ON THE SOLITARY WAVE SOLUTIONS OF DIFFERENT VERSIONS OF FRACTIONAL 3D- WAZWAZ -BENJAMIN-BONA-MAHONY EQUATIONS

Öz

Nonlinear fractional Wazwaz -Benjamin-Bona-Mahony (WBBM) equations play an important role in physics. The equations form an important model for studying the approximately unidirectional propagation of small amplitude long waves in certain nonlinear distribution systems as an alternative to Kortweg and de Vries (KDV). In this study, the fractional 3D-WBBM equations are solved by using the Improved Bernoulli Sub-Equation Function (IBSEF) method. 3D, 2D and contour plots are given to show the physical properties of the solutions. The main aim of this method is to clarify obvious the exact solutions to the equations. Moreover, the effectiveness of the method is demonstrated by the findings presented in this paper.

Anahtar Kelimeler

• Fractional derivative
• WBBM equations
• Exact solution

KESİRLİ 3D- WAZWAZ -BENJAMIN-BONA-MAHONY DENKLEMLERİNİN FARKLI VERSİYONLARININ SOLİTARY DALGA ÇÖZÜMLERİ ÜZERİNE

Öz

Lineer olmayan kesirli Wazwaz-Benjamin-Bona-Mahony (WBBM) denklemleri fizikte önemli bir rol oynar. Bu denklemler, Kortweg ve de Vries'e (KDV) alternatif olarak belirli doğrusal olmayan dağıtım sistemlerinde küçük genlikli uzun dalgaların yaklaşık olarak tek yönlü yayılmasını incelemek için önemli bir model oluşturur. Çalışmada, kesirli 3D-WBBM denklemleri, Geliştirilmiş Bernoulli Alt Denklem Fonksiyonu (IBSEF) yöntemi kullanılarak çözülmüştