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Approximation of Eigenvalues of Sturm‐Liouville Problems by Using Hermite Interpolation

Author

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  • M. M. Tharwat
  • S. M. Al-Harbi

Abstract

Eigenvalue problems with eigenparameter appearing in the boundary conditions usually have complicated characteristic determinant where zeros cannot be explicitly computed. In this paper, we use the derivative sampling theorem “Hermite interpolations” to compute approximate values of the eigenvalues of Sturm‐Liouville problems with eigenvalue parameter in one or two boundary conditions. We use recently derived estimates for the truncation and amplitude errors to compute error bounds. Also, using computable error bounds, we obtain eigenvalue enclosures. Also numerical examples, which are given at the end of the paper, give comparisons with the classical sinc method and explain that the Hermite interpolations method gives remarkably better results.

Suggested Citation

  • M. M. Tharwat & S. M. Al-Harbi, 2013. "Approximation of Eigenvalues of Sturm‐Liouville Problems by Using Hermite Interpolation," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    • Handle: RePEc:wly:jnlaaa:v:2013:y:2013:i:1:n:412028
    DOI: 10.1155/2013/412028
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    References listed on IDEAS

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    1. D. Baleanu & A. H. Bhrawy & T. M. Taha, 2013. "Two Efficient Generalized Laguerre Spectral Algorithms for Fractional Initial Value Problems," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    2. Ahmad Imani & Azim Aminataei & Ali Imani, 2011. "Collocation Method via Jacobi Polynomials for Solving Nonlinear Ordinary Differential Equations," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2011, pages 1-11, May.
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    4. D. Baleanu & A. H. Bhrawy & T. M. Taha, 2013. "A Modified Generalized Laguerre Spectral Method for Fractional Differential Equations on the Half Line," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    5. D. Baleanu & A. H. Bhrawy & T. M. Taha, 2013. "Two Efficient Generalized Laguerre Spectral Algorithms for Fractional Initial Value Problems," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-10, June.
    6. M. M. Tharwat & A. H. Bhrawy & A. S. Alofi, 2013. "Computing Eigenvalues of Discontinuous Sturm-Liouville Problems with Eigenparameter in All Boundary Conditions Using Hermite Approximation," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-14, August.
    7. Ferial Ghaemi & Robiah Yunus & Ali Ahmadian & Soheil Salahshour & Mohamed Suleiman & Shanti Faridah Saleh, 2013. "Application of Fuzzy Fractional Kinetic Equations to Modelling of the Acid Hydrolysis Reaction," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    8. Ferial Ghaemi & Robiah Yunus & Ali Ahmadian & Soheil Salahshour & Mohamed Suleiman & Shanti Faridah Saleh, 2013. "Application of Fuzzy Fractional Kinetic Equations to Modelling of the Acid Hydrolysis Reaction," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-19, September.
    9. M. M. Tharwat & A. H. Bhrawy & A. S. Alofi, 2013. "Computing Eigenvalues of Discontinuous Sturm‐Liouville Problems with Eigenparameter in All Boundary Conditions Using Hermite Approximation," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
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