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Sinc approximation of eigenvalues of Sturm–Liouville problems with a Gaussian multiplier

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Abstract

Eigenvalue problems with eigenparameter appearing in the boundary conditions usually have complicated characteristic determinant where zeros cannot be explicitly computed. Sampling theory is one of the most important mathematical tools used in communication engineering since it enables engineers to reconstruct signals from some of their sampled data. The sinc Gaussian sampling technique derived by Qian (Proc Am Math Soc 131:1169–1176, 2002) establishes a sampling technique which converges faster than the classical sampling technique. Schmeisser and Stenger (Sampl Theory Signal Image Process 6:199–221, 2007) studied the associated error analysis. In the present paper we apply a sinc Gaussian technique to compute approximate values of the eigenvalues of Sturm–Liouville problems with eigenvalue parameter in one or two boundary conditions. The error of this method decays exponentially in terms of the number of involved samples. Therefore the accuracy of the new technique is higher than the classical sinc method. Numerical worked examples with tables and illustrative figures are given at the end of the paper.

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References

  1. Annaby, M.H., Asharabi, R.M.: Computing eigenvalues of boundary-value problems using sinc-Gaussian method. Sampl. Theory Signal Image Process. 7, 293–312 (2008)

    MATH  MathSciNet  Google Scholar 

  2. Annaby, M.H., Tharwat, M.M.: A sinc-Gaussian technique for computing eigenvalues of second-order linear pencils. Appl. Numer. Math. 63, 129–137 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  3. Annaby, M.H., Tharwat, M.M.: On computing eigenvalues of second-order linear pencils. IMA J. Numer. Anal. 27, 366–380 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  4. Annaby, M.H., Tharwat, M.M.: On sampling theory and eigenvalue problems with an eigenparameter in the boundary conditions. SUT J. Math. 42, 157–176 (2006)

    MATH  MathSciNet  Google Scholar 

  5. Annaby, M.H., Tharwat, M.M.: Sinc-based computations of eigenvalues of Dirac systems. BIT 47, 699–713 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  6. Annaby, M.H., Tharwat, M.M.: On the computation of the eigenvalues of Dirac systems. Calcolo 49, 221–240 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  7. Binding, P.A., Browne, P.J., Watson, B.A.: Sturm–Liouville problems with boundary conditions rationally dependent on the eigenparameter II. J. Comput. Appl. Math. 148, 147–169 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  8. Boumenir, A.: Higher approximation of eigenvalues by sampling. BIT 40, 215–225 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  9. Boumenir, A.: Sampling and eigenvalues of non-self-adjoint Sturm–Liouville problems. SIAM. J. Sci. Comput. 23, 219–229 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  10. Butzer, P.L., Stens, R.L.: A modification of the Whittaker–Kotel’nikov–Shannon sampling series. Aequationes Math. 28, 305–311 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  11. Butzer, P.L., Schmeisser, G., Stens, R.L.: An introduction to sampling analysis. In: Marvasti, F. (ed.) Non Uniform Sampling: Theory and Practices, pp. 17–121. Kluwer, New York (2001)

  12. Chadan, K., Sabatier, P.C.: Inverse Problems in Quantum Scattering Theory, 2nd edn. Springer, Berlin (1989)

  13. Chanane, B.: Computing the spectrum of non-self-adjoint Sturm–Liouville problems with parameter-dependent boundary conditions. J. Comput. Appl. Math. 206, 229–237 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  14. Eastham, M.S.P.: Theory of Ordinary Differential Operators. Van Nostrand Reinhold, London (1970)

    Google Scholar 

  15. Fulton, C.T.: Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions. Proc. R. Soc. Edinb. 77 A, 293–308 (1977)

    Article  MathSciNet  Google Scholar 

  16. Gervais, R., Rahman, Q.I., Schmeisser, G.: A bandlimited function simulating a duration-limited one. In: Butzer, P.L., Stens, R.L. (eds.) Approximation Theory and Functional Analysis, pp. 355–362. Birkhäuser, Basel (1984)

  17. Hinton, D.B.: An expansion theorem for an eigenvalue problem with eigenvalue parameter in the boundary condition. Q. J. Math. Oxf. 30, 33–42 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  18. Kotelnikov, V.: On the carrying capacity of the “ether” and wire in telecommunications, material for the first All-Union Conference on Questions of Communications. Izd. Red. Upr. Svyazi RKKA Mosc. Russ. 55, 55–64 (1933)

