This paper develops an ansatz method to derive the analytic solution of the neutron flux system under general initial conditions. Explicit closed series forms are established for the neutron flux and the delayed neutron concentration in terms of exponential functions. Existing results in the literature are recovered as special cases.

neutron diffusion, partial differential equation, analytic solution, ansatz method.

Received: October 25, 2023; Accepted: December 13, 2023; Published: February 12, 2024

How to cite this article: Tami M. T. Alsubie, Abdelhalim Ebaid, Amjad S. S. Albalawi, Saeed A. Alghamdi, Fawzi F. M. Alhamdi, Omar S. H. Alhamd, Salman M. M. Al-Anzi and Mona Aljoufi, Developing ansatz method for solving the neutron diffusion system under general physical conditions, Advances in Differential Equations and Control Processes 31(1) (2024), 15-26. http://dx.doi.org/10.17654/0974324324002

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:
[1] C. Ceolin, M. T. Vilhena, S. B. Leite and C. Z. Petersen, An analytical solution of the one-dimensional neutron diffusion kinetic equation in Cartesian geometry, International Nuclear Atlantic Conference – INAC 2009, Janerio, Brazil, 2009.
[2] S. M. Khaled, Exact solution of the one-dimensional neutron diffusion kinetic equation with one delayed precursor concentration in Cartesian geometry, AIMS Mathematics 7(7) (2022), 12364-12373. doi: 10.3934/math.2022686.
[3] A. A. Nahla and M. F. Al-Ghamdi, Generalization of the analytical exponential model for homogeneous reactor kinetics equations, J. Appl. Math. Volume 2012, Article ID 282367. https://doi.org/10.1155/2012/282367.
[4] F. Tumelero, C. M. F. Lapa, B. E. J. Bodmann and M. T. Vilhena, Analytical representation of the solution of the space kinetic diffusion equation in a one-dimensional and homogeneous domain, Braz. J. Radiat. Sci. 7 (2019), 1-13. https://doi.org/10.15392/bjrs.v7i2B.389.
[5] A. A. Nahla, F. Al-Malki and M. Rokaya, Numerical techniques for the neutron diffusion equations in the nuclear reactors, Adv. Stud. Theor. Phys. 6 (2012), 649 664.
[6] S. M. Khaled, Power excursion of the training and research reactor of Budapest University, Int. J. Nucl. Energy Sci. Technol. 3 (2007), 42-62.
https://doi.org/10.1504/IJNEST.2007.012440.
[7] S. M. Khaled and F. A. Mutairi, The influence of different hydraulics models in treatment of some physical processes in super critical states of light water reactors, Int. J. Nucl. Energy Sci. Technol. 8 (2014), 290-309. https://doi.org/10.1504/IJNEST.2014.064940.
[8] S. Dulla, P. Ravetto, P. Picca and D. Tomatis, Analytical benchmarks for the kinetics of accelerator driven systems, Joint International Topical Meeting on Mathematics and Computation and Supercomputing in Nuclear Applications, Monterey-California, on CD-ROM, 2007.
[9] M. A. Al-Sharif, A. Ebaid, H. S. Alrashdi, A. H. Alenazy and N. E. Kanaan, A novel ansatz method for solving the neutron diffusion system in Cartesian geometry, Journal of Advances in Mathematics and Computer Science 37(11) (2022), 90-99.
[10] Abdulrahman F. Aljohani, Abdelhalim Ebaid, Emad H. Aly, Ioan Pop, Ahmed O. M. Abubaker and Dalal J. Alanazi, Explicit solution of a generalized mathematical model for the solar collector/photovoltaic applications using nanoparticles, Alexandria Engineering Journal 67 (2023), 447-459. https://doi.org/10.1016/j.aej.2022.12.044.
[11] S. M. Khaled, The exact effects of radiation and Joule heating on magnetohydrodynamic Marangoni convection over a flat surface, Therm. Sci. 22 (2018), 63-72.
[12] Mona D. Aljoufi, Exact solution of a solar energy model using four different kinds of nanofluids: advanced application of Laplace transform, Case Studies in Thermal Engineering 50 (2023), 103396.
