Two Energy groups Neutron Diffusion Model in Spherical Reactors
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Abstract
This paper investigates the neutron diffusion model with two energy groups in spherical reactors. In particular, the integer-order two energy groups neutron diffusion model in spherical reactors is resolved using the Laplace transform method by regarding the spherical radius r as a time domain. Next, we transform the neutron diffusion model into fractional-order versions using the Caputo differentiator, resulting in what is referred to as the fractional-order two-energy-group neutron diffusion model. To address this fractional-order system, we introduce a novel approach to reduce a system of 2α-order to a duplicated system of α-order, where 0 < α ≤ 1. This converted system is then solved using one of the recent modifications of the fractional Euler method called the Modified Fractional Euler Method (MFEM). Several numerical simulations are depicted to verify our findings.
References
Duderstadt, J.J.; Hamilton, L.J. Nuclear Reactor Analysis; John Wiley, Sons: Boston, MA, USA, 1976.
Lamarsh, J.R. Introduction to Nuclear Engineering, 2nd ed.; Addison-Wesley: Boston, MA, USA, 1983.
The stability of a nuclear power plant, Journal of Nuclear Energy. Part A. Reactor Science, Volume 12, Issue 4,1960.
Nairat, M., Shqair, M., & Alhalholy, T. (2019). Cylindrically symmetric fractional Helmholtz equation. Appl. Math.
E-Notes, 19, 708–717.
S. Dababneh, K. Khasawneh, and Z. Odibat, An alternative solution of the neutron diffusion equation in cylindrical
symmetry, Ann. Nucl. Energy, 38 (2010) 1140–1143.
M. Shqair, Developing a new approaching technique of homotopy perturbation method to solve two groups reflected
cylindrical reactor. Results Phys. 12 (2019) 1880–1887.
Shqair, M., Farrag, E. A., & Al-Smadi, M. (2022). Solving Multigroup Reflected Spherical Reactor System of Equations
Using the Homotopy Perturbation Method. Mathematics, 10 (10), 1784.
Shqair, M.; Ghabar, I.; Burqan, A. Using Laplace Residual Power Series Method in Solving Coupled Fractional
Neutron Diffusion Equations with Delayed Neutrons System. Fractal Fract. 2023, 7, 219. https://doi.org/10.3390/
fractalfract7030219
Khaled, S. Exact solution of the one-dimensional neutron diffusion kinetic equation with one delayed precursor concentration in Cartesian geometry, AIMS Math. 2022, 7, 12364–12373.
Aboanber, A.; Nahla, A.; Aljawazneh, S. Fractional two energy groups matrix representation for nuclear reactor
dynamics with an external source. Ann. Nucl. Energy 2021, 153, 108062.
Sardar, T.; Ray, S.; Bera, R. The solution of coupled fractional neutron diffusion equations with delayed neutrons. Int.
J. Nuclear Energy Sci. Technol. 2010, 5, 105–133.
R. A. El-Nabulsi, Nonlocal effects to neutron diffusion equation in a nuclear reactor, Journal of Computational and
Theoretical Transport, 49(6), 2020, 267–281.
R. A. El-Nabulsi, Neutrons diffusion variable coefficient advection in nuclear reactors, International Journal of
Advanced Nuclear Reactor Design and Technology, 3, 2021, 102–107.
A. Nahla, F. Al-Malki, M. Rokaya, Numerical Techniques for the Neutron Diffusion Equations in the Nuclear Reactors,
Adv. Studies Theor. Phys., 6 (2012) 649–664.
K. Khasawneh, S. Dababneh, Z. Odibat, A solution of the neutron diffusion equation in hemispherical symmetry using
the homotopy perturbation method. Annals of Nuclear Energy, 36 (2009) 1711–1717.
A. Sood, R. Forster, D. Parsons, Analytical Benchmark Test Set for Criticality Code Verification. Los Alamos National
Laboratory, US (1999).
A. Burqan, M. Shqair, A. El-Ajou, S. M. E. Ismaeel and Z. Al-Zhour, Analytical solutions to the coupled fractional
neutron diffusion equations with delayed neutrons system using Laplace transform method, AIMS Mathematics, 8(8):
–19312 (2023).
Dababneh, Amer, Amjed Zraiqat, Abdulkarim Farah, Hassan Al-Zoubi, and Ma’Mon Abu Hammad. “Numerical methods for finding periodic solutions of ordinary differential equations with strong nonlinearity.” J. Math. Comput. Sci.
, no. 6 (2021): 6910–6922.