    Google Scholar 

  19. Kowalski, M., Sikorski, K., Stenger, F.: Selected Topics in Approximation and Computation, SIAM, Philadelphia. Oxford University Press, Oxford (1995)

  20. Levitan, B.M., Sargsjan, I.S.: Introduction to spectral theory: self adjoint ordinary differential operators. In: Translation of Mathematical Monographs, vol. 39. American Mathematical Society, Providence, RI (1975)

  21. Likov, A.V., Mikhailov, Yu.A.: The Theory of Heat and Mass Transfer. Qosenergaizdat (1963) (Russian)

  22. Lund, J., Bowers, K.: Sinc Methods for Quadrature and Differential Equations. SIAM, Philadelphia (1992)

    Book  MATH  Google Scholar 

  23. Qian, L.: On the regularized Whittaker–Kotel’nikov–Shannon sampling formula. Proc. Am. Math. Soc. 131, 1169–1176 (2002)

    Article  Google Scholar 

  24. Qian, L., Creamer, D.B.: A modification of the sampling series with a Gaussian multiplier. Sampl. Theory Signal Image Process. 5, 1–20 (2006)

    MATH  MathSciNet  Google Scholar 

  25. Qian, L., Creamer, D.B.: Localized sampling in the presence of noise. Appl. Math. Lett. 19, 351–355 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  26. Schmeisser, G., Stenger, F.: Sinc approximation with a Gaussian multiplier. Sampl. Theory Signal Image Process. 6, 199–221 (2007)

    MATH  MathSciNet  Google Scholar 

  27. Shannon, C.E.: Communications in the presence of noise. Proc. I.R.E 37, 10–21 (1949)

    Article  MathSciNet  Google Scholar 

  28. Shkalikov, A.A.: Boundary value problems for ordinary diffierential equations with a parameter in boundary conditions. Trydy Sem. Imeny I. G. Petrowsgo 9, 190–229 (1983) (Russian)

    Google Scholar 

  29. Stenger, F.: Numerical methods based on Whittaker cardinal, or sinc functions. SIAM Rev. 23, 156–224 (1981)

    MathSciNet  Google Scholar 

  30. Stenger, F.: Numerical Methods Based on Sinc and Analytic Functions. Springer, New York (1993)

    Book  MATH  Google Scholar 

  31. Stens, R.L.: Sampling by generalized kernels. In: Higgins, J.R., Stens, R.L. (eds.) Sampling Theory in Fourier and Signal Analysis: Advanced Topics, pp. 130–157. Oxford University Press, Oxford (1999)

  32. Tharwat, M.M., Bhrawy, A.H., Yildirim, A.: Numerical computation of eigenvalues of discontinuous Sturm–Liouville problems with parameter dependent boundary conditions using Sinc method. Numer. Algorithms 63, 27–48 (2013). doi:10.1007/s11075-012-9609-3

    Google Scholar 

  33. Tharwat, M.M., Bhrawy, A.H., Yildirim, A.: Numerical computation of eigenvalues of discontinuous Dirac system using Sinc method with error analysis. Int. J. Comput. Math. 89, 2061–2080 (2012). doi:10.1080/00207160.2012.700112

    Google Scholar 

  34. Tikhonov, A.N., Samarskii, A.A.: Equations of Mathematical Physics. Pergamon, Oxford (1963)

    MATH  Google Scholar 

  35. Titchmarsh, E.C.: Eigenfunction Expansions Associated with Second Order Differential Equations, Part I. Clarendon Press, Oxford (1962)

    MATH  Google Scholar 

  36. Walter, J.: Regular eigenvalue problems with eigenvalue parameter in the boundary condition. Math. Z. 133, 301–312 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  37. Whittaker, E.T.: On the functions which are represented by the expansion of the interpolation theory. Proc. R. Soc. Edinb. Sect. A 35, 181–194 (1915)

    Google Scholar 

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Acknowledgments

This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The author, therefore, acknowledges with thanks DSR technical and financial support.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia

    M. M. Tharwat

  2. Department of Mathematics, Faculty of Science, Beni-Suef University, Beni Suef, Egypt

    M. M. Tharwat

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Tharwat, M.M. Sinc approximation of eigenvalues of Sturm–Liouville problems with a Gaussian multiplier. Calcolo 51, 465–484 (2014). https://doi.org/10.1007/s10092-013-0095-3

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