Numerical Computation of Spectral Solutions for Sturm-Liouville Eigenvalue Problems

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Sameh Gana


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.

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  1. S. Flügge, Practical Quantum Mechanics, Springer, Berlin, 1994.
  2. J.D Pryce, Numerical Solution of Sturm–liouville Problems, Oxford University Press, Oxford, 1993.
  3. G.W. Hanson, A.B. Yakovlev, Operator Theory for Electromagnetics, Springer, New York, 2002.
  4. C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Methods in Fluid Dynamics, Springer, Berlin, 1988.
  5. B. Fornberg, A Practical Guide to Pseudospectral Methods, Cambridge University Press, 1996.
  6. L.N. Trefethen, Spectral Methods in MATLAB, SIAM, 2000.
  7. J.A. Weideman, S.C. Reddy, A MATLAB Differentiation Matrix Suite, ACM Trans. Math. Softw. 26 (2000), 465-519.
  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.
  9. J.L. Aurentz, L.N. Trefethen, Block Operators and Spectral Discretizations, SIAM Rev. 59 (2017), 423-446.
  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.
  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.
  13. S. Pruess, C.T. Fulton, Mathematical Software for Sturm-Liouville Problems, ACM Trans. Math. Softw. 19 (1993), 360-376.
  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.
  15. L.N. Trefethen, T.A. Driscoll, N. Hale, Chebfun-Numerical Computing With Functions.
  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.