Fractional Approach to Two-Group Neutron Diffusion in Slab Reactors
(1) * Iqbal M. Batiha Mail (1) Department of Mathematics, Al Zaytoonah University of Jordan, Amman 11733, Jordan. 2) Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman, United Arab Emirates)
(2) Nadia Allouch Mail (University of Mostaganem, Algeria)
(3) Mohammed Shqair Mail (Zarqa University, Jordan)
(4) Iqbal H. Jebril Mail (Al Zaytoonah University of Jordan, Jordan)
(5) Shawkat Alkhazaleh Mail (Jadara University, Jordan)
(6) Shaher Momani Mail (1) Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman, United Arab Emirates. 2) Department of Mathematics, The University of Jordan, Amman, Jordan)
*corresponding author

Abstract


The two-energy neutron diffusion model in slab reactors characterizes neutron behavior across two energy groups: fast and thermal. Fast neutrons, generated by fission, decelerate through collisions, transitioning into thermal neutrons. This model employs diffusion equations to compute neutron flux distributions and reactor parameters, thereby optimizing reactor design and safety to ensure efficient neutron utilization and stable, sustained nuclear reactions. The primary objective of this research is to explore both analytical and numerical solutions to the two-energy neutron diffusion model in slab reactors. Specifically, we will utilize the Laplace transform method for an analytical solution of the two-energy neutron diffusion model. Subsequently, employing the Caputo differentiator, we transform the original neutron diffusion model into its fractional-order equivalents, yielding the fractional-order two-energy group neutron diffusion model in slab reactors. To address the resulting fractional-order system, we develop a novel approach aimed at reducing the 2β-order system to a β-order system, where β ∈ (0, 1]. This transformed system is then solved using the Modified Fractional Euler Method (MFEM), an advanced variation of the fractional Euler method. Finally, we present numerical simulations that validate our results and demonstrate their applicability.


Keywords


Two-Energy Group Neutron Diffusion Model; Slab Reactors; Fractional Calculus; Modified Fractional Euler Method; Numerical Simulations

   

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https://doi.org/10.31763/ijrcs.v5i1.1524 		      

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References


[1] J. J. Duderstadt and L. J. Hamilton, Nuclear Reactor Analysis, John Wiley & Sons: Hoboken, 1976, https://www.wiley.com/en-us/exportProduct/pdf/9780471223634.

[2] J. S. Cassell and M. M. R. Williams, “A solution of the neutron diffusion equation for a hemisphere with mixed boundary conditions,†Annals of Nuclear Energy, vol. 31, no. 17, pp. 1987-2004, 2004, https://doi.org/10.1016/j.anucene.2004.04.008.

[3] J. R. Lamarsh, Introduction to Nuclear Engineering, U.S. Department of Energy Office of Scientific and Technical Information, 1975, https://www.osti.gov/biblio/5272097.

[4] A. Nahla, F. Al-Malki, and M. Rokaya, “Numerical techniques for the neutron diffusion equations in the nuclear reactors,†Advanced Studies in Theoretical Physics, vol. 6, no. 14, pp. 649-664, 2012, https://www.m-hikari.com/astp/astp2012/astp13-16-2012/nahlaASTP13-16-2012.pdf.

[5] S. Ray and 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, vol. 1, pp. 1-12, 2011, https://www.researchgate.net/publication/281335348_Application_of_homotopy_analysis_method_and_adomian_decomposition_method_for_the_solution_of_neutron_diffusion_equation_in_the_hemisphere_and_cylindrical_reactors.

[6] M. Shqair, “Developing a new approaching technique of homotopy perturbation method to solve two group reflected cylindrical reactor,†Results in Physics, vol. 12, pp. 1880-1887, 2019, https://doi.org/10.1016/j.rinp.2019.01.063.

[7] 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 and Fractional, vol. 7, no. 3, p. 219, 2023, https://doi.org/10.3390/fractalfract7030219.

[8] 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, vol. 8, no. 8, pp. 19297-19312, 2023, https://doi.org/10.3934/math.2023984.

[9] A. Aboanber, A. Nahla, and S. Aljawazneh, “Fractional two energy groups matrix representation for nuclear reactor dynamics with an external source,†Annals of Nuclear Energy, vol. 153, p. 108062, 2021, https://doi.org/10.1016/j.anucene.2020.108062.

