Analytical solutions to the coupled fractional neutron diffusion equations with delayed neutrons system using Laplace transform method

Aliaa Burqan; Mohammed Shqair; Ahmad El-Ajou; Sherif M. E. Ismaeel; Zeyad AlZhour; Aliaa Burqan; Mohammed Shqair; Ahmad El-Ajou; Sherif M. E. Ismaeel; Zeyad AlZhour

doi:10.3934/math.2023984

AIMS Mathematics

2023, Volume 8, Issue 8: 19297-19312. doi: 10.3934/math.2023984

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Research article

Analytical solutions to the coupled fractional neutron diffusion equations with delayed neutrons system using Laplace transform method

1.
College of Science, Zarqa University, Zarqa 13110, Jordan
2.
Department of Mathematics, Faculty of Science, Al-Balqa Applied University, Salt 19117, Jordan
3.
Department of Physics College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia
4.
Department of Physics, Faculty of Science, Ain Shams University, Cairo, Egypt
5.
Department of Basic Engineering Sciences, College of Engineering, Imam Abdulrahman Bin Faisal University, Dammam, Saudi Arabia

Received: 27 February 2023 Revised: 14 April 2023 Accepted: 19 April 2023 Published: 07 June 2023
MSC : 35R11, 44A10, 82D75

The neutron diffusion equation (NDE) is one of the most important partial differential equations (PDEs), to describe the neutron behavior in nuclear reactors and many physical phenomena. In this paper, we reformulate this problem via Caputo fractional derivative with integer-order initial conditions, whose physical meanings, in this case, are very evident by describing the whole-time domain of physical processing. The main aim of this work is to present the analytical exact solutions to the fractional neutron diffusion equation (F-NDE) with one delayed neutrons group using the Laplace transform (LT) in the sense of the Caputo operator. Moreover, the poles and residues of this problem are discussed and determined. To show the accuracy, efficiency, and applicability of our proposed technique, some numerical comparisons and graphical results for neutron flux simulations are given and tested at different values of time $ t $ and order $ \alpha $ which includes the exact solutions (when $ \alpha = 1). $ Finally, Mathematica software (Version 12) was used in this work to calculate the numerical quantities.
- diffusion equation,
- kinetic exact solution,
- Laplace transform,
- Caputo factional operator
Citation: Aliaa Burqan, Mohammed Shqair, Ahmad El-Ajou, Sherif M. E. Ismaeel, Zeyad AlZhour. Analytical solutions to the coupled fractional neutron diffusion equations with delayed neutrons system using Laplace transform method[J]. AIMS Mathematics, 2023, 8(8): 19297-19312. doi: 10.3934/math.2023984

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Abstract

The neutron diffusion equation (NDE) is one of the most important partial differential equations (PDEs), to describe the neutron behavior in nuclear reactors and many physical phenomena. In this paper, we reformulate this problem via Caputo fractional derivative with integer-order initial conditions, whose physical meanings, in this case, are very evident by describing the whole-time domain of physical processing. The main aim of this work is to present the analytical exact solutions to the fractional neutron diffusion equation (F-NDE) with one delayed neutrons group using the Laplace transform (LT) in the sense of the Caputo operator. Moreover, the poles and residues of this problem are discussed and determined. To show the accuracy, efficiency, and applicability of our proposed technique, some numerical comparisons and graphical results for neutron flux simulations are given and tested at different values of time $ t $ and order $ \alpha $ which includes the exact solutions (when $ \alpha = 1). $ Finally, Mathematica software (Version 12) was used in this work to calculate the numerical quantities.

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