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Soft Computing

, Volume 22, Issue 8, pp 2683–2693 | Cite as

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

  • Xu-Qiong Luo
  • Li-Bin Liu
  • Aijia Ouyang
  • Guangqing Long
Methodologies and Application

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.

Keywords

B-spline collocation method Self-adapting differential evolution Singularly perturbed Optimization problem Shishkin mesh 

Notes

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).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Xu-Qiong Luo
    • 1
  • Li-Bin Liu
    • 2
  • Aijia Ouyang
    • 3
    • 4
  • Guangqing Long
    • 2
  1. 1.School of Mathematics and Computing ScienceChangsha University of Science and TechnologyChangshaChina
  2. 2.School of Mathematics and StatisticsGuangxi Teachers Education UniversityNanningChina
  3. 3.Department of Information EngineeringZunyi Normal CollegeZunyiChina
  4. 4.Guangxi High School Key Laboratory of Complex System and Computational IntelligenceNanningChina

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