[10] R. A. El-Nabulsi, “Nonlocal effects to neutron diffusion equation in a nuclear reactor,†Journal of Computational and Theoretical Transport, vol. 49, no. 6, pp. 267-281, 2020, https://doi.org/10.1080/23324309.2020.1816551.

[11] S. M. Khaled, “Exact solution of the one-dimensional neutron diffusion kinetic equation with one delayed precursor concentration in Cartesian geometry,†AIMS Mathematics, vol. 7, no. 7, pp. 12364-12373, 2022, https://doi.org/10.3934/math.2022686.

[12] K. Khasawneh, S. Dababneh, and Z. Odibat, “A solution of the neutron diffusion equation in hemispherical symmetry using the homotopy perturbation method,†Annals of Nuclear Energy, vol. 36, no. 11-12, pp. 1711-1717, 2009, https://doi.org/10.1016/j.anucene.2009.09.001.

[13] T. Sardar, S. Ray, and R. Bera, “The solution of coupled fractional neutron diffusion equations with delayed neutrons,†International Journal of Nuclear Energy Science and Technology, vol. 5, no. 2, pp. 105-133, 2009, https://doi.org/10.1504/IJNEST.2010.030552.

[14] W. G. Alshanti, “Solutions of linear and non-linear partial differential equations by means of tensor product theory of Banach space,†Electronic Journal of Differential Equations, vol. 2024, no. 01-83, pp. 1-8, 2024, https://doi.org/10.58997/ejde.2024.28.

[15] S. Momani, I. M. Batiha, A. Abdelnebi, and I. H. Jebril, “A powerful tool for dealing with high- dimensional fractional-order systems with applications to fractional Emden–Fowler systems,†Chaos, Solitons & Fractals: X, vol. 12, p. 100110, 2024, https://doi.org/10.1016/j.csfx.2024.100110.

[16] I. M. Batiha, I. H. Jebril, A. Abdelnebi, Z. Dahmani, S. Alkhazaleh, and N. Anakira, “A new fractional representation of the higher order Taylor scheme,†Computational and Mathematical Methods, vol. 2024, p. 2849717, 2024, https://doi.org/10.1155/2024/2849717.

[17] I. M. Batiha, J. Oudetallah, A. Ouannas, A. A. Al-Nana, and I. H. Jebril, “Tuning the fractional-order PID-controller for blood glucose level of diabetic patients,†International Journal of Advances in Soft Computing and its Applications, vol. 13, no. 2, pp. 1-10, 2021, http://www.i-csrs.org/Volumes/ijasca/2021.2.1.pdf.

[18] I. M. Batiha, S. A. Njadat, R. M. Batyha, A. Zraiqat, A. Dababneh, and S. Momani, “Design fractional order PID controllers for single-joint robot arm model,†International Journal of Advances in Soft Computing and its Applications, vol. 14, no. 2, pp. 96-114, 2022, https://doi.org/10.15849/IJASCA.220720.07.

[19] I. M. Batiha, A. Benguesmia, T. E. Oussaeif, I. H. Jebril, A. Ouannas, and S. Momani, “Study of a superlinear problem for a time fractional parabolic equation under integral over-determination condition,†IAENG International Journal of Applied Mathematics, vol. 54, no. 2, pp. 187-193, 2024, https://www.iaeng.org/IJAM/issues_v54/issue_2/IJAM_54_2_05.pdf.

[20] I. M. Batiha, N. Allouch, I. H. Jebril, and S. Momani, “A robust scheme for reduction of higher fractional order systems,†Journal of Engineering Mathematics, vol. 144, no. 4, 2024, https://doi.org/10.1007/s10665-023-10310-6.

[21] I. M. Batiha, A. Bataihah, A. A. Al-Nana, S. Alshorm, I. H. Jebril, and A. Zraiqat, “A numerical scheme for dealing with fractional initial value problem,†International Journal of Innovative Computing, Information and Control, vol. 19, no. 3, pp. 763–774, 2023, https://doi.org/10.24507/ijicic.19.03.763.

[22] I. M. Batiha, S. Alshorm, A. Al-Husban, R. Saadeh, G. Gharib, and S. Momani, “The n-point composite fractional formula for approximating Riemann–Liouville integrator,†Symmetry, vol. 15, no. 4, p. 938, 2023, https://doi.org/10.3390/sym15040938.

