AN EFFECTIVE SHOOTING PIECEWISE ANALYTICAL INTEGRATION METHOD FOR SINGULAR PERTURBATION TWO-POINT BOUNDARY VALUE PROBLEMS

Haifa S. Al-Juaydi; Essam R. El-Zahar

Haifa S. Al-Juaydi Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, P.O. Box 83, Al-Kharj 11942, Saudi Arabia
Essam R. El-Zahar Department of Basic Engineering Science, Faculty of Engineering, Menoufia University, Shebin El-Kom 32511, Egypt; Department of Basic Engineering Science, Faculty of Engineering, Menoufia University, Shebin El-Kom 32511, Egypt

Article ID: 2400

Keywords: singular perturbation, boundary value problems, shooting method, locally exact integration, stability, asymptotic approximations.

Abstract

In this paper, an effective semi-analytical-numerical method is proposed for solving singular perturbation two-point boundary value problems (SPBVPs). Firstly, the original problem is replaced by an equivalent singular perturbation initial value problems (SPIVPs) of first-order with an unknown initial condition that can be determined iteratively using shooting method. Then, an adaptive one-step explicit piecewise analytical integration scheme over a special non-uniform mesh is presented to integrate these SPIVPs. The accuracy and stability properties of the scheme are investigated and shown to yield at least second-order of accuracy and L-stability property. A good estimation of the missed initial condition is obtained and suggested as a starting initial guess to ensure accelerated convergence of the shooting method. To demonstrate the applicability of the method, we have applied it to linear and nonlinear test problems at different values of the perturbation parameter. The method can be extended to higher-order SPBVPs. We have applied it to the well-known third-order Blasius’ viscous flow problem for a large suction case. The results indicate that the method approximates the solution very well not only over the boundary layer region but also overall the problem domain. Moreover, the method is more accurate and has a higher computational efficiency compared to other methods in the literature.

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