A new eighth order exponentially fitted explicit Numerov-type method for solving oscillatory problems

Original Paper
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Abstract

In this work we derive a new family of seven stages, eighth order, exponential fitted, Numerov (i.e. two-step) type methods. A particular variable coefficient method is constructed. The numerical results show that it outperforms exponential fitted two step hybrid method found in the literature sharing similar characteristics. Moreover, the presented method outperforms Numerov type methods with minimized phase-lag for problems in consideration share a non dominant linear part.

Keywords

Initial value problems Hybrid method Variable coefficients 

Mathematics Subject Classification

65L05 65L06 68W30 

References

  1. 1.
    I. Alolyan, Z.A. Anastassi, T.E. Simos, Appl. Math. Comput. 218, 5370 (2012)Google Scholar
  2. 2.
    I. Alolyan, T.E. Simos, Comput. Math. Appl. 62, 3756 (2011)CrossRefGoogle Scholar
  3. 3.
    Z.A. Anastassi, T.E. Simos, J. Comput. Appl. Math. 236, 3880 (2012)CrossRefGoogle Scholar
  4. 4.
    D.B. Berg, T.E. Simos, C. Tsitouras, Math. Methods Appl. Sci. (2017).  https://doi.org/10.1002/mma.4711
  5. 5.
    M. Dong, T.E. Simos, Filomat 31, 4999 (2017)Google Scholar
  6. 6.
    F. Hui, T.E. Simos, Appl. Comput. Math. 15, 220 (2016)Google Scholar
  7. 7.
    Z. Kalogiratou, T. Monovasilis, T.E. Simos, Comput. Math. Appl. 60, 1639 (2010)CrossRefGoogle Scholar
  8. 8.
    Z. Kalogiratou, T. Monovasilis, H. Ramos, T.E. Simos, J. Comput. Appl. Math. 303, 146 (2016)CrossRefGoogle Scholar
  9. 9.
    A.A. Kosti, Z.A. Anastassi, T.E. Simos, Comput. Math. Appl. 61, 3381 (2011)CrossRefGoogle Scholar
  10. 10.
    T. Monovasilis, Z. Kalogiratou, T.E. Simos, Appl. Math. Comput. 209, 91 (2009)Google Scholar
  11. 11.
    T. Monovasilis, Z. Kalogiratou, T.E. Simos, Appl. Math. Inf. Sci. 7, 81 (2013)CrossRefGoogle Scholar
  12. 12.
    T. Monovasilis, Z. Kalogiratou, T.E. Simos, Mediterr. J. Math. 13, 2271 (2016)CrossRefGoogle Scholar
  13. 13.
    T. Monovasilis, Z. Kalogiratou, H. Ramos, T.E. Simos, Math. Methods Appl. Sci. 40, 5286 (2017)CrossRefGoogle Scholar
  14. 14.
    G.A. Panopoulos, T.E. Simos, Appl. Math. Inf. Sci. 7, 73 (2013)CrossRefGoogle Scholar
  15. 15.
    G.A. Panopoulos, T.E. Simos, Appl. Math. Inf. Sci. 8, 703 (2014)CrossRefGoogle Scholar
  16. 16.
    G.A. Panopoulos, T.E. Simos, J. Comput. Appl. Math. 290, 1 (2015)CrossRefGoogle Scholar
  17. 17.
    D.F. Papadopoulos, T.E. Simos, Appl. Math. Inf. Sci. 7, 433 (2013)CrossRefGoogle Scholar
  18. 18.
    D.F. Papadopoulos, T.E. Simos, Abstract and Applied Analysis (2013)Google Scholar
  19. 19.
    H. Ramos, Z. Kalogiratou, T. Monovasilis, T.E. Simos, Numer. Algorithm 72, 1089 (2016)CrossRefGoogle Scholar
  20. 20.
    T.E. Simos, Appl. Math. Comput. 209, 137 (2009)Google Scholar
  21. 21.
    T.E. Simos, Appl. Math. Lett. 22, 1616 (2009)CrossRefGoogle Scholar
  22. 22.
    T.E. Simos, Acta Appl. Math. 110, 1331 (2010)CrossRefGoogle Scholar
  23. 23.
    T.E. Simos, Abstr. Appl. Anal. 2012, 182536 (2012)Google Scholar
  24. 24.
    T.E. Simos, J. Appl. Math. 2012, 420387 (2012)Google Scholar