[23] T. Hamadneh et al., “General methods to synchronize fractional discrete reaction–diffusion systems applied to the glycolysis model,†Fractal and Fractional, vol. 7, no. 11, p. 828, 2023, https://doi.org/10.3390/fractalfract7110828.

[24] I. M. Batiha, O. Talafha, O. Y. Ababneh, S. Alshorm, and S. Momani, “Handling a commensurate, in commensurate, and singular fractional-order linear time-invariant system,†Axioms, vol. 12, no. 18, p. 771, 2023, https://doi.org/10.3390/axioms12080771.

[25] N. R. Anakira, O. Ababneh, A. S. Heilat, and I. Batiha, “A new accurate approximate solution of singular two-point boundary value problems,†General Letters in Mathematics, vol. 12, no. 1, pp. 31–39, 2022, https://doi.org/10.31559/glm2022.12.1.4.

[26] R. B. Albadarneh, A. K. Alomari, N. Tahat, and I. M. Batiha, “Analytic solution of nonlinear singular BVP with multi-order fractional derivatives in electrohydrodynamic flows,†Turkish World Mathematical Society Journal of Applied and Engineering Mathematics, vol. 11, no. 4, pp. 1125–1137, 2021, http://jaem.isikun.edu.tr/web/index.php/archive/113-vol11-no4/770.

[27] I. M. Batiha, R. El-Khazali, A. AlSaedi, and S. Momani, “The general solution of singular fractional- order linear time-invariant continuous systems with regular pencils,†Entropy, vol. 20, no. 6, p. 400, 2018, https://doi.org/10.3390/e20060400.

[28] I. M. Batiha, N. Barrouk, A. Ouannas, and A. Farah, “A study on invariant regions, existence and uniqueness of the global solution for tridiagonal reaction-diffusion systems,†Journal of Applied Mathematics and Informatics, vol. 41, no. 4, pp. 893–906, 2023, https://doi.org/10.14317/jami.2023.893.

[29] I. Batiha, A. Ouannas, and J. Emwas, “A stabilization approach for a novel chaotic fractional-order discrete neural network,†Journal of Mathematical and Computational Science, vol. 11, no. 5, pp. 5514–5524, 2021, https://doi.org/10.28919/jmcs/6004.

[30] Z. Chebana, T. E. Oussaeif, S. Dehilis, A. Ouannas, and I. M. Batiha, “On nonlinear Neumann integral condition for a semilinear heat problem with blowup simulation,†Palestine Journal of Mathematics, vol. 12, no. 3, pp. 389-394, 2023, https://www.researchgate.net/publication/375084322.

[31] R. B. Albadarneh, A. Abbes, A. Ouannas, I. M. Batiha, and T. E. Oussaeif, “On chaos in the fractional order discrete-time macroeconomic systems,†AIP Conference Proceedings, vol. 2849, no. 1, p. 030014, 2023, https://doi.org/10.1063/5.0162686.

[32] I. M. Batiha, N. Djenina, and A. Ouannas, “A stabilization of linear incommensurate fractional-order difference systems,†AIP Conference Proceedings, vol. 2849, no. 1, p. 030013, 2023, https://doi.org/10.1063/5.0164866.

[33] N. Abdelhalim, A. Ouannas, I. Rezzoug, and I. M. Batiha, “A study of a high–order time–fractional partial differential equation with purely integral boundary conditions,†Fractional Differential Calculus, vol. 13, no. 2, pp. 199—210, 2023, https://doi.org/10.7153/fdc-2023-13-13.

[34] A. A. Khennaoui, A. O. Almatroud, A. Ouannas, M. M. Al-sawalha, G. Grassi, V. T. Pham, and I. M. Batiha, “An unprecedented 2-dimensional discrete-time fractional-order system and its hidden chaotic attractors,†Mathematical Problems in Engineering, vol. 2021, p. 6768215, 2021, https://doi.org/10.1155/2021/6768215.

[35] A. Bouchenak, I. M. Batiha, M. Aljazzazi, I. H. Jebril, M. Al-Horani, and R. Khalil, “Atomic exact solution for some fractional partial differential equations in Banach spaces,†Partial Differential Equations in Applied Mathematics, vol. 9, p. 100626, 2024, https://doi.org/10.1016/j.padiff.2024.100626.

[36] I. M. Batiha, I. H. Jebril, A. A. Al-Nana, and S. Alshorm, “A simple harmonic quantum oscillator: fractionalization and solution,†Mathematical Models in Engineering, vol. 10, no. 1, pp. 26–34, 2024, https://doi.org/10.21595/mme.2024.23904.

