Table of Contents
International Journal of Nuclear Energy
Volume 2013 (2013), Article ID 903904, 12 pages
http://dx.doi.org/10.1155/2013/903904
Research Article

Generalized and Stability Rational Functions for Dynamic Systems of Reactor Kinetics

1Department of Mathematics, Faculty of Science and Arts, Qassim University, P.O. Box 1300 Buraidah, Saudi Arabia
2Department of Mathematics, Faculty of Science, Tanta University, Tanta 31527, Egypt

Received 27 February 2013; Revised 2 June 2013; Accepted 7 June 2013

Academic Editor: Arkady Serikov

Copyright © 2013 Ahmed E. Aboanber. 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. A. E. Aboanber, “Analytical solution of the point kinetics equations by exponential mode analysis,” Progress in Nuclear Energy, vol. 42, no. 2, pp. 179–197, 2003. View at Publisher · View at Google Scholar · View at Scopus
  2. A. E. Aboanber, “An efficient analytical form for the period-reactivity relation of beryllium and heavy-water moderated reactors,” Nuclear Engineering and Design, vol. 224, no. 3, pp. 279–292, 2003. View at Publisher · View at Google Scholar · View at Scopus
  3. F. Zhang, W.-Z. Chen, and X.-W. Gui, “Analytic method study of point-reactor kinetic equation when cold start-up,” Annals of Nuclear Energy, vol. 35, no. 4, pp. 746–749, 2008. View at Publisher · View at Google Scholar · View at Scopus
  4. W. Z. Chen, B. Kuang, L. F. Guo, Z. Y. Chen, and B. Zhu, “New analysis of prompt supercritical process with temperature feedback,” Nuclear Engineering and Design, vol. 236, no. 12, pp. 1326–1329, 2006. View at Publisher · View at Google Scholar · View at Scopus
  5. W. Z. Chen, L. F. Guo, B. Zhu, and H. F. Li, “Accuracy of analytical methods for obtaining supercritical transients with temperature feedback,” Progress in Nuclear Energy, vol. 49, no. 4, pp. 290–302, 2007. View at Publisher · View at Google Scholar · View at Scopus
  6. H. F. Li, W. Z. Chen, F. Zhang, and L. Luo, “Approximate solutions of point kinetics equations with one delayed neutron group and temperature feedback during delayed supercritical process,” Annals of Nuclear Energy, vol. 34, no. 6, pp. 521–526, 2007. View at Publisher · View at Google Scholar · View at Scopus
  7. G. R. Keepin and C. W. Cox, “General solution of the reactor kinetic equations,” Nuclear Science and Engineering, vol. 8, pp. 670–690, 1960. View at Google Scholar
  8. C. M. Kang and K. F. Hansen, “Finite element methods for reactor analysis,” Nuclear Science and Engineering, vol. 51, no. 4, pp. 456–495, 1973. View at Google Scholar · View at Scopus
  9. J. C. Allerd and D. S. Carter, “Kinetics of homogeneous power reactors of the LAPRE Type,” Nuclear Science and Engineering, vol. 3, pp. 482–503, 1958. View at Google Scholar
  10. J. Sanchez, “On the numerical solution of the point reactor kinetics equations by generalized Runge-Kutta methods,” Nuclear Science and Engineering, vol. 103, no. 1, pp. 94–99, 1989. View at Google Scholar · View at Scopus
  11. C. M. Kang, “Piecewise polynomial approximations for the point kinetics equations,” Transactions of the American Nuclear Society, vol. 14, pp. 201–202, 1971. View at Google Scholar
  12. J. P. Hennart, “Piecewise polynomial approximations for nuclear reactor point and space kinetics,” Nuclear Science and Engineering, vol. 64, no. 4, pp. 875–901, 1977. View at Google Scholar · View at Scopus
  13. W. L. Hendry and G. I. Bell, “An analysis of the time-dependent neutron transport equation with delayed neutrons by the method of matched asymptotic expansions,” Nuclear Science and Engineering, vol. 35, pp. 240–248, 1969. View at Google Scholar
  14. R. Goldstein and L. M. Shotkin, “Use of the prompt-jump approximation in fast reactor kinetics,” Nuclear Science and Engineering, vol. 38, no. 2, pp. 94–103, 1969. View at Google Scholar · View at Scopus
  15. T. Blenski, A. Gadomski, and J. Mika, “Higher order prompt-jump approximation in reactor kinetics,” Nuclear Science and Engineering, vol. 66, no. 3, pp. 277–283, 1978. View at Google Scholar · View at Scopus
  16. H. D. Brown, “A general treatment of flux transients,” Nuclear Science and Engineering, vol. 2, pp. 687–693, 1957. View at Google Scholar
  17. F. T. Adler, “Reactor kinetics: integral equation formulation,” Journal of Nuclear Energy Parts A/B, vol. 15, no. 2-3, pp. 81–85, 1961. View at Google Scholar · View at Scopus
  18. K. F. Hansen, B. V. Koen, and W. W. Little, “Stable numerical solutions of the reactor kinetics equations,” Nuclear Science and Engineering, vol. 22, pp. 51–59, 1965. View at Google Scholar
