Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2016, Article ID 4356371, 11 pages
http://dx.doi.org/10.1155/2016/4356371
Research Article

Fractional Mathematical Operators and Their Computational Approximation

Escuela Técnica Superior de Ingeniería Aeronáutica y del Espacio, Universidad Politécnica de Madrid, Plaza Cardenal Cisneros 3, 28040 Madrid, Spain

Received 27 January 2016; Revised 30 July 2016; Accepted 9 August 2016

Academic Editor: Manuel Doblaré

Copyright © 2016 José Crespo and Francisco Javier Montáns. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. H. Kober, “On fractional integrals and derivatives,” The Quarterly Journal of Mathematics, vol. 11, no. 1, pp. 193–211, 1940. View at Google Scholar · View at MathSciNet
  2. R. L. Bagley and P. J. Torvik, “Fractional calculus-a different approach to the analysis of viscoelastically damped structures,” AIAA Journal, vol. 21, no. 5, pp. 741–748, 1983. View at Publisher · View at Google Scholar · View at Scopus
  3. P. J. Torvik and R. L. Bagley, “On the appearance of the fractional derivative in the behavior of real materials,” Journal of Applied Mechanics, vol. 51, no. 2, pp. 294–298, 1984. View at Publisher · View at Google Scholar · View at Scopus
  4. G. Peano, Arithmetices Principia, Nova Methodo Exposita, Fratres Bocca, 1889.
  5. W. Ackermann, “Zum Hilbertschen Aufbau der reellen Zahlen,” Mathematische Annalen, vol. 99, no. 1, pp. 118–133, 1928. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. R. L. Goodstein, “Transfinite ordinals in recursive number theory,” The Journal of Symbolic Logic, vol. 12, pp. 123–129, 1947. View at Publisher · View at Google Scholar · View at MathSciNet
  7. K. A. Rutsov and G. F. Romerio, Ackermann's Function and New Arithmetical Operations, Rotary Saluzzo, 2004.
  8. H. P. Williams, What Lies between + and ×, The London School of Economics and Political Science, London, UK, 2010.
  9. Q. Yuan, Answer to: Is There a Natural Way to Extend Repeated Exponentiation Beyond Integers? In Mathematics Stack Exchange, an Stackoverflow web site, 2016, http://math.stackexchange.com/questions/56663/is-there-a-natural-way-to-extend-repeated-exponentiation-beyond-integers/56710#56710.
  10. M. L. Campagnolo, C. Moore, and J. F. Costa, “An analog characterization of the Grzegorczyk hierarchy,” Journal of Complexity, vol. 18, no. 4, pp. 977–1000, 2002. View at Publisher · View at Google Scholar · View at Scopus
  11. M. Latorre and F. J. Montáns, “Extension of the Sussman-Bathe spline-based hyperelastic model to incompressible transversely isotropic materials,” Computers and Structures, vol. 122, pp. 13–26, 2013. View at Publisher · View at Google Scholar · View at Scopus
  12. M. Latorre and F. J. Montáns, “What-you-prescribe-is-what-you-get orthotropic hyperelasticity,” Computational Mechanics, vol. 53, no. 6, pp. 1279–1298, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. M. Miñano and F. J. Montáns, “A new approach to modeling isotropic damage for Mullins effect in hyperelastic materials,” International Journal of Solids and Structures, vol. 67-68, pp. 272–282, 2015. View at Publisher · View at Google Scholar · View at Scopus
  14. Y. Ling, “Uniaxial true stress—strain after necking,” AMP Journal of Technology, vol. 5, 1996. View at Google Scholar
  15. L. R. G. Treloar, “Stress-strain data for vulcanised rubber under various types of deformation,” Transactions of the Faraday Society, vol. 40, pp. 59–70, 1944. View at Publisher · View at Google Scholar · View at Scopus
  16. R. L. Eubank, Nonparametric Regression and Spline Smoothing, vol. 157, Marcel Dekker, New York, NY, USA, 2nd edition, 1999. View at MathSciNet
  17. H. L. Weinert, Fast Compact Algorithms and Software for Spline Smoothing, Springer, New York, NY, USA, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  18. P. Dierckx, Curve and Surface Fitting with Splines, Monographs on Numerical Analysis, Oxford University Press, Oxford, United Kingdom, 1993. View at MathSciNet
  19. R. P. Brent, “Fast algorithms for high-precision computation of elementary functions,” in Proceedings of the 7th Conference on Real Numbers and Computers (RNC '06), Nancy, France, July 2006.