Piecewise reproducing kernel-based symmetric collocation approach for linear stationary singularly perturbed problems

F. Z. Geng; F. Z. Geng

doi:10.3934/math.2020385

AIMS Mathematics

2020, Volume 5, Issue 6: 6020-6029. doi: 10.3934/math.2020385

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Research article

Piecewise reproducing kernel-based symmetric collocation approach for linear stationary singularly perturbed problems

F. Z. Geng ^{1,2
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1.
School of Mathematics and Statistics, Changshu Institute of Technology, Changshu, Jiangsu 215500, P R China
2.
Department of Mathematics, Nanjing University, Nanjing, Jiangsu 210093, PR China

Received: 05 April 2020 Accepted: 03 July 2020 Published: 23 July 2020
MSC : 65L60, 65R20

The aim of this paper is to develop an accurate symmetric collocation scheme for a class of linear stationary singular perturbation problems with two boundary layers. To adapt to the character of solutions, piecewise reproducing kernels is constructed. In the boundary layers intervals, inverse multiquadrics kernel function is employed. In the regular interval, exponential kernel function is used. On the basis of the piecewise reproducing kernels, a new symmetric collocation technique is presented for the considered linear stationary singular perturbation problems. Results of numerical tests illustrate that our method is easy to implement and is uniformly effective for any small ε.
- reproducing kernel method,
- singular perturbation problems,
- piecewise kernels
Citation: F. Z. Geng. Piecewise reproducing kernel-based symmetric collocation approach for linear stationary singularly perturbed problems[J]. AIMS Mathematics, 2020, 5(6): 6020-6029. doi: 10.3934/math.2020385

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Abstract

The aim of this paper is to develop an accurate symmetric collocation scheme for a class of linear stationary singular perturbation problems with two boundary layers. To adapt to the character of solutions, piecewise reproducing kernels is constructed. In the boundary layers intervals, inverse multiquadrics kernel function is employed. In the regular interval, exponential kernel function is used. On the basis of the piecewise reproducing kernels, a new symmetric collocation technique is presented for the considered linear stationary singular perturbation problems. Results of numerical tests illustrate that our method is easy to implement and is uniformly effective for any small ε.

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