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A Pseudo-Spectral Fourier Collocation Method for Inhomogeneous Elliptical Inclusions with Partial Differential Equations

Author

Listed:
  • Xiao Wang

    (School of Mathematics and Statistics, Qingdao University, Qingdao 266071, China)

  • Juan Wang

    (School of Mathematics and Statistics, Qingdao University, Qingdao 266071, China)

  • Xin Wang

    (School of Foreign Languages, Qingdao University, Qingdao 266071, China)

  • Chujun Yu

    (Directly Affiliated College, Shandong Open University, Jinan 250014, China)

Abstract

Inhomogeneous elliptical inclusions with partial differential equations have aroused appreciable concern in many disciplines. In this paper, a pseudo-spectral collocation method, based on Fourier basis functions, is proposed for the numerical solutions of two- (2D) and three-dimensional (3D) inhomogeneous elliptic boundary value problems. We describe how one can improve the numerical accuracy by making some extra “reconstruction techniques” before applying the traditional Fourier series approximation. After the particular solutions have been obtained, the resulting homogeneous equation can then be calculated using various boundary-type methods, such as the method of fundamental solutions (MFS). Using Fourier basis functions, one does not need to use large matrices, making accrual computations relatively fast. Three benchmark numerical examples involving Poisson, Helmholtz, and modified-Helmholtz equations are presented to illustrate the applicability and accuracy of the proposed method.

Suggested Citation

  • Xiao Wang & Juan Wang & Xin Wang & Chujun Yu, 2022. "A Pseudo-Spectral Fourier Collocation Method for Inhomogeneous Elliptical Inclusions with Partial Differential Equations," Mathematics, MDPI, vol. 10(3), pages 1-18, January.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:3:p:296-:d:727966
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    Citations

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    Cited by:

    1. Musawenkhosi Patson Mkhatshwa & Melusi Khumalo, 2022. "Trivariate Spectral Collocation Approach for the Numerical Solution of Three-Dimensional Elliptic Partial Differential Equations," Mathematics, MDPI, vol. 10(13), pages 1-23, June.
    2. Jianliang Chen & Qinghai Zhao & Liang Zhang, 2022. "Multi-Material Topology Optimization of Thermo-Elastic Structures with Stress Constraint," Mathematics, MDPI, vol. 10(8), pages 1-18, April.
    3. Wang, Taishan & Zhang, Junlong & You, Li & Zeng, Xueting & Ma, Yuan & Li, Yongping & Huang, Guohe, 2023. "Optimal design of two-dimensional water trading considering hybrid “three waters”-government participation for an agricultural watershed," Agricultural Water Management, Elsevier, vol. 288(C).
    4. Zhenjie Liang & Futian Weng & Yuanting Ma & Yan Xu & Miao Zhu & Cai Yang, 2022. "Measurement and Analysis of High Frequency Assert Volatility Based on Functional Data Analysis," Mathematics, MDPI, vol. 10(7), pages 1-11, April.
    5. Zhang, Junlong & Li, Yongping & You, Li & Huang, Guohe & Xu, Xiaomei & Wang, Xiaoya, 2022. "Optimizing effluent trading and risk management schemes considering dual risk aversion for an agricultural watershed," Agricultural Water Management, Elsevier, vol. 269(C).
    6. Liang Zhang & Qinghai Zhao & Jianliang Chen, 2022. "Reliability-Based Topology Optimization of Thermo-Elastic Structures with Stress Constraint," Mathematics, MDPI, vol. 10(7), pages 1-22, March.

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