Analytical solutions to the coupled fractional neutron diffusion equations with delayed neutrons system using Laplace transform method
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...
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AIMS Press
2023-06-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.2023984?viewType=HTML |
| _version_ | 1797794063845425152 |
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| author | Aliaa Burqan Mohammed Shqair Ahmad El-Ajou Sherif M. E. Ismaeel Zeyad AlZhour |
| author_facet | Aliaa Burqan Mohammed Shqair Ahmad El-Ajou Sherif M. E. Ismaeel Zeyad AlZhour |
| author_sort | Aliaa Burqan |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-13T02:57:23Z |
| format | Article |
| id | doaj.art-3930a0b282964ff08b4989f5bbc2f827 |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-03-13T02:57:23Z |
| publishDate | 2023-06-01 |
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| spelling | doaj.art-3930a0b282964ff08b4989f5bbc2f8272023-06-28T00:54:01ZengAIMS PressAIMS Mathematics2473-69882023-06-0188192971931210.3934/math.2023984Analytical solutions to the coupled fractional neutron diffusion equations with delayed neutrons system using Laplace transform methodAliaa Burqan 0Mohammed Shqair1Ahmad El-Ajou 2Sherif M. E. Ismaeel 3Zeyad AlZhour 41. College of Science, Zarqa University, Zarqa 13110, Jordan1. College of Science, Zarqa University, Zarqa 13110, Jordan2. Department of Mathematics, Faculty of Science, Al-Balqa Applied University, Salt 19117, Jordan3. 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, Egypt5. Department of Basic Engineering Sciences, College of Engineering, Imam Abdulrahman Bin Faisal University, Dammam, Saudi ArabiaThe 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.https://www.aimspress.com/article/doi/10.3934/math.2023984?viewType=HTMLdiffusion equationkinetic exact solutionlaplace transformcaputo factional operator |
| spellingShingle | 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 AIMS Mathematics diffusion equation kinetic exact solution laplace transform caputo factional operator |
| title | Analytical solutions to the coupled fractional neutron diffusion equations with delayed neutrons system using Laplace transform method |
| title_full | Analytical solutions to the coupled fractional neutron diffusion equations with delayed neutrons system using Laplace transform method |
| title_fullStr | Analytical solutions to the coupled fractional neutron diffusion equations with delayed neutrons system using Laplace transform method |
| title_full_unstemmed | Analytical solutions to the coupled fractional neutron diffusion equations with delayed neutrons system using Laplace transform method |
| title_short | Analytical solutions to the coupled fractional neutron diffusion equations with delayed neutrons system using Laplace transform method |
| title_sort | analytical solutions to the coupled fractional neutron diffusion equations with delayed neutrons system using laplace transform method |
| topic | diffusion equation kinetic exact solution laplace transform caputo factional operator |
| url | https://www.aimspress.com/article/doi/10.3934/math.2023984?viewType=HTML |
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