Calogero-Bogoyavlenskii-Schiff denkleminin analitik çözümleri

Öz

Kısmi Diferansiyel denklemlerin analitik çözümleri, kuantum mekanik ve plazma fiziği gibi alanlarda, açığa çıkmamış olayların açıklanmasında fayda sağlamaktadır. Bu çalışmada, lineer olmayan kısmi türevli Calogero-Bogoyavlenskii-Schiff (CBS) diferansiyel denkleminin analitik çözümlerini bulmak için Homojen Denge Metodundan yararlanılmıştır. Homojen Denge Metodunun Calogero-Bogoyavlenskii-Schiff denklemine uygulanmasıyla elde edilen analitik çözümler literatürde bulunan sonuçlarla karşılaştırılmış ve literatürde bulunan çözümlerle uyumlu hiperbolik ve trigonometrik tipten fonksiyonlar elde edilmiştir.

Anahtar Kelimeler

• Calogero–Bogoyavlenskii-Schiff (CBS) denklemi
• homojen denge metodu
• analitik çözüm

Analytical solutions of Calogero-Bogoyavlenskii-Schiff equation

Öz

Analytical solutions of Partial Differential equations are useful in explaining unrevealed events in areas such as quantum mechanics and plasma physics. In this study, the Homogeneous Balance Method is used to find analytical solutions of the nonlinear partial differential Calogero-Bogoyavlenskii-Schiff (CBS) differential equation. Analytical solutions obtained by applying the Homogeneous Balance Method to the Calogero-Bogoyavlenskii-Schiff equation are compared with the results found in the literature, and hyperbolic and trigonometric type functions that are compatible with the solutions in the literature are obtained.

Anahtar Kelimeler

• Calogero–Bogoyavlenskii-Schiff (CBS) equation
• homogeneous balance method
• analytical solution

Kaynakça

1. Goldston, R. J. ve Rutherford, P. H., Introduction to Plasma Physics, CRC Press, (1995).
2. Dönmez, O. ve Dağhan, D., Analytic Solutions of the Schamel-KdV Equation by Using Different Methods: Application to a Dusty Space Plasma, Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 21, 208-215, (2017).
3. Çengel, Y. A. ve Cımbala J. M., Akışkanlar Mekaniği, Güven Bilimsel, (2008).
4. Anderson, D., Variational approach to nonlinear pulse propagation in fibers, Physics Letters A, 27, 3135-3145, (1983).
5. Yavuz, M. ve Yokus A., Analytical and numerical approaches to nerve impulse model fractional‐order, Numerical Methods for Partial Differential Equations, 36(6), 1348-1368, (2020).
6. Evirgen, F., Analyze the optimal solutions of optimization problems by means of fractional gradient based system using VIM, An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 6, 75-83, (2016).
7. Evirgen, F. ve Yavuz M., An Alternative Approach for Nonlinear Optimization Problem with Caputo - Fabrizio Derivative, ITM Web of Conferences 22, 01009, (2018).
8. Yavuz, M. ve Yaşkıran B., Homotopy methods for fractional linear/nonlinear differential equations with a local derivative operator, Journal of Balıkesir University Institute of Science and Technology, 20(3) Special Issue, 75-89, (2018).