[37] I. M. Batiha, Z. Chebana, T. E. Oussaeif, A. Ouannas, and I. H. Jebril, “On a weak Solution of a fractional order temporal equation,†Mathematics and Statistics, vol. 10, no. 5, pp. 1116–1120, 2022, https://doi.org/10.13189/ms.2022.100522.

[38] N. Anakira, Z. Chebana, T. E. Oussaeif, I. M. Batiha, and A. Ouannas, “A study of a weak solution of a diffusion problem for a temporal fractional differential equation,†Nonlinear Functional Analysis and Applications, vol. 27, no. 3, pp. 679–689, 2022, https://doi.org/10.22771/nfaa.2022.27.03.14.

[39] I. M. Batiha, I. Rezzoug, T. E. Oussaeif, A. Ouannas, and I. H. Jebril, “Pollution detection for the singular linear parabolic equation,†Journal of Applied Mathematics and Informatics, vol. 41, no. 3, pp. 647–656, 2023, https://doi.org/10.14317/jami.2023.647.

[40] Z. Chebana, T. E. Oussaeif, A. Ouannas, and I. M. Batiha, “Solvability of Dirichlet problem for a fractional partial differential equation by using energy inequality and Faedo-Galerkin method,†Innovative Journal of Mathematics, vol. 1, no. 1, pp. 34–44, 2022, https://doi.org/10.55059/ijm.2022.1.1/4.

[41] I. M. Batiha, “Solvability of the solution of superlinear hyperbolic Dirichlet problem,†International Journal of Analysis and Applications, vol. 20, p. 62, 2022, https://doi.org/10.28924/2291-8639-20-2022-62.

[42] I. M. Batiha, Z. Chebana, T. E. Oussaeif, A. Ouannas, S. Alshorm, and A. Zraiqat, “Solvability and dynamics of superlinear reaction diffusion problem with integral condition,†IAENG International Journal of Applied Mathematics, vol. 53, no. 1, pp. 113–121, 2023, https://www.iaeng.org/IJAM/issues v53/issue1/IJAM_53_1_15.pdf.

[43] I. M. Batiha, Z. Chebana, T. E. Oussaeif, A. Ouannas, I. H. Jebril, and M. Shatnawi, “Solvability of nonlinear wave equation with nonlinear integral Neumann conditions,†International Journal of Analysis and Applications, vol. 21, p. 34, 2023, https://doi.org/10.28924/2291-8639-21-2023-34.

[44] I. M. Batiha, A. Ouannas, R. Albadarneh, A. A. Al-Nana, and S. Momani, “Existence and uniqueness of solutions for generalized Sturm–Liouville and Langevin equations via Caputo–Hadamard fractional order operator,†Engineering Computations, vol. 39, no. 7, pp. 2581–2603, 2022, https://doi.org/10.1108/EC-07-2021-0393.

[45] T. E. Oussaeif, B. Antara, A. Ouannas, I. M. Batiha, K. M. Saad, H. Jahanshahi, A. M. Aljuaid, and A. A. Aly, “Existence and uniqueness of the solution for an inverse problem of a fractional diffusion equation with integral condition,†Journal of Function Spaces, vol. 2022, p. 7667370, 2022, https://doi.org/10.1155/2022/7667370.

[46] I. M. Batiha, N. Barrouk, A. Ouannas, and W. G. Alshanti, “On global existence of the fractional reaction diffusion system’s solution,†International Journal of Analysis and Applications, vol. 11, 2023, https://doi.org/10.28924/2291-8639-21-2023-11.

[47] I. M. Batiha , A. Bataiha, A. Al-Nana, S. Alshorm, I. H. Jebril, and A. Zraiqat, “A numerical scheme for dealing with fractional initial value problem,†International Journal of Innovative Computing, Information and Control, vol. 19, no. 3, pp. 763-774, 2023, https://doi.org/10.24507/ijicic.19.03.763.

[48] R. B. Albadarneh, I. Batiha, A. K. Alomari, and N. Tahat, “Numerical approach for approximating the Caputo fractional-order derivative operator,†AIMS Mathematics, vol. 6, no. 11, pp. 12743-12756, 2021, https://doi.org/10.3934/math.2021735.

