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BIT Numerical Mathematics

, Volume 58, Issue 1, pp 199–220 | Cite as

Theoretical analysis of Sinc-collocation methods and Sinc-Nyström methods for systems of initial value problems

  • Tomoaki Okayama
Article
  • 180 Downloads

Abstract

A Sinc-collocation method was proposed by Stenger, who also gave a theoretical analysis of the method in the case of a “scalar” equation. This paper extends the theoretical results to the case of a “system” of equations. Furthermore, this paper proposes a more efficient method by replacing the variable transformation employed in Stenger’s method. The efficiency was confirmed by both a theoretical analysis and some numerical experiments. In addition to the existing and newly proposed Sinc-collocation methods, this paper also gives similar theoretical results for the Sinc-Nyström methods proposed by Nurmuhammad et al. In terms of computational cost, the newly proposed Sinc-collocation method is the most efficient among these methods.

Keywords

Sinc approximation Sinc indefinite integration Differential equation Volterra integral equation tanh transformation Double-exponential transformation 

Mathematics Subject Classification

65L05 65R20 65D30 

References

  1. 1.
    Carlson, T.S., Dockery, J., Lund, J.: A sinc-collocation method for initial value problems. Math. Comput. 66, 215–235 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Nurmuhammad, A., Muhammad, M., Mori, M.: Numerical solution of initial value problems based on the double exponential transformation. Publ. Res. Inst. Math. Sci. Kyoto Univ. 41, 937–948 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Stenger, F.: Numerical Methods Based on Sinc and Analytic Functions. Springer, New York (1993)CrossRefzbMATHGoogle Scholar
  4. 4.
    Mori, M., Sugihara, M.: The double-exponential transformation in numerical analysis. J. Comput. Appl. Math. 127, 287–296 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Sugihara, M., Matsuo, T.: Recent developments of the Sinc numerical methods. J. Comput. Appl. Math. 164–165, 673–689 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Muhammad, M., Mori, M.: Double exponential formulas for numerical indefinite integration. J. Comput. Appl. Math. 161, 431–448 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Okayama, T., Matsuo, T., Sugihara, M.: Improvement of a Sinc-collocation method for Fredholm integral equations of the second kind. BIT Numer. Math. 51, 339–366 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Tanaka, K., Sugihara, M., Murota, K.: Function classes for successful DE-Sinc approximations. Math. Comput. 78, 1553–1571 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Okayama, T.: A note on the Sinc approximation with boundary treatment. JSIAM Lett. 5, 1–4 (2013)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Okayama, T., Matsuo, T., Sugihara, M.: Error estimates with explicit constants for Sinc approximation, Sinc quadrature and Sinc indefinite integration. Numer. Math. 124, 361–394 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Stenger, F., Gustafson, S.Å., Keyes, B., O’Reilly, M., Parker, K.: ODE-IVP-PACK via Sinc indefinite integration and Newton’s method. Numer. Algorithms 20, 241–268 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Polyanin, A.D., Zaitsev, V.F.: Handbook of Exact Solutions for Ordinary Differential Equations, 2nd edn. CRC, Boca Raton (2003)zbMATHGoogle Scholar
  13. 13.
    Okayama, T., Tanaka, K., Matsuo, T., Sugihara, M.: DE-Sinc methods have almost the same convergence property as SE-Sinc methods even for a family of functions fitting the SE-Sinc methods. Part I: definite integration and function approximation. Numer. Math. 125, 511–543 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Tanaka, K., Okayama, T., Matsuo, T., Sugihara, M.: DE-Sinc methods have almost the same convergence property as SE-Sinc methods even for a family of functions fitting the SE-Sinc methods. Part II: indefinite integration. Numer. Math. 125, 545–568 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Okayama, T., Matsuo, T., Sugihara, M.: Theoretical analysis of Sinc-Nyström methods for Volterra integral equations. Math. Comput. 20, 1189–1215 (2015)zbMATHGoogle Scholar
  16. 16.
    Okayama, T., Matsuo, T., Sugihara, M.: Sinc-collocation methods for weakly singular Fredholm integral equations of the second kind. J. Comput. Appl. Math. 234, 1211–1227 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Atkinson, K.E.: The Numerical Solution of Integral Equations of the Second Kind. Cambridge University Press, Cambridge (1997)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Graduate School of Information SciencesHiroshima City UniversityHiroshimaJapan

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