9. Yıldız, G. ve Dağhan D., Solution of the (2+1) Dimensional Breaking Soliton Equation by Using Two Different Methods, Journal of Engineering Technology and Applied Sciences, 1, 13-18, (2016).
10. Dağhan D., Yavuz H. ve Yıldız G., Lineer Olmayan Kısmi Türevli Denklemlere Homotopi Pertürbasyon Yönteminin Uygulanması, Ömer Halisdemir Üniversitesi Mühendislik Bilimleri Dergisi, 6, 290-301, (2017).
11. Yavuz M. ve Ozdemir N., Numerical Inverse Laplace Homotopy Technique for Fractional Heat Equations, Thermal Science, 22, 185-194, (2018).
12. Yavuz M., Novel solution methods for initial boundary value problems of fractional order with conformable differentiation, An International Journal of Optimization and Control: Theories & Applications, 8, 1-7, (2018).
13. Wang, M., Zhou, Y., ve Li, Z., Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics, Physics Letters A, 216, 67-75, (1996).
14. Fan, E. ve Zhang, H., A note on the homogeneous balance method, Physics Letters A, 246, 403-406, (1998).
15. Fan, E., Two new applications of the homogeneous balance method, Physics Letters A, 265, 353-357, (2000).
16. Senthilvelan, M., On the extended applications of Homogenous Balance method, Applied Mathematics and Computation, 123, 381-388, (2001).
17. Zhao, X. ve Tang, D., A new note on a homogeneous balance method, Physics Letters A, 297, 59-67, (2002).
18. Zhao, X., Wang, Li. ve Sun W., The repeated homogeneous balance method and its applications to nonlinear partial differential equations, Chaos, Solitons and Fractals, 28, 448–453, (2006).
19. Jie, J., Wu, J. ve Zhang, J., Homogeneous Balance method for an Inhomogeneous KdV equation: Böcklund Transformation and Lax Pair, International Journal of Nonlinear Science, 9, 69-71, (2010).
20. Abdel Rady, A.S., Osman, E.S., ve Khalfallah, M., The homogeneous balance method and its application to the Benjamin–Bona–Mahoney (BBM) equation, Applied Mathematics and Computation, 217, 1385-1390, (2010).
21. Zayed, M.E., ve Alurrfi, K.A.E., The Homogeneous Balance method and its applications for nonlinear evolution equations, Italian Journal of the Applied Mathematics, 33, 307-318, (2014).
22. Yi, W., He, X-D. ve Yang X-F., The homogeneous balance of undetermined coefficients method and its application, Open Mathematics, 14, 816-826, (2016).
23. Türkmen, Ç., Lineer Olmayan Kısmi Türevli Diferansiyel Denklemlere Kudryashov Metodu ve Homojen Denge Metodunun Uygulanması, Yüksek Lisans Tezi, Niğde Ömer Halisdemir Üniversitesi, (2019).
24. Radha, B. ve Duraisamy, C., The homogeneous balance method and applications for finding the exact solutions for nonlinear equations, Journal of Ambient Intelligence and Humanized Computing, (2020).
25. Yu, SJ., Toda, K.ve Fukuyama T., N-soliton solutions to a (2+1)-dimensional integrable equation, Journal of Physics A: Mathematical and General, 31, 10181–10186, (1998).
26. Bruzon, M.S., Gandarias, M.L., Muriel, C., Ramierez, J., Saez, S. ve Romero, F.R., The Calogero–Bogoyavlenskii–Schiff equation in 2+1 dimensions, Theoretical and Mathematical Physics, 137 (1), 1367–1377, (2003).
27. Kobayashi, T. ve Toda K., The Painlevé test and reducibility to the canonical forms for higher-dimensional soliton equations with variable-coefficients, Symmetry, Integrability and Geometry: Methods and Applications, 2, 1–10, (2006).
28. Peng, Y., New types of localized coherent structures in the Bogoyavlenskii–Schiff equation, International Journal of Theoretical Physics, 45 (9), 1779–1783, (2006).
29. Wazwaz, A.M., New solutions of distinct physical structures to high-dimensional nonlinear evolution equations, Applied Mathematics and Computation, 196, 363–370, (2008).
30. Kaplan, M., Bekir, A. ve Akbulut, A., A generalized Kudryashov method to some nonlinear are evolution equations in mathematical physics, Nonlinear Dynamics, 85, 2843–2850, (2016).
31. Başkonuş, H. M., Sulaiman, T. A. ve Bulut, H., New solitary wave solutions to the (2+1) dimensional Calogero–Bogoyavlenskii–Schiff and the Kadomtsev Petviashvilihierarchy equations, Indian Journal of Physics, 91, 1237-1243, (2017).
32. Tahami, M. ve Najafi, M., Multi-wave solutions for the generalized (2+1)-dimensional nonlinear evolution equations, Optik, 136, 228-236, (2017).
33. Salem, S., Kassem, M., Mohamed ve Mabrouk, S. M., Similarity Solution of (2+1)-Dimensional Calogero-Bogoyavlenskii-Schiff Equation Lax Pair, American Journal of Applied Mathematics, 7 (5), 137-144, (2019).
34. Saleh, R., Kassem, M. ve Mabrouk, S. M., Investigation of breaking dynamics for Riemann waves in shallow water, Chaos, Solitons and Fractals, 132, 109571, (2020).