[49] R. B. Albadarneh, I. M. Batiha, A. Adwai, N. Tahat, and A. K. Alomari, “Numerical approach of Riemann-Liouville fractional derivative operator,†International Journal of Electrical and Computer Engineering, vol. 11, no. 6, pp. 5367-5378, 2021, https://doi.org/10.11591/ijece.v11i6.pp5367-5378.

[50] R. B. Albadarneh, A. M. Adawi, S. Al-Sa’di, I. M. Batiha, and S. Momani, “A pro rata definition of the fractional-order derivative,†in Mathematics and Computation, vol. 418, pp. 65–79, 2023, https://doi.org/10.1007/978-981-99-0447-1_6.

[51] I. H. Jebril, M. S. El-Khatib, A. A. Abubaker, S. B. Al-Shaikh, and I. M. Batiha, “Results on Katugampola fractional derivatives and integrals,†International Journal of Analysis and Applications, vol. 21, p. 113, 2023, https://doi.org/10.28924/2291-8639-21-2023-113.

[52] I. M. Batiha, S. Alshorm, A. Ouannas, S. Momani, O. Y. Ababneh, and M. Albdareen, “Modified three point fractional formulas with Richardson extrapolation,†Mathematics, vol. 10, no. 19, p. 3489, 2022, https://doi.org/10.3390/math10193489.

[53] I. M. Batiha, S. Alshorm, I. Jebril, A. Zraiqat, Z. Momani, and S. Momani, “Modified 5-point fractional formula with Richardson extrapolation,†AIMS Mathematics, vol. 8, no. 4, pp. 9520-9534, 2023, https://doi.org/10.3934/math.2023480.

[54] I. M. Batiha, A. A. Abubaker, I. H. Jebril, S. B. Al-Shaikh, and K. Matarneh, “New algorithms for dealing with fractional initial value problems,†Axioms, vol. 12, no. 15, p. 488, 2023, https://doi.org/10.3390/axioms12050488.

[55] A. A. Al-Nana, I. M. Batiha, and S. Momani, “A numerical approach for dealing with fractional boundary value problems,†Mathematics, vol. 11, no. 19, p. 4082, 2023, https://doi.org/10.3390/math11194082.

[56] I. M. Batiha, I. H. Jebril, S. Alshorm, A. A. Al-nana, S. Alkhazaleh, and S. Momani, “Handling systems of fractional stochastic differential equations using modified fractional Euler method,†Global and Stochastic Analysis, vol. 11, no. 1, pp. 95-105, 2024, https://openurl.ebsco.com/EPDB%3Agcd%3A16%3A20121664/detailv2?sid=ebsco%3Aplink%3Ascholar&id=ebsco%3Agcd%3A176088420&crl=c&link_origin=scholar.google.com.

[57] A. A. Kilbas, Theory and Application of Fractional Differential Equations, Elsevier: Amsterdam, 2006, https://books.google.co.id/books?id=uxANOU0H8IUC&hl=id&source=gbs_navlinks_s.

[58] B. Jin, Fractional Differential Equations, Spring Cham, 2021, https://doi.org/10.1007/978-3-030-76043-4.

[59] I. M. Batiha, S. Alshorm, I. H. Jebril, and M. Hammad, “A brief review about fractional calculus,†International Journal of Open Problems in Computer Science and Mathematics, vol. 15, no. 4, pp. 39-56, 2022, https://www.researchgate.net/profile/Iqbal-Batiha/publication/366839100_A_Brief_Review_about_Fractional_Calculus/links/63f4a312574950594531b537/A-Brief-Review-about-FractionalCalculus.pdf.

[60] Z. M. Odibat and S. Momani, “An algorithm for the numerical solution of differential equations of fractional order,†The Korean Society for Computational and Applied Mathematics, vol. 26, no. 1 2, pp. 15-27, 2008, https://koreascience.kr/article/JAKO200833338752380.pdf.

[61] R. E. Bellman and R. S. Roth, The Laplace Transform, World Scientific: Singapore, 1984, https://books.google.co.id/booksid=roFjH53691wC&dq=The+Laplace+Transform&lr=&hl=id&source=gbs_navlinks_s.

[62] A. Sood, R. D. Forster, and R. Parsons, “Analytical benchmark test set for criticality code verification,†Progress in Nuclear Energy, vol. 42, no. 1, pp. 55-106, 2003, https://doi.org/10.1016/S0149-1970(02)00098-7.


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