Füzyon ve Fizyon Fenomenleri İçin Doğrusal Olmayan Dalga Çözümlerinin İncelenmesi

Abstract

Bu çalışmada, (3+1)boyutlu Jimbo-Miwa denkleminin dalga çözümleri ve buna bağlı olarak da çözümün füzyon ve fisyon olmak üzere iki farklı olgusu modifiye üstel fonksiyon yöntemi kullanılarak elde edilmiştir. Daha olası çözümler elde etmek için modifiye edilmiş üstel fonksiyon yönteminin doğası gereği iki farklı durum incelenmiştir. Ortaya çıkan çözümler incelendiğinde trigonometrik, hiperbolik ve rasyonel fonksiyonlar elde edilmiştir. Wolfram Mathematica yazılımı tarafından bulunan çözüm fonksiyonlarının (3+1) boyutlu potansiyel Jimbo-Miwa denklemini sağlayıp sağlamadığı kontrol edildi. Uygun parametreler belirlenerek çözüm fonksiyonunun iki ve üç boyutlu grafikleri, kontur ve hassasiyet grafikleri elde edildi.

Keywords

• Geliştirilmiş Üstel Fonksiyon Metodu
• (3+1)-boyutlu Jimbo-Miwa denklemi
• Dalga çözümleri

Investigation of Nonlinear Wave Solutions for Fusion and Fission Phenomenas

Abstract

In this study, wave solutions of the (3+1) dimensional Jimbo-Miwa equation and two different phenomena of the solution, fusion and fission, are obtained using the modified exponential functionmethod. In order to get more possible solutions, two different cases are investigated due to the nature of the modified exponential function method. When the resulting solutions are analyzed, trigonometric, hyperbolic and rational functions are obtained. It was checked whether the solution functions found by the Wolfram Mathematica software provided the (3+1) dimensional potential Jimbo-Miwa equation. Two and three dimensional graphs, contour and density graphs of the solution function were get by determining the appropriate parameters.

Keywords

• Modified Exponential Function Method
• The (3+1)-dimensional Jimbo-Miwa equation
• Wave solutions

References

1. [1] Zheng X, Chen Y , Zhang H. Generalized extended tanh-function method and its application to (1+ 1)-dimensional dispersive long wave equation. Physics Letters A.2003; 311(2-3):145-157.
2. [2] Elwakil SA, El-Labany SK, Zahran MA , Sabry R. Modified extended tanh-function method for solving nonlinear partial differential equations. Physics Letters A.2002; 299(2-3):179-188.
3. [3] Fan E , Hon YC. Applications of extended tanh method to ‘special’ types of nonlinear equations. Applied Mathematics and Computation. 2003;141(2-3):351-358.
4. [4] Yang XF, Deng ZC , Wei YA. Riccati-Bernoulli sub-ODE method for nonlinear partial differential equations and its application. Advances in Difference equations.2015;2015(1):1-17.
5. [5] Baskonus HM , Bulut H. Regarding on the prototype solutions for the nonlinear fractional-order biological population model. In AIP Conference Proceedings AIP Publishing LLC.2016;1738:1.
6. [6] Abdelrahman MA. A note on Riccati-Bernoulli Sub-ODE method combined with complex transform method applied to fractional differential equations. Nonlinear Engineering.2018;7(4):279-285.
7. [7] Liu CS. Trial equation method and its applications to nonlinear evolution equations. Acta. Phys. Sin..2005;54(6):2505-2509.
8. [8] Liu CS. Trial equation method to nonlinear evolution equations with rank inhomogeneous: mathematical discussions and its applications. CoTPh.2006;45(2): 219-223.

9. [9] Bulut H, Baskonus HM , Pandir Y. The modified trial equation method for fractional wave equation and time fractional generalized Burgers equation. In Abstract and Applied Analysis Hindawi.2013; Vol. 2013.
10. [10] Gurefe Y, Misirli E, Sonmezoglu A, Ekici M. Extended trial equation method to generalized nonlinear partial differential equations. Applied Mathematics and Computation.2013; 219(10): 5253-5260.
11. [11] Pandir Y, Gurefe Y , Misirli E. A multiple extended trial equation method for the fractional Sharma-Tasso-Olver equation. In AIP Conference Proceedings American Institute of Physics.2013; 1558(1): 1927-1930.
12. [12] Hosseini K, Gholamin P. Feng’s first integral method for analytic treatment of two higher dimensional nonlinear partial differential equations. Differential Equations and Dynamical Systems.2015; 23(3): 317-325.
13. [13] He JH, Wu XH. Exp-function method for nonlinear wave equations, Chaos, Solitons & Fractals. 2006; 30(3):700-708.
14. [14] Baskonus HM, Askin M. Travelling wave simulations to the modified Zakharov-Kuzentsov model arising in plasma physics. In Litteris et Artibus, Lviv Polytechnic Publishing House. 2016.
15. [15] Gurefe Y, Misirli E. Exp-function method for solving nonlinear evolution equations with higher order nonlinearity.Computers & Mathematics with Applications.2011; 61(8): 2025-2030.
16. [16] Misirli E, Gurefe Y. The Exp-function method to solve the generalized Burgers-Fisher equation. Nonlinear Sci. Lett. A. 2010;1: 323-328.
17. [17] Misirli E, Gurefe Y. Exact solutions of the Drinfel’d–Sokolov–Wilson equation using the exp-function method. Applied Mathematics and Computation.2010; 216(9): 2623-2627.
18. [18] Ma WX, Lee JH. A transformed rational function method and exact solutions to the 3+ 1 dimensional Jimbo–Miwa equation. Chaos, Solitons & Fractals.2009; 42(3): 1356-1363.
19. [19] Zhang Y , Sun S, Dong H. Hybrid solutions of (3+ 1)-dimensional Jimbo-Miwa equation. Mathematical Problems in Engineering. 2017.
20. [20] Tang Y, Ma WX, Xu W, Gao L. Wronskian determinant solutions of the (3+1)-dimensional Jimbo–Miwa equation. Applied Mathematics and Computation. 2011; 217(21): 8722-8730.
21. [21] Ma WX. Lump-type solutions to the (3+1)-dimensional Jimbo-Miwa equation. International Journal of Nonlinear Sciences and Numerical Simulation.2016; 17(7-8): 355-359.
22. [22] Yue Y, Huang L, Chen Y. Localized waves and interaction solutions to an extended (3+1)-dimensional Jimbo–Miwa equation. Applied Mathematics Letters.2019;89: 70-77.
23. [23] Zhang X, Chen Y. Rogue wave and a pair of resonance stripe solitons to a reduced (3+1)-dimensional Jimbo–Miwa equation. Communications in Nonlinear Science and Numerical Simulation.2017; 52: 24-31.
24. [24] Öziş T, Aslan I . Exact and explicit solutions to the (3+ 1)-dimensional Jimbo–Miwa equation via the Exp-function method. Physics Letters A, 2008;372(47): 7011-7015.