An initial-value approach for solving singularly perturbed two-point boundary value problems

This paper presents a precise method for solving singularly perturbed boundary-value problems with the boundary layer at one end. The method divides the interval evenly and gives a set of algebraic equations in a matrix form by the precise integration relationship of each segment. Substituting the boundary conditions into the algebraic equations, the coefficient matrix can be transformed to the block tridiagonal matrix. Considering the nature of the problem, an efficient reduction method is given for solving singular perturbation problems. Since the precise integration relationship introduces no discrete error in the discrete process, the present method has high precision. Numerical examples show the validity of the present method.

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“…In the following part, we give an efficient method to solve (8). Here, we employ a reduction procedure for m = 2 M + 1 (M =0, 1, 2, · · · ).…”

Section: Precise Methods For Solving Singularly Perturbed Boundary-valmentioning

confidence: 99%

“…Various methods were developed during the last few years. The notable methods are asymptotic expansion approximations [1][2] , finite-difference methods [3][4][5] , finite element methods [6] , boundary-value techniques [7] , initial-value techniques [8][9] , spline techniques [10][11] , and so on.…”

Section: Introductionmentioning

confidence: 99%

Precise integration method for solving singular perturbation problems

Cheung

Sheshenin

2010

Appl. Math. Mech.-Engl. Ed.

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“…Reddy and Chakravarthy [14] presented a method of reduction of order for solving linear and a class of nonlinear SPBVPs. Then Reddy and Chakravarthy [15] presented three initial-value problems instead of the linear second-order SPBVP. Habib and El-Zahar [16] considered a class of nonlinear SPBVP which was replaced by an asymptotically equivalent first order IVP and was solved using locally exact integration.…”

Section: Introductionmentioning

confidence: 99%

An Algorithm for Solving Second Order Nonlinear Singular Perturbation Boundary Value Problems

El‐Zahar

Hassanein

2011

J. of Mod. Meth. in Numer. Math.

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We consider nonlinear singular perturbation problems of the formwith a boundary layer at one end point. Using singular perturbation analysis the original problem is replaced by an asymptotically equivalent first-order initial-value problem (IVP). Then, a variable step size initial-value algorithm is applied to solve this IVP in the boundary layer region. The algorithm is derived based on the piecewise linearization of the obtained IVP and its analytical integration over a non-uniform mesh. The step size is determined directly according to the variation of the solution within a step in the boundary layer region and results in two-term recurrence relation with controlled step size. Some numerical examples are given to illustrate the validity of the given method. It is observed that the present method approximates the exact solution very well.

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“…Y.N. Reddy, P. Pramod Chakravarthy [14] treated the linear case by three equivalents IVP. Habib and El-Zahar [15] considered a semi-linear SPP which was integrated to obtain a scalar first-order initial-value problem.…”

Section: Introductionmentioning

confidence: 99%

Parallel Initial-Value Algorithm for Quasilinear Shock Problems with Turning Points

Habib¹,

El‐Zahar²

2008

The International Conference on Mathematics and Engineering Phy

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In this paper, a parallel initial value algorithm is presented for quasilinear stationary shock problems with turning points exhibiting two-boundary layers or an internal layer. The method is based on obtaining two independent asymptotically equivalent first-order singularly-perturbed initial-value problems (SPIVPs) of the original problem. The error is estimated to be of order ε . The two SPIVPs are modified to obtain two boundary-layer correction problems. These non-stiff initial-value problems are solved simultaneously in parallel process using (RKV45) non-stiff code integrator. The obtained solutions are combined to approximate the solution of the original problem. Numerical experiments indicate the high accuracy and the efficiency of the method. Furthermore, the accuracy of numerical results improves as the small parameter ε tends to zero Keywords: quasilinear singular perturbation problems; turning points problems; two boundary layer; internal layer; Initial value methods, parallel algorithms.

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An initial-value approach for solving singularly perturbed two-point boundary value problems

Cited by 29 publications

References 5 publications

Precise integration method for solving singular perturbation problems

Precise integration method for solving singular perturbation problems

An Algorithm for Solving Second Order Nonlinear Singular Perturbation Boundary Value Problems

Parallel Initial-Value Algorithm for Quasilinear Shock Problems with Turning Points

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