Computational and Applied Mathematics

, Volume 37, Issue 3, pp 3473–3502 | Cite as

Generalized trigonometric Fourier-series method with automatic time step control for non-linear point kinetics equations

  • Yasser Mohamed HamadaEmail author


The trigonometric Fourier-series method (TFS) is generalized to provide approximate solutions for non-linear point kinetics equations with feedback using varying step sizes. This method can provide a very stable solution against the size of the discrete time step allowing much larger step sizes to be used. Systems of the point kinetics equations are solved using Fourier-series expansion over a partition of the total time interval. The approximate solution requires determining the series coefficients over each time step in that partition. These coefficients are determined using the high-order derivatives of the solution vector at the beginning of the time step introducing a system of linear algebraic equations to be solved at each step. This system is similar to the Vandermonde system. Two successive orders of the partial sums are used to estimate the local truncation error. This error and some other constrains are used to produce the largest step size allowable at each step while keeping the error within a specific tolerance. The process of calculating suitable step sizes should be automatic and inexpensive. Convergence and stability of the proposed method are discussed and a new formula is introduced to maintain stability. The proposed method solves the general linear and non-linear kinetics problems. The method has been applied to five different types of reactivities including step/ramp insertions with temperature feedback. The method is seemed to be valid for larger time intervals than those used in the conventional numerical integration, and is thus useful in reducing computing time. Computational results are found to be consistent with the analysis, they demonstrate that the convergence of the iteration scheme can be accelerated and the resulting computing times can be greatly reduced while maintain computational accuracy.


Fourier series method Reactor dynamics Feedback Step size control Numerical solutions 

Mathematics Subject Classification



  1. Aboanber AE (2006) Stability of generalized Runge-Kutta methods for stiff kinetics coupled differential equations. J Phys A: Math Gen 39:1859–1876MathSciNetCrossRefzbMATHGoogle Scholar
  2. Aboanber AE, Hamada YM (2003) Power series solution (PWS) of nuclear reactor dynamics with newtonian temperature feedback. Ann Nucl Eng 30:111–1122CrossRefGoogle Scholar
  3. Alfio Q, Riccardo S, Fausto S (2007) Numerical Mathematics. 2th edn, New York, NY 10029-6574 USAGoogle Scholar
  4. Brown JW, Churchill RV (1993) Fourier series and boundary value problems, 5th edn. McGraw-Hill, New YorkGoogle Scholar
  5. Canuto C, Hussaini MY, Quarteroni A, Zang TA (2006) Spectral methods. fundamentals in single domains. Springer, Berlin HeidelbergzbMATHGoogle Scholar
  6. Charalambos DA, Owen B (1998) Principle of real analysis, 3d edn. Academic, New YorkzbMATHGoogle Scholar
  7. Erwin K (2011) Adv Eng Math, 10th edn. Wiley, HobokenGoogle Scholar
  8. Froehlich R, Johnson SR, Merrill MH (1968) GAKIT—a one-dimensional multigroup kinetics code with temperature feedback. GA-8576, General AtomicGoogle Scholar
  9. Ganapol BD et al (2012) The solution of the point kinetics equation: a converged accelerated Taylor series (CATS), PHYSOR 2012, Knoxville, TN, USA, April 15–20 on CD-ROM. American Nuclear Society, LaGrange Park, ILGoogle Scholar
  10. Ganapol BD (2013) A highly accurate algorithm for the solution of the point kinetics equations. Ann Nucl Energy 62:564–571CrossRefGoogle Scholar
  11. Hamada YM (2013) Confirmation of accuracy of generalized power series method for the solution of point kinetics equations with feedback. Ann Nucl Energy 55:184–193CrossRefGoogle Scholar
  12. Hamada YM (2014) Liapunov’s stability on autonomous nuclear dynamical systems. Prog Nuclear Energy 73:11–20CrossRefGoogle Scholar
  13. Hamada YM (2015) Trigonometric Fourier- series solutions of the point reactor kinetics equations systems. Nucl Sci Des 281:142–153CrossRefGoogle Scholar
  14. Hennart JP (1977) Piecewise polynomial approximations for nuclear reactor point and space kinetics. Nucl Sci Eng 64:875–901CrossRefGoogle Scholar
  15. Hetrick DL (1993) Dynamics of Nuclear Reactors. American Nuclear Society (JBC), Illinois, USAGoogle Scholar
  16. Howell Kenneth B (2001) Principles of Fourier analysis. CRC Press, Boca Raton, FloridaCrossRefzbMATHGoogle Scholar
  17. Iserles A (2009) A first course in the numerical analysis of differential equations, 2nd edn. Cambridge University, CambridgezbMATHGoogle Scholar
  18. Konrd K (1954) Theory and application of infinite series. 4th edn, German Edition. Blackie & Son GlasgowGoogle Scholar
  19. Lawrence ES, Arnold JI, Stephen HF (2008) Elementary linear algebra: a matrix approach, 2nd edn. Pearson Education Inc, Upper Saddle River, NJ, p 7458Google Scholar
  20. Nahla AA (2010) Analytical solution to solve the point reactor kinetics equations. Nucl Sci Des 240:1622–1629CrossRefGoogle Scholar
  21. Nahla AA (2011) An efficient technique for the point reactor kinetics equations with Newtonian temperature feedback effects. Ann Nucl Energy 38:2810–2817CrossRefGoogle Scholar
  22. Nobrega J (1971) A new solution of the point kinetics. Nucl Sci Eng 46:366–375CrossRefGoogle Scholar
  23. Trefethen Lloyd N (2000) Spectral methods in MATLAB. SIAM, Philadelphia, PACrossRefzbMATHGoogle Scholar

Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2017

Authors and Affiliations

  1. 1.Department of Basic Science, Faculty of Computers and InformaticsSuez Canal UniversityIsmailiaEgypt

Personalised recommendations