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Probabilistic evolution theory for the solution of explicit autonomous ordinary differential equations: squarified telescope matrices

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Abstract

Probabilistic evolution theory (PREVTH) is used for the solution of initial value problems of first order explicit autonomous ordinary differential equation sets with second degree multinomial right hand side functions. It is an approximation method based on Kronecker power series: a rewriting of multivariate Taylor series using matrices having certain flexible parameters. Kronecker power series have matrices which are called telescope matrices: \(n \times n^{j+1}\) matrices where j is the index of summation. The additive terms of each telescope matrix is formed through Kronecker product from both sides by Kronecker powers of identity matrices. Recently, squarification is proposed in order to avoid the growing of the matrices in size at each additive term of the series. This paper explains the squarification procedure: the procedure used in order to avoid Kronecker multiplications within PREVTH so that the sizes of the matrices do not grow and so that the amount of necessary computation is reduced. The recursion between squarified matrices is also given. As a numerical application, the solution of a Hénon–Heiles system is provided.

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References

  1. M. Demiralp, H. Rabitz, Int. J. Eng. Sci. 31(2), 307 (1993). doi:10.1016/0020-7225(93)90043-T

    Article  Google Scholar 

  2. M. Demiralp, H. Rabitz, Int. J. Eng. Sci. 31(2), 333 (1993). doi:10.1016/0020-7225(93)90044-U

    Article  Google Scholar 

  3. E. Demiralp, M. Demiralp, L. Hernandez-Garcia, J. Math. Chem. 50, 870 (2012). doi:10.1007/s10910-011-9930-4

    Article  CAS  Google Scholar 

  4. M. Demiralp, E. Demiralp, L. Hernandez-Garcia, J. Math. Chem. 50, 850 (2012). doi:10.1007/s10910-011-9929-x

    Article  CAS  Google Scholar 

  5. M. Demiralp, J. Math. Chem. 51(4), 1170 (2012). doi:10.1007/s10910-012-0079-6

    Article  Google Scholar 

  6. M. Demiralp, N. Baykara, J. Math. Chem. 51(4), 1187 (2012). doi:10.1007/s10910-012-0080-0

    Article  Google Scholar 

  7. M. Demiralp, B. Tunga, J. Math. Chem. 51(4), 1198 (2012). doi:10.1007/s10910-012-0081-z

    Article  Google Scholar 

  8. M. Demiralp, E. Demiralp, J. Math. Chem. 51(1), 58 (2012). doi:10.1007/s10910-012-0070-2

    Article  Google Scholar 

  9. M. Demiralp, E. Demiralp, J. Math. Chem. 51(1), 38 (2012). doi:10.1007/s10910-012-0064-0

    Google Scholar 

  10. C. Gözükırmızı, M. Demiralp, J. Math. Chem. 52(3), 866 (2014). doi:10.1007/s10910-013-0298-5

    Article  Google Scholar 

  11. C. Gözükırmızı, M. Demiralp, J. Math. Chem. 52(3), 881 (2014). doi:10.1007/s10910-013-0299-4

    Article  Google Scholar 

  12. F. Hunutlu, N. Baykara, M. Demiralp, Int. J. Comput. Math. 90(11), 2326 (2013)

    Article  Google Scholar 

  13. E. Tataroğlu, M. Demiralp, Int. J. Math. Comput. Methods 1, 201 (2016)

    Google Scholar 

  14. M.E. Kırkın, M. Demiralp, Int. J. Comput. 1, 158 (2016)

    Google Scholar 

  15. M. Ayvaz, M. Demiralp, J. Math. Chem. 52(8), 2161 (2014). doi:10.1007/s10910-014-0371-8

    Article  CAS  Google Scholar 

  16. M. Ayvaz, M. Demiralp, J. Math. Chem. 52(8), 2294 (2014). doi:10.1007/s10910-014-0381-6

    Article  CAS  Google Scholar 

  17. M. Demiralp, in Proceedings of the 19th International Conference on Applied Mathematics (AMATH’14) (WSEAS Press, İstanbul, 2014), ISBN:978-1-61804-258-3, pp. 99–104

  18. M.E. Kırkın, C. Gözükırmızı, in AIP Conference Proceedings, vol. 1702 (2015). doi:10.1063/1.4938947

  19. C. Gözükırmızı, M. Demiralp, in AIP Conference Proceedings, vol. 1702 (2015). doi:10.1063/1.4938937

  20. The OEIS Foundation Inc. The Online Encyclopedia of Integer Sequences (2012). http://oeis.org

  21. C. Gözükırmızı. Sequence A262180 in [20] (2015)

  22. J.C. Butcher, Numerical Methods for Ordinary Differential Equations (Wiley Online Library, Hoboken, 2008)

    Book  Google Scholar 

  23. H.Z. Munthe-Kaas, A. Lundervold, Found. Comput. Math. 13(4), 583 (2013). doi:10.1007/s10208-013-9167-7

    Article  Google Scholar 

  24. R.I. McLachlan, K. Modin, H. Munthe-Kaas, O. Verdier, What are Butcher series, really? The story of rooted trees and numerical methods for evolution equations (2016). arXiv:1512.00906

  25. A. Iserles, H.Z. Munthe-Kaas, S.P. Nørsett, A. Zanna, Acta Numer. 2000(9), 215 (2000)

    Article  Google Scholar 

  26. A. Iserles, G. Quispel, Why Geometric Numerical Integration? (2016). arXiv:1602.07755

  27. K. Ebrahimi-Fard, D. Manchon, Found. Comput. Math. 14(1), 1 (2014). doi:10.1007/s10208-013-9172-x

    Article  Google Scholar 

Download references

Acknowledgments

First author would like to thank Richard J. Fateman for an e-mail communication on arbitrary precision arithmetic and exact arithmetic.

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Authors and Affiliations

  1. Computer Engineering Department, Beykent University, Ayazağa, Maslak, Istanbul, Turkey

    Coşar Gözükırmızı

  2. Informatics Institute, Istanbul Technical University, Maslak, Istanbul, Turkey

    Melike Ebru Kırkın & Metin Demiralp

Authors
  1. Coşar Gözükırmızı
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  2. Melike Ebru Kırkın
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  3. Metin Demiralp
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Correspondence to Coşar Gözükırmızı.

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Gözükırmızı, C., Kırkın, M.E. & Demiralp, M. Probabilistic evolution theory for the solution of explicit autonomous ordinary differential equations: squarified telescope matrices. J Math Chem 55, 175–194 (2017). https://doi.org/10.1007/s10910-016-0678-8

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  • DOI: https://doi.org/10.1007/s10910-016-0678-8

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