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A Boundary Shape Function Method for Computing Eigenvalues and Eigenfunctions of Sturm–Liouville Problems

Author

Listed:
  • Chein-Shan Liu

    (Center of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung 202301, Taiwan)

  • Jiang-Ren Chang

    (Department of Systems Engineering and Naval Architecture, National Taiwan Ocean University, Keelung 202301, Taiwan)

  • Jian-Hung Shen

    (Department of Marine Engineering, National Taiwan Ocean University, Keelung 202301, Taiwan)

  • Yung-Wei Chen

    (Department of Marine Engineering, National Taiwan Ocean University, Keelung 202301, Taiwan)

Abstract

In the paper, we transform the general Sturm–Liouville problem (SLP) into two canonical forms: one with the homogeneous Dirichlet boundary conditions and another with the homogeneous Neumann boundary conditions. A boundary shape function method (BSFM) was constructed to solve the SLPs of these two canonical forms. Owing to the property of the boundary shape function, we could transform the SLPs into an initial value problem for the new variable with initial values that were given definitely. Meanwhile, the terminal value at the right boundary could be entirely determined by using a given normalization condition for the uniqueness of the eigenfunction. In such a manner, we could directly determine the eigenvalues as the intersection points of an eigenvalue curve to the zero line, which was a horizontal line in the plane consisting of the zero values of the target function with respect to the eigen-parameter. We employed a more delicate tuning technique or the fictitious time integration method to solve an implicit algebraic equation for the eigenvalue curve. We could integrate the Sturm–Liouville equation using the given initial values to obtain the associated eigenfunction when the eigenvalue was obtained. Eight numerical examples revealed a great advantage of the BSFM, which easily obtained eigenvalues and eigenfunctions with the desired accuracy.

Suggested Citation

  • Chein-Shan Liu & Jiang-Ren Chang & Jian-Hung Shen & Yung-Wei Chen, 2022. "A Boundary Shape Function Method for Computing Eigenvalues and Eigenfunctions of Sturm–Liouville Problems," Mathematics, MDPI, vol. 10(19), pages 1-22, October.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:19:p:3689-:d:936682
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    References listed on IDEAS

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    1. Qura Tul Ain & T. Sathiyaraj & Shazia Karim & Muhammad Nadeem & Patrick Kandege Mwanakatwe & C. Rajivganthi, 2022. "ABC Fractional Derivative for the Alcohol Drinking Model using Two-Scale Fractal Dimension," Complexity, Hindawi, vol. 2022, pages 1-11, June.
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    Cited by:

    1. Liu, Chein-Shan & Chen, Yung-Wei & Chang, Chih-Wen, 2024. "Precise eigenvalues in the solutions of generalized Sturm–Liouville problems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 217(C), pages 354-373.
    2. Liu, Chein-Shan & Li, Botong, 2023. "Solving Sturm–Liouville inverse problems by an orthogonalized enhanced boundary function method and a product formula for symmetric potential," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 210(C), pages 640-660.

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