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A Perturbative Approach for the Solution of Sturm-Liouville Problems

Received: 24 March 2023     Accepted: 27 April 2023     Published: 13 June 2023
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Abstract

Spectral analysis extends the theory of eigenvectors and eigenvalues of a square matrix to a broader theory involving operators. In particular, a branch of spectral analysis is devoted to Sturm-Liouville (SL) problems, which are eigenvalue problems for differential operators. In this study, we propose a numerical method to solve SL problems. This method uses a simple perturbative approach. Starting from an SL problem having differential operator L0 and known eigensystem, the proposed iterative algorithm considers M SL problems having differential operators Lm, m = 1, 2,..., M, such that Lm is a perturbation of Lm−1, and LM is the differential operator of the SL problem that we want to solve. Each step of this algorithm is based on the well-known Jacobi orthogonal component correction method, which acts on the refinement of approximated eigensystems. Moreover, the proposed method depends on the choice of L0 and the representation basis for the eigenfunctions, thus giving rise to different approximation schemes. We show the performance of the proposed method both in the solution of some selected SL problems and the refinement of approximated eigensystems computed by other numerical methods. In these numerical experiments, the perturbative method is compared with a classical approximation technique and the obtained results are strongly promising in terms of accuracy.

Published in Applied and Computational Mathematics (Volume 12, Issue 3)
DOI 10.11648/j.acm.20231203.11
Page(s) 46-54
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2023. Published by Science Publishing Group

Keywords

Sturm-Liouville Problem, Eigenvalue Problem, Perturbative Approach

References
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[3] E. A. Coddington and N. Levinson. Theory of ordinary differential equations. New York: McGraw-Hill Book Company, 1955.
[4] B. N. Datta. Numerical Linear Algebra and Applications. Pacific Grove, California: Brook / Cole Publishing Co., 1995.
[5] B. N. Datta. Numerical methods for linear control systems. California: Elsevier, 2003.
[6] N. Egidi and P. Maponi. “A Preliminary Study of a Perturbative Approach for the Computation of Matrix Eigenvalues”. In: Proceedings of MASCOT12 and ISGG12 Meeting on Applied Scientific Computing and Tools, Grid Generation, Approximation and Visualization. IMACS Series in Computational and Applied Mathematics.
[7] Mohammed M Eladly and Benjamin W Schafer. “Numerical and analytical study of stainless steel beam-to-column extended end-plate connections”. In: Engineering Structures 240 (2021), p. 112392.
[8] G. H. Golub and C. F. Van Loan. Matrix Computations (third edition). Baltimore: The Johns Hopkins University Press, 1996.
[9] V. E. Howle and L. N. Trefethen. “Eigenvalues and musical instruments”. In: Journal of Computational and Applied Mathematics. 135 (2001), pp. 23-40.
[10] D. J. Inman. Vibrations: Control, Measurement and Stability. Englewood Cliffs, NJ: Prentice Hall, 1989.
[11] C. G. J. Jacobi. “Über ein leichtes Verfahren die in der Theorie der Säcularstörungen vorkommenden Gleichungen numerisch aufzulösen”. In: J. reine angew. Math. 30 (1846), pp. 51-95.
[12] Mohammad Jafari and Alice Alipour. “Methodologies to mitigate windinduced vibration of tall buildings: A state- of-the-art review”. In: Journal of Building Engineering. 33 (2021), p. 101582.
[13] H. B. Keller. Numerical methods for two-point boundary- value problems. Waltham (Massachusetts): Blaisdell, 1968.
[14] Rami Ahmad El-Nabulsi and Alireza Khalili Golmankhaneh. “Generalized heat diffusion equations with variable coefficients and their fractalization from the Black-Scholes equation”. In: Communications in Theoretical Physics. 73.5 (2021), p. 055002.
[15] NAG, Fortran Library Manual, Mark 19, Numerical Algorithm Group. Oxford UK: 1999.
[16] Jean-Jacques Sinou and B Chomette. “Active vibration control and stability analysis of a time-delay system subjected to friction-induced vibration”. In: Journal of Sound and Vibration. 500 (2021), p. 116013.
[17] G. L. G. Sleijpen and H. A. van der Vorst. “A Jacobi-Davidson iteration method for linear eigenvalue problems”. In: SIAM J. Matrix Anal. Appl. 17 (1996), pp. 401-425.
[18] Mariana Domnica Stanciu et al. “Changing the vibrational behavior of the wooden thin arched plates - the maestro violins experimental study case”. In: Thin- Walled Structures. 174 (2022), p. 109042.
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    Nadaniela Egidi, Josephin Giacomini, Pierluigi Maponi. (2023). A Perturbative Approach for the Solution of Sturm-Liouville Problems. Applied and Computational Mathematics, 12(3), 46-54. https://doi.org/10.11648/j.acm.20231203.11