  25. 25.
    T.E. Simos, Appl. Comput. Math. 14, 296 (2015)Google Scholar
  26. 26.
    T.E. Simos, Appl. Math. Inf. Sci. 8, 447 (2014)CrossRefGoogle Scholar
  27. 27.
    T.E. Simos, C. Tsitouras, Math. Methods Appl. Sci. 40, 6276 (2017)CrossRefGoogle Scholar
  28. 28.
    S. Stavroyiannis, T.E. Simos, Appl. Numer. Math. 59, 2467 (2009)CrossRefGoogle Scholar
  29. 29.
    C. Tsitouras, Appl. Math. Lett. 11, 65 (1998)CrossRefGoogle Scholar
  30. 30.
    C. Tsitouras, I.T. Famelis, T.E. Simos, Comput. Math. Appl. 62, 2101 (2011)CrossRefGoogle Scholar
  31. 31.
    C. Tsitouras, I. Famelis, T.E. Simos, J. Comput. Appl. Math. 321, 226–231 (2017)CrossRefGoogle Scholar
  32. 32.
    C. Tsitouras, G. Papageorgiou, Computing 43, 255 (1990)CrossRefGoogle Scholar
  33. 33.
    W. Zhang, T.E. Simos, Mediterr. J. Math. 13, 5177 (2016)CrossRefGoogle Scholar
  34. 34.
    R. Thomas, T.E. Simos, J. Comput. Appl. Math. 87, 215 (1997)CrossRefGoogle Scholar
  35. 35.
    C. Tsitouras, T.E. Simos, J. Comput. Appl. Math. 147, 397 (2002)CrossRefGoogle Scholar
  36. 36.
    C. Tsitouras, in Proceedings of 1st International Symposium Nonlinear Problems, ed. by vrakakis (TU Athens, 2000), p. 429Google Scholar
  37. 37.
    C. Tsitouras, Comput. Math. Appl. 45, 37 (2003)CrossRefGoogle Scholar
  38. 38.
    J.P. Coleman, IMA J. Numer. Anal. 23, 197 (2003)CrossRefGoogle Scholar
  39. 39.
    I.T. Famelis, C. Tsitouras, J. Comput. Appl. Math. 218, 543 (2008)CrossRefGoogle Scholar
  40. 40.
    T.E. Simos, C. Tsitouras, Math. Methods Appl. Sci. 40, 7867 (2017)CrossRefGoogle Scholar
  41. 41.
    J.M. Franco, L. Randez, Numer. Algorithm (2017).  https://doi.org/10.1007/s11075-017-0374-1
  42. 42.
    J.C. Butcher, Math. Comput. 18, 50 (1964)CrossRefGoogle Scholar
  43. 43.
    J.C. Butcher, J. Aust. Math. Soc. 4, 179 (1994)CrossRefGoogle Scholar
  44. 44.
    M.P. Calvo, J.M. Sanz-Serna, SIAM J. Sci. Comput. 14, 1237 (1993)CrossRefGoogle Scholar
  45. 45.
    J.M. Franco, L. Randez, Appl. Math. Comput. 273, 493 (2016)Google Scholar
  46. 46.
    J.C. Butcher, Numerical Methods for Ordinary Differential Equations (John Wiley & Sons Inc, Chichester, 2008)CrossRefGoogle Scholar
  47. 47.
    P.W. Sharp, J.C. Castillo-Rogezb, K.R. Grazier, J. Comput. Appl. Math. 236, 2378 (2012)CrossRefGoogle Scholar
  48. 48.
    E. Hairer, C. Lubich, G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations (Springer, Berlin, 2002)CrossRefGoogle Scholar
  49. 49.
    C. Tsitouras, Int. J. Mod. Phys. C 17, 861 (2006)CrossRefGoogle Scholar
  50. 50.
    C. Tsitouras, T.E. Simos, Appl. Math. Comput. 95, 15 (1998)CrossRefGoogle Scholar
  51. 51.
    J.M. Franco, I. Gomez, Appl. Math. Comput. 232, 643 (2014)Google Scholar
  52. 52.
    Mathematica, Version 11.2 (Wolfram Research, Inc., Champaign, IL, 2017)Google Scholar
  53. 53.
    MATLAB, version 7.8.0 (R2009a) (The MathWorks Inc., Natick, Massachusetts, 2009)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Automation EngineeringTEI of Sterea HellasPsahna CampusGreece
  2. 2.microSENSES Laboratory, Department of Electronic EngineeringTEI of AthensEgaleoGreece

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