Solving the 1D neutron diffusion kinetic equation under mixed boundary conditions: Explicit solution

Essam R. El-Zahar; Abdelhalim Ebaid; Laila F. Seddek; S. M. Khaled; Essam R. El-Zahar; Abdelhalim Ebaid; Laila F. Seddek; S. M. Khaled

doi:10.3934/math.20251207

AIMS Mathematics

2025, Volume 10, Issue 11: 27462-27479. doi: 10.3934/math.20251207

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Solving the 1D neutron diffusion kinetic equation under mixed boundary conditions: Explicit solution

1.
Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, P.O. Box 83, Al-Kharj 11942, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, University of Tabuk, P.O. Box 741, Tabuk 71491, Saudi Arabia
3.
Department of Mathematics, Faculty of Science, Helwan University, P.O. Box 11795, Cairo, Egypt

Received: 09 August 2025 Revised: 03 November 2025 Accepted: 10 November 2025 Published: 25 November 2025
MSC : 83C15, 44A10, 35Q70

Obtaining accurate solutions for mathematical models of neutron diffusion systems may lead to a deeper understanding of processes in reactor physics. The present paper applies the Laplace transform to the time-dependent neutron diffusion equation (together with the delayed neutron precursor equation) under a reflective boundary condition at one edge. The residue theorem is employed to obtain the inverse transform, leading to a series solution structured as a modal expansion associated with the eigenvalues of a transcendental equation. Moreover, the obtained series solution is theoretically proven to converge. The numerical results show acceptable accuracy based on residual errors. Physically, the neutron flux exhibits oscillatory behavior within the spatial domain, resulting in a wave-like alternating surface. Additionally, the delayed neutron precursor concentration stabilizes over time, gradually approaching a stationary profile, which is consistent with the physical expectations. The results also support the effectiveness of the Laplace transform technique in capturing the early-time behavior of the system. Differences between the present results and those reported in the relevant literature are explained.
- neutron diffusion,
- reactor physics,
- partial differential equation,
- analytic solution,
- Laplace transform
Citation: Essam R. El-Zahar, Abdelhalim Ebaid, Laila F. Seddek, S. M. Khaled. Solving the 1D neutron diffusion kinetic equation under mixed boundary conditions: Explicit solution[J]. AIMS Mathematics, 2025, 10(11): 27462-27479. doi: 10.3934/math.20251207

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Abstract

Obtaining accurate solutions for mathematical models of neutron diffusion systems may lead to a deeper understanding of processes in reactor physics. The present paper applies the Laplace transform to the time-dependent neutron diffusion equation (together with the delayed neutron precursor equation) under a reflective boundary condition at one edge. The residue theorem is employed to obtain the inverse transform, leading to a series solution structured as a modal expansion associated with the eigenvalues of a transcendental equation. Moreover, the obtained series solution is theoretically proven to converge. The numerical results show acceptable accuracy based on residual errors. Physically, the neutron flux exhibits oscillatory behavior within the spatial domain, resulting in a wave-like alternating surface. Additionally, the delayed neutron precursor concentration stabilizes over time, gradually approaching a stationary profile, which is consistent with the physical expectations. The results also support the effectiveness of the Laplace transform technique in capturing the early-time behavior of the system. Differences between the present results and those reported in the relevant literature are explained.

References

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