Numerical Computation of Spectral Solutions for Sturm-Liouville Eigenvalue Problems

Article Sidebar

PDF
DOI: https://doi.org/10.28924/2291-8639-21-2023-86
Published: Aug 9, 2023
Cite as:
Sameh Gana, Numerical Computation of Spectral Solutions for Sturm-Liouville Eigenvalue Problems, Int. J. Anal. Appl., 21 (2023), 86.

Main Article Content

Sameh Gana

Abstract

This paper focuses on the study of Sturm-Liouville eigenvalue problems. In the classical Chebyshev collocation method, the Sturm-Liouville problem is discretized to a generalized eigenvalue problem where the functions represent interpolants in suitably rescaled Chebyshev points. We are concerned with the computation of high-order eigenvalues of Sturm-Liouville problems using an effective method of discretization based on the Chebfun software algorithms with domain truncation. We solve some numerical Sturm-Liouville eigenvalue problems and demonstrate the efficiency of computations.

Article Details

References

  1. S. Flügge, Practical Quantum Mechanics, Springer, Berlin, 1994. https://doi.org/10.1007/978-3-642-61995-3.
  2. J.D Pryce, Numerical Solution of Sturm–liouville Problems, Oxford University Press, Oxford, 1993. https://orca.cardiff.ac.uk/id/eprint/101057.
  3. G.W. Hanson, A.B. Yakovlev, Operator Theory for Electromagnetics, Springer, New York, 2002. https://doi.org/10.1007/978-1-4757-3679-3.
  4. C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Methods in Fluid Dynamics, Springer, Berlin, 1988. https://doi.org/10.1007/978-3-642-84108-8.
  5. B. Fornberg, A Practical Guide to Pseudospectral Methods, Cambridge University Press, 1996. https://doi.org/10.1017/CBO9780511626357.
  6. L.N. Trefethen, Spectral Methods in MATLAB, SIAM, 2000. https://doi.org/10.1137/1.9780898719598.
  7. J.A. Weideman, S.C. Reddy, A MATLAB Differentiation Matrix Suite, ACM Trans. Math. Softw. 26 (2000), 465-519. https://doi.org/10.1145/365723.365727.
  8. T.A. Driscoll, F. Bornemann, L.N. Trefethen, The Chebop System for Automatic Solution of Differential Equations, Bit Numer. Math. 48 (2008), 701-723. https://doi.org/10.1007/s10543-008-0198-4.
  9. J.L. Aurentz, L.N. Trefethen, Block Operators and Spectral Discretizations, SIAM Rev. 59 (2017), 423-446. https://doi.org/10.1137/16m1065975.
  10. C.M. Bender, S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York, 1978.
  11. L.D. Akulenko, S.V. Nesterov, High-Precision Methods in Eigenvalue Problems and Their Applications, Chapman and Hall/CRC, 2004. https://doi.org/10.4324/9780203401286.
  12. V. Ledoux, M.V. Daele, G.V. Berghe, MATSLISE: A MATLAB Package for the Numerical Solution of SturmLiouville and Schrödinger Equations, ACM Trans. Math. Softw. 31 (2005), 532-554. https://doi.org/10.1145/1114268.1114273.
  13. S. Pruess, C.T. Fulton, Mathematical Software for Sturm-Liouville Problems, ACM Trans. Math. Softw. 19 (1993), 360-376. https://doi.org/10.1145/155743.155791.
  14. P.B. Bailey, M.K. Gordon, L.F. Shampine, Automatic Solution of the Sturm-Liouville Problem, ACM Trans. Math. Softw. 4 (1978), 193-208. https://doi.org/10.1145/355791.355792.
  15. L.N. Trefethen, T.A. Driscoll, N. Hale, Chebfun-Numerical Computing With Functions. http://www.chebfun.org.
  16. T.A. Driscoll, N. Hale, L.N. Trefethen, Chebfun Guide, Pafnuty publications, Oxford, 2014.
  17. E. Tadmor, The Exponential Accuracy of Fourier and Chebyshev Differencing Methods, SIAM J. Numer. Anal. 23 (1986), 1-10. https://doi.org/10.1137/0723001.