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Fast and Accurate Calculations of Fourth-Order Non-self-adjoint Sturm–Liouville Eigenvalues for Problems in Physics and Engineering

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Abstract

Finding the eigenvalues of non-self-adjoint boundary value problems is a very difficult task, especially when the problems are of higher-order or when high-index eigenvalues are required. In fact, the lack of oscillation theorems of non-self-adjoint problems as well as the distribution and scatteration of the eigenvalues in the complex plane, makes the computational process of the eigenvalues a strong and difficult challenge. In this paper, we propose a fast and accurate numerical technique based on the Chebyshev spectral collocation method for approximating the eigenvalues of fourth-order non-self-adjoint Sturm–Liouville boundary value problems. This technique transforms the non-self-adjoint problem into a generalized eigenvalue problem by employing the spectral differentiation matrices to determine the derivatives of Chebyshev polynomials at the Chebyshev–Gauss–Lobatto nodes. The excellent performance of the suggested technique is investigated by considering three numerical examples among which singular ones. The numerical results and comparison with other methods indicate that this technique is easy to implement, considerably accurate and requires less computational costs even when high-index eigenvalues are required.

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Acknowledgements

I would like to express my gratitude and appreciation for Editorial Submission Advisors at the Springer Nature Transfer Desk for their valuable help. The author also thank Editors and Reviewers of International Journal of Applied and Computational Mathematics for their review efforts.

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  1. Department of Mathematics, University of Aden, Khormaksar, P.O. Box 6312, Aden, Yemen

    Anis Haytham Saleh Taher

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Taher, A.H.S. Fast and Accurate Calculations of Fourth-Order Non-self-adjoint Sturm–Liouville Eigenvalues for Problems in Physics and Engineering. Int. J. Appl. Comput. Math 7, 212 (2021). https://doi.org/10.1007/s40819-021-01151-x

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