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    ACS Style

    Nadaniela Egidi; Josephin Giacomini; Pierluigi Maponi. A Perturbative Approach for the Solution of Sturm-Liouville Problems. Appl. Comput. Math. 2023, 12(3), 46-54. doi: 10.11648/j.acm.20231203.11

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    AMA Style

    Nadaniela Egidi, Josephin Giacomini, Pierluigi Maponi. A Perturbative Approach for the Solution of Sturm-Liouville Problems. Appl Comput Math. 2023;12(3):46-54. doi: 10.11648/j.acm.20231203.11

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  • @article{10.11648/j.acm.20231203.11,
      author = {Nadaniela Egidi and Josephin Giacomini and Pierluigi Maponi},
      title = {A Perturbative Approach for the Solution of Sturm-Liouville Problems},
      journal = {Applied and Computational Mathematics},
      volume = {12},
      number = {3},
      pages = {46-54},
      doi = {10.11648/j.acm.20231203.11},
      url = {https://doi.org/10.11648/j.acm.20231203.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20231203.11},
      abstract = {Spectral analysis extends the theory of eigenvectors and eigenvalues of a square matrix to a broader theory involving operators. In particular, a branch of spectral analysis is devoted to Sturm-Liouville (SL) problems, which are eigenvalue problems for differential operators. In this study, we propose a numerical method to solve SL problems. This method uses a simple perturbative approach. Starting from an SL problem having differential operator L0 and known eigensystem, the proposed iterative algorithm considers M SL problems having differential operators Lm, m = 1, 2,..., M, such that Lm is a perturbation of Lm−1, and LM is the differential operator of the SL problem that we want to solve. Each step of this algorithm is based on the well-known Jacobi orthogonal component correction method, which acts on the refinement of approximated eigensystems. Moreover, the proposed method depends on the choice of L0 and the representation basis for the eigenfunctions, thus giving rise to different approximation schemes. We show the performance of the proposed method both in the solution of some selected SL problems and the refinement of approximated eigensystems computed by other numerical methods. In these numerical experiments, the perturbative method is compared with a classical approximation technique and the obtained results are strongly promising in terms of accuracy.},
     year = {2023}
    }
    

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    T1  - A Perturbative Approach for the Solution of Sturm-Liouville Problems
    AU  - Nadaniela Egidi
    AU  - Josephin Giacomini
    AU  - Pierluigi Maponi
    Y1  - 2023/06/13
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    DO  - 10.11648/j.acm.20231203.11
    T2  - Applied and Computational Mathematics
    JF  - Applied and Computational Mathematics
    JO  - Applied and Computational Mathematics
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    EP  - 54
    PB  - Science Publishing Group
    SN  - 2328-5613
    UR  - https://doi.org/10.11648/j.acm.20231203.11
    AB  - Spectral analysis extends the theory of eigenvectors and eigenvalues of a square matrix to a broader theory involving operators. In particular, a branch of spectral analysis is devoted to Sturm-Liouville (SL) problems, which are eigenvalue problems for differential operators. In this study, we propose a numerical method to solve SL problems. This method uses a simple perturbative approach. Starting from an SL problem having differential operator L0 and known eigensystem, the proposed iterative algorithm considers M SL problems having differential operators Lm, m = 1, 2,..., M, such that Lm is a perturbation of Lm−1, and LM is the differential operator of the SL problem that we want to solve. Each step of this algorithm is based on the well-known Jacobi orthogonal component correction method, which acts on the refinement of approximated eigensystems. Moreover, the proposed method depends on the choice of L0 and the representation basis for the eigenfunctions, thus giving rise to different approximation schemes. We show the performance of the proposed method both in the solution of some selected SL problems and the refinement of approximated eigensystems computed by other numerical methods. In these numerical experiments, the perturbative method is compared with a classical approximation technique and the obtained results are strongly promising in terms of accuracy.
    VL  - 12
    IS  - 3
    ER  - 

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Author Information
  • School of Science and Technology, University of Camerino, Camerino, Italy

  • School of Science and Technology, University of Camerino, Camerino, Italy

  • School of Science and Technology, University of Camerino, Camerino, Italy

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