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B-spline collocation and self-adapting differential evolution (jDE) algorithm for a singularly perturbed convection–diffusion problem

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Abstract

Many numerical methods applied on a Shishkin mesh are very popular in solving the singularly perturbed problems. However, few approaches are used to obtain the Shishkin mesh transition parameter. Thus, in this paper, we first use the cubic B-spline collocation method on a Shishkin mesh to solve the singularly perturbed convection–diffusion problem with two small parameters. Then, we transform the Shishkin mesh transition parameter selection problem into a nonlinear unconstrained optimization problem which is solved by using the self-adapting differential evolution (jDE) algorithm. To verify the performance of our presented method, a numerical example is employed. It is shown from the experiment results that our approach is efficient. Compared with other evolutionary algorithms, the jDE algorithm performs better and with more stability.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (11301044, 11401054, 61662090, 11461011), the general Project of Hunan provincial education department (14C0047), the Natural Science Foundation of Guizhou Provincial Education Department (No. KY[2016]018), the Scientific Research Fund of Hunan Provincial Education Department (No. 13C333), the Doctoral Foundation of Zunyi Normal College (No. BS[2015]13), the open fund of Key Laboratory of Guangxi High Schools for Complex System and Computational Intelligence (No. 15CI03D), Natural Science Foundation of Guangxi Education Department (No. ZD2014080).

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Authors and Affiliations

  1. School of Mathematics and Computing Science, Changsha University of Science and Technology, Changsha, 410004, Hunan, China

    Xu-Qiong Luo

  2. School of Mathematics and Statistics, Guangxi Teachers Education University, Nanning, 530001, China

    Li-Bin Liu & Guangqing Long

  3. Department of Information Engineering, Zunyi Normal College, Zunyi, 563002, Guizhou, China

    Aijia Ouyang

  4. Guangxi High School Key Laboratory of Complex System and Computational Intelligence, Nanning, 530006, China

    Aijia Ouyang

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  1. Xu-Qiong Luo
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  2. Li-Bin Liu
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  3. Aijia Ouyang
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  4. Guangqing Long
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Correspondence to Li-Bin Liu.

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Communicated by V. Loia.

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Luo, XQ., Liu, LB., Ouyang, A. et al. B-spline collocation and self-adapting differential evolution (jDE) algorithm for a singularly perturbed convection–diffusion problem. Soft Comput 22, 2683–2693 (2018). https://doi.org/10.1007/s00500-017-2523-9

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