  
Present Method ◽  
Boundary Value ◽  
Transformation Method ◽  
High Accuracy ◽  
Singularly Perturbed ◽  
Point Boundary ◽  
Transform Method ◽  
Differential Transform ◽  
First Time

Differential transform method is adopted, for the first time, for solving linear singularly perturbed two-point boundary value problems. Four numerical examples are given to demonstrate the effectiveness of the present method. Results show that the numerical scheme is very effective and convenient for solving a large number of linear singularly perturbed two-point boundary value problems with high accuracy.

scholarly journals A Numerical Method for a Singularly Perturbed Three-Point Boundary Value Problem

2010 ◽  
Vol 2010 ◽  
pp. 1-17 ◽  
Author(s):  
Musa Çakır ◽  
Gabil M. Amiraliyev
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Numerical Method ◽  
Numerical Experiments ◽  
Diffusion Problem ◽  
Singularly Perturbed ◽  
Point Boundary ◽  
Convection Diffusion ◽  
First Order ◽  
Type Boundary

The purpose of this paper is to present a uniform finite difference method for numerical solution of nonlinear singularly perturbed convection-diffusion problem with nonlocal and third type boundary conditions. The numerical method is constructed on piecewise uniform Shishkin type mesh. The method is shown to be convergent, uniformly in the diffusion parameterε, of first order in the discrete maximum norm. Some numerical experiments illustrate in practice the result of convergence proved theoretically.

Numerical solutions of singularly perturbed convection-diffusion problems

2014 ◽  
Vol 24 (6) ◽  
pp. 1268-1274 ◽  
Author(s):  
Fazhan Geng ◽  
Suping Qian ◽  
Shuai Li
Keyword(s):  
Asymptotic Expansion ◽  
Present Method ◽  
Numerical Solutions ◽  
Variational Iteration Method ◽  
Expansion Method ◽  
Singularly Perturbed ◽  
Content Type ◽  
Convection Diffusion ◽  
Asymptotic Expansion Method ◽  
Diffusion Problems

Purpose – The purpose of this paper is to find an effective numerical method for solving singularly perturbed convection-diffusion problems. Design/methodology/approach – The present method is based on the asymptotic expansion method and the variational iteration method (VIM). First a zeroth order asymptotic expansion for the solution of the given singularly perturbed convection-diffusion problem is constructed. Then the reduced terminal value problem is solved by using the VIM. Findings – Two numerical examples are introduced to show the validity of the present method. Obtained numerical results show that the present method can provide very accurate analytical approximate solutions not only in the boundary layer, but also away from the layer. Originality/value – The combination of the asymptotic expansion method and the VIM is applied to singularly perturbed convection-diffusion problems.

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