https://doi.org/10.1016/j.csite.2023.103396.
[13] A. Ebaid, W. Alharbi, M. D. Aljoufi and E. R. El-Zahar, The exact solution of the falling body problem in three-dimensions: comparative study, Mathematics 8(10) (2020), 1726. https://doi.org/10.3390/math8101726.
[14] A. Ebaid, C. Cattani, A. S. Al. Juhani and E. R. El-Zahar, A novel exact solution for the fractional Ambartsumian equation, Adv. Differ. Equ. Volume 2021, Article number 88. https://doi.org/10.1186/s13662-021-03235-w.
[15] A. Ebaid and H. K. Al-Jeaid, The Mittag-Leffler functions for a class of first-order fractional initial value problems: dual solution via Riemann-Liouville fractional derivative, Fractal Fract. 6 (2022), 85.
https://doi.org/10.3390/fractalfract6020085.
[16] A. F. Aljohani, A. Ebaid, E. A. Algehyne, Y. M. Mahrous, C. Cattani and H. K. Al-Jeaid, The Mittag-Leffler function for re-evaluating the chlorine transport model: comparative analysis, Fractal Fract. 6 (2022), 125.
https://doi.org/10.3390/fractalfract6030125.
[17] A. F. Aljohani, A. Ebaid, E. A. Algehyne, Y. M. Mahrous, P. Agarwal, M. Areshi and H. K. Al-Jeaid, On solving the chlorine transport model via Laplace transform, Sci. Rep. 12 (2022), 12154.
https://doi.org/10.1038/s41598-022-14655-3.
[18] R. Haberman, Elementary Applied Partial Differential Equations, 2nd ed., Prentice-Hall, Inc., USA, 1987.
[19] A. Ebaid, Approximate periodic solutions for the non-linear relativistic harmonic oscillator via differential transformation method, Commun. Nonlinear Sci. Numer. Simul. 15 (2010), 1921-1927.
[20] Y. Liu and K. Sun, Solving power system differential algebraic equations using differential transformation, IEEE Trans. Power Syst. 35 (2020), 2289-2299. https://doi.org/10.1109/TPWRS.2019.2945512.
[21] S. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, CRC Press, Boca Raton, FL, USA, 2003.
[22] A. Chauhan and R. Arora, Application of homotopy analysis method (HAM) to the non-linear KdV equations, Commun. Math. 31 (2023), 205-220. https://doi.org/10.46298/cm.10336.
[23] A. Ebaid, Remarks on the homotopy perturbation method for the peristaltic flow of Jeffrey fluid with nano-particles in an asymmetric channel, Comput. Math. Appl. 68 (2014), 77-85.
[24] M. Nadeem and F. Li, He-Laplace method for nonlinear vibration systems and nonlinear wave equations, J. Low Freq. Noise Vib. Act. Control. 38 (2019), 1060 1074.
[25] A. Ebaid, A. F. Aljohani and E. H. Aly, Homotopy perturbation method for peristaltic motion of gold-blood nanofluid with heat source, Int. J. Numer. Meth. Heat Fluid Flow 30 (2020), 3121-3138.
https://doi.org/10.1108/HFF-11-2018-0655.
[26] M. Nadeem, S. A. Edalatpanah, I. Mahariq and W. H. F. Aly, Analytical view of nonlinear delay differential equations using Sawi iterative scheme, Symmetry 14 (2022), 2430. https://doi.org/10.3390/sym14112430.
[27] Z. Ayati and J. Biazar, On the convergence of Homotopy perturbation method, J. Egyptian Math. Soc. 23 (2015), 424-428.
[28] J. S. Duan and R. Rach, A new modification of the Adomian decomposition method for solving boundary value problems for higher order nonlinear differential equations, Appl. Math. Comput. 218 (2011), 4090-4118.
[29] A. Ebaid, A new analytical and numerical treatment for singular two-point boundary value problems via the Adomian decomposition method, J. Comput. Appl. Math. 235 (2011), 1914-1924.
[30] S. Bhalekar and J. Patade, An analytical solution of Fisher’s equation using decomposition method, Am. J. Comput. Appl. Math. 6 (2016), 123-127.
[31] A. H. S. Alenazy, A. Ebaid, E. A. Algehyne and H. K. Al-Jeaid, Advanced study on the delay differential equation Mathematics 10 (2022), 4302. https://doi.org/10.3390/math10224302.