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A Quadratic B-Spline Collocation Method for a Singularly Perturbed Semilinear Reaction–Diffusion Problem with Discontinuous Source Term

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Abstract

In this paper, a quadratic B-spline collocation method is developed to solve a singularly perturbed semilinear reaction–diffusion problem with a discontinuous source term. The discontinuous source term leads to a jump in the second-order derivative of the exact solution at the discontinuous point. A quadratic B-spline collocation method on a Shishkin-type mesh is used to discretized the singularly perturbed problem on the left and right sides of the discontinuous point, respectively. The collocation equations at the discontinuous point are obtained using the conditions satisfied at the discontinuous point. It is shown that the scheme is stable and almost second-order uniformly convergent. Numerical experiments support the theoretical results.

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Acknowledgements

We would like to thank the anonymous reviewer for some suggestions for the improvement of this paper.

Funding

The work was supported by Ningbo Municipal Natural Science Foundation (Grant Nos. 2023J302, 2021J179) and Zhejiang Provincial Natural Science Foundation of China (Grant Nos. LGF22H260003, LTGY23H240002).

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Authors and Affiliations

  1. Institute of Mathematics, Zhejiang Wanli University, Ningbo, 315100, People’s Republic of China

    Zhongdi Cen, Jian Huang & Aimin Xu

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  1. Zhongdi Cen
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  2. Jian Huang
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  3. Aimin Xu
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Contributions

The first author carried out the literature review, designed the numerical algorithm, and wrote the main manuscript text. The second author conducted numerical experiments and participated in designing the numerical algorithm, and the third author participated in analyzing the error and designing the numerical algorithm. All authors read and approved the final manuscript.

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Correspondence to Jian Huang.

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Cen, Z., Huang, J. & Xu, A. A Quadratic B-Spline Collocation Method for a Singularly Perturbed Semilinear Reaction–Diffusion Problem with Discontinuous Source Term. Mediterr. J. Math. 20, 269 (2023). https://doi.org/10.1007/s00009-023-02473-4

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  • DOI: https://doi.org/10.1007/s00009-023-02473-4

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