  19. T. A. Porsching, “Numerical solution of the reactor kinetics equations by approximate exponentials,” Nuclear Science and Engineering, vol. 25, pp. 183–188, 1966. View at Google Scholar
  20. H. Li, W. Chen, L. Luo, and Q. Zhu, “A new integral method for solving the point reactor neutron kinetics equations,” Annals of Nuclear Energy, vol. 36, no. 4, pp. 427–432, 2009. View at Publisher · View at Google Scholar · View at Scopus
  21. J. C. Vigil, “Solution of the reactor kinetics equations by analytic continuation,” Nuclear Science and Engineering, vol. 29, pp. 392–401, 1967. View at Google Scholar
  22. Y.-A. Chao and A. Attard, “A resolution of the stiffness problem of reactor kinetics,” Nuclear Science and Engineering, vol. 90, no. 1, pp. 40–46, 1985. View at Google Scholar · View at Scopus
  23. J. Basken and J. D. Lewins, “Power series solutions of the reactor kinetics equations,” Nuclear Science and Engineering, vol. 122, no. 3, pp. 407–416, 1996. View at Google Scholar · View at Scopus
  24. A. E. Aboanber and Y. M. Hamada, “PWS: an efficient code system for solving space-independent nuclear reactor dynamics,” Annals of Nuclear Energy, vol. 29, no. 18, pp. 2159–2172, 2002. View at Publisher · View at Google Scholar · View at Scopus
  25. A. E. Aboanber and Y. M. Hamada, “Power series solution (PWS) of nuclear reactor dynamics with newtonian temperature feedback,” Annals of Nuclear Energy, vol. 30, no. 10, pp. 1111–1122, 2003. View at Publisher · View at Google Scholar · View at Scopus
  26. T. Sathiyasheela, “Power series solution method for solving point kinetics equations with lumped model temperature and feedback,” Annals of Nuclear Energy, vol. 36, no. 2, pp. 246–250, 2009. View at Publisher · View at Google Scholar · View at Scopus
  27. J. A. W. da Nóbrega, “A new solution of the point kinetics equations,” Nuclear Science and Engineering, vol. 46, pp. 366–370, 1971. View at Google Scholar
  28. R. D. Lawrence and J. J. Doring, “A smoothing and extrapolation method for point kinetics,” Transactions of the American Nuclear Society, vol. 24, pp. 199–201, 1976. View at Google Scholar
  29. A. E. Aboanber and A. A. Nahla, “On pade' approximations to the exponential function and application to the point kinetics equations,” Progress in Nuclear Energy, vol. 44, no. 4, pp. 347–368, 2004. View at Publisher · View at Google Scholar · View at Scopus
  30. A. E. Aboanber and A. A. Nahla, “Generalization of the analytical inversion method for the solution of the point kinetics equations,” Journal of Physics A, vol. 35, no. 14, pp. 3245–3263, 2002. View at Publisher · View at Google Scholar · View at Scopus
  31. B. Quintero-Leyva, “CORE: a numerical algorithm to solve the point kinetics equations,” Annals of Nuclear Energy, vol. 35, no. 11, pp. 2136–2138, 2008. View at Publisher · View at Google Scholar · View at Scopus
  32. B. D. Ganapol, “The refined way to solve the reactor point kinetics equations for imposed reactivity insertions,” Nuclear Technology and Radiation Protection, vol. 24, no. 3, pp. 157–166, 2009. View at Google Scholar
  33. B. D. Ganapol and P. Picca, “A highly accurate benchmark for reactor point kinetics with feedback,” in Proceedings of the 17th Pacific Basin Nuclear Conference, Cancún, México, October 2010.
  34. B. D. Ganapol, “A highly accurate algorithm for the solution of the point kinetics equations,” Annals of Nuclear Energy, 2013. View at Publisher · View at Google Scholar
  35. J. J. Duderstadt and L. J. Hamilton, Nuclear Reactor Analysis, John Wiley & Sons, New York, NY, USA, 1976.
  36. A. F. Henry, Nuclear Reactor Analysis, The MIT Press, Cambridge, Mass, USA, 1975.
  37. C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations, Prentice Hall, Englewood Cliffs, NJ, USA, 1971.
  38. R. S. Varga, “On higher order stable implicit methods for solving parabolic partial differential equations,” Journal of Mathematical Physics, vol. 40, pp. 220–231, 1961. View at Google Scholar
  39. S. P. Nørsett, “C-Polynomials for rational approximation to the exponential function,” Numerische Mathematik, vol. 25, no. 1, pp. 39–56, 1975. View at Publisher · View at Google Scholar · View at Scopus
  40. B. D. Ganapol, P. Picca, A. Previti, and D. Mostacci, The Solution of the Point Kinetics Equations Via Acceleration Taylor Series (CATS), Knoxville, Tenn, USA, 2012.
  41. D. L. Hetrick, Dynamics of Nuclear Reactor, American Nuclear Society, La Grange Park, Ill, USA, 1993.
  42. A. E. Aboanber, “Stability of generalized Runge-Kutta methods for stiff kinetics coupled differential equations,” Journal of Physics A, vol. 39, no. 8, pp. 1859–1876, 2006. View at Publisher · View at Google Scholar · View at Scopus