Amal Benguesmia, Iqbal M. Batiha, Taki-Eddine Oussaeif, Adel Ouannas, Waseem G. Alshanti, Inverse Problem of
a Semilinear Parabolic Equation with an Integral Overdetermination Condition, Nonlinear Dynamics and Systems
Theory, Vol. 23, Issue 3, Pages 249–260, 2023.
S. Ray, A. Patra, Application of homotopy analysis method and Adomian decomposition method for the solution of neutron diffusion equation in the hemisphere and cylindrical reactors, Journal of Nuclear Engineering and Technology,
(2011), 1–14.
S. M. Khaled, Exact solution of the one-dimensional neutron diffusion kinetic equation with one delayed precursor concentration in Cartesian geometry, AIMS Mathematics, 2022, Volume 7, Issue 7: 12364-12373. doi: 10.3934/
math.2022686
Cassell, J.S., Williams, M.M.R., 2004. A solution of the neutron diffusion equation for a hemisphere with mixed boundary conditions. Annals of Nuclear Energy 31, 1987–2004.
M. Shqair, I. Ghabar and A. Burqan, Using Laplace Residual Power Series Method in Solving Coupled Fractional
Neutron Diffusion Equations with Delayed Neutrons System, Fractal Fract., 7, 219, (2023).
S. Gimnitz Simon, B. Bira, Dia Zeidan, Optimal systems, series solutions and conservation laws for a time fractional
cancer tumor model, Chaos, Solitons & Fractals, Volume 169, April 2023, 113311.
Ahmed Salem Heilat, Belal Batiha, Tariq Qawasmeh, Raed Hatamleh, Hybrid Cubic B -spline Method for Solving A
Class of Singular Boundary Value Problems, European Journal of Pure and Applied Mathematics, Volume 16, Issue
, Pages 751–762, 2023. Journal
Qawasmeh T, Tallafha A, Shatanawi W. Fixed Point Theorems through Modified ω-Distance and Application to
Nontrivial Equations. Axioms. 2019; 8(2):57. https://doi.org/10.3390/axioms8020057
Bataihah A, Shatanawi W, Qawasmeh T, Hatamleh R. On -Simulation Functions and Fixed Point Results in the
Setting of ωt-Distance Mappings with Application on Matrix Equations. Mathematics. 2020; 8(5):837. https://doi.
org/10.3390/math8050837
Batiha Iqbal, Bataih Anwar, Al-Nana Abeer, Alshorm Shameseddin, Jebril Iqbal, Zraiqat Amjed, A Numerical Scheme
for Dealing with Fractional Initial Value Problem, International Journal of Innovative Computing, Information and
Control, Vol. 19, Issue 3, pp. 763–774, 2023.
Batiha, I.M., Abubaker, A.A., Jebril, I.H., Al-Shaikh, S.B., Matarneh, K. New Algorithms for Dealing with Fractional
Initial Value Problems. Axioms 2023, 12, 488. https://doi.org/10.3390/axioms12050488
S. Alshorm, I. M. Batiha, I. Jebril and A. Dababneh, Handling Systems of Incommensurate Fractional Order Equations
Using Improved Fractional Euler Method, 2023 International Conference on Information Technology (ICIT), Amman,
Jordan, 2023, pp. 657–660, doi: 10.1109/ICIT58056.2023.10226115.
I. M. Batiha, S. Alshorm, A. Zraiqat, M. Alia and I. Jebril, Numerical Solution for Incommensurate System of Fractional
Order Differential Equations, 2023 International Conference on Information Technology (ICIT), Amman, Jordan,
, pp. 652–656, doi: 10.1109/ICIT58056.2023.10225807.
Kilbas, A. A. (2006) Theory and Application of Fractional Differential Equations. Elsevier, Amsterdam.
Batiha, I.M., Talafha, O., Ababneh, O.Y., Alshorm, S., Momani, S. Handling a Commensurate, Incommensurate,
and Singular Fractional-Order Linear Time-Invariant System. Axioms 2023, 12, 771. https://doi.org/10.3390/
axioms12080771
Batiha, Iqbal M., Shameseddin Alshorm, IqbalL H. Jebril, and Mamon Abu Hammad. “A Brief Review about Fractional
Calculus.” Int. J. Open Problems Compt Math 15, no. 4 (2022).
Batiha, Iqbal M., Shameseddin Alshorm, Abdallah Al-Husban, Rania Saadeh, Gharib Gharib, and Shaher Momani.
“The n-Point Composite Fractional Formula for Approximating Riemann–Liouville Integrator.” Symmetry 15, no. 4
(2023): 938.
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