A Perturbative Approach for the Solution of Sturm-Liouville Problems

Nadaniela Egidi; Josephin Giacomini; Pierluigi Maponi

doi:doi:10.11648/j.acm.20231203.11

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A Perturbative Approach for the Solution of Sturm-Liouville Problems

Nadaniela Egidi, Josephin Giacomini, Pierluigi Maponi

Published in Applied and Computational Mathematics (Volume 12, Issue 3)

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 L_m, m = 1, 2,..., M, such that L_m is a perturbation of L_m₋₁, and L_M 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), 2024. Published by Science Publishing Group

Keywords

Sturm-Liouville Problem, Eigenvalue Problem, Perturbative Approach

References

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[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.
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[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.
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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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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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TY  - JOUR
T1  - A Perturbative Approach for the Solution of Sturm-Liouville Problems
AU  - Nadaniela Egidi
AU  - Josephin Giacomini
AU  - Pierluigi Maponi
Y1  - 2023/06/13
PY  - 2023
N1  - https://doi.org/10.11648/j.acm.20231203.11
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
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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

Nadaniela Egidi

School of Science and Technology, University of Camerino, Camerino, Italy
Josephin Giacomini

School of Science and Technology, University of Camerino, Camerino, Italy
Pierluigi Maponi

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

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APA Style

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

Copy | Download

@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}
}

Copy | Download

TY  - JOUR
T1  - A Perturbative Approach for the Solution of Sturm-Liouville Problems
AU  - Nadaniela Egidi
AU  - Josephin Giacomini
AU  - Pierluigi Maponi
Y1  - 2023/06/13
PY  - 2023
N1  - https://doi.org/10.11648/j.acm.20231203.11
DO  - 10.11648/j.acm.20231203.11
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 46
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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