Application of The Homotopy Perturbation Method to the Neutron Diffusion Equation

Öz

The Homotopy Perturbation Method (HPM) has been shown to be effective in solving both linear and nonlinear differential equations in mathematics, making it useful in a wide range of applications in the fields of physics and engineering. In this study, the Homotopy Perturbation Method was applied to the neutron diffusion equation for a one-dimensional time-independent approach. The Laplace operator of the neutron diffusion equation was considered for Cartesian, spherical and cylindrical coordinates. The critical radius values obtained for three different systems were calculated for all possible values of the relevant material parameter B. The results show that the solution of the neutron diffusion equation is agree with the literature.

Anahtar Kelimeler

• Homotopy Perturbation Method
• neutron diffusion equation
• second order differential equations
• spherical and cylindrical coordinates

Homotopi Perturbasyon Metodunun Nötron Difuzyon Denklemine Uygulanması

Öz

Homotopi Pertürbasyon Yönteminin (HPM), matematikte hem doğrusal hem de doğrusal olmayan diferansiyel denklemlerin çözümünde etkili olduğu, fizik ve mühendislik alanlarındaki geniş bir uygulama yelpazesinde faydalı olduğu gösterilmiştir. Bu çalışmada, tek boyutlu zamandan bağımsız yaklaşım için nötron difüzyon denklemine Homotopi Pertürbasyon Yöntemi uygulanmıştır. Nötron difüzyon denkleminin Laplace operatörü kartezyen, küresel ve silindirik koordinatlar için dikkate alındı. Üç farklı sistem için elde edilen kritik yarıçap değerleri, ilgili malzeme parametresi B’nin tüm olası değerleri için hesaplandı. Sonuçlar nötron difüzyon denkleminin çözümünün literatürle uyumlu olduğunu göstermektedir.

Anahtar Kelimeler

• Homotopi Pertürbasyon Yöntemi
• nötron difüzyon denklemi
• ikinci derece diferansiyel denklemler
• küresel ve silindirik koordinatlar

Kaynakça

1. Çiçek H., Mondalı M., (2022). Approximate Solution of Fractional Order Pseudo-Hyperbolic Partial Differential Equation Using Homotopy Perturbation Method. Uşak University Journal of Science and Natural Sciences, 67-75.
2. Dağhan D., Yavuz H.M., Yıldız G., (2017). Application of Homotopy Perturbation Method to Nonlinear Partial Differential Equations. Ömer Halisdemir University, Journal of Science and Engineering, 6(1), 290-301.
3. Dababneh S., Khasawneh K., Odibot Z., (2011). An alternative solution of the neutron diffusion equation in cylindrical symmetry. Annals of Nuclear Energy, 1142-1143.
4. Dababneh S., Khasawneh K., Odibat Z., (2009). A solution of the neutron diffusion equation in hemispherical symmetry using the homotopy perturbation method. Annals of Nuclear Energy, 1711-1717.
5. Eş H., (2022). Solution of Third Order Linear Partial Differential Equations with Homotopy Perturbation Method. Master’s Thesis, Harran University, Institute of. Science and Technology, Şanlıurfa.
6. He J.H., (2000). A review on some new recently developed nonlinear analytical techniques. International Journal of Nonlinear Sciences and Numerical Simulation 1, 51-70.
7. Koklu H., Ersoy A., Özer O., (2016). Calculation of the neutron diffusion equation by using Homotopy Perturbation Method. Department of Engineering Physics, 2-3.
8. Lamarsh J.R., Baratta A.J., (2001). Introduction to Nuclear Engineering, NJ, Prentice Hall, 252.

9. Mondalı M., Eş H., (2021). Solution of Third Order Partial Differential Equation with Homotopy Perturbation Method. BEU Journal of Science, 10 (4), 1527-1534.
10. Özpınar F., (2020). Solution of Fractional Order Partial Differential Equations with Discrete Homotoy Perturbation Method. Afyon Kocatepe University, Journal of Scienceand Engineering, 213-221.
11. Shqair M., (2019). Solution of different geometries reflected reactors neutron diffusion equation using the homotopy perturbation method. Results in Physics, 61-66.
12. Shqair M., Farrag A.E., Al-Smadi M., (2022). Solving Multi-Group Reflected Spherical Reactor System of Equations Using the Homotopy Perturbation Method. Physics Department, 10(10), 1784-1795.
13. Yener G., (2009). Solution of Some Differential Equations Using Homotopy Perturbation Method. Master’s Thesis, Yıldız Teknik University, Institute of Science and Technology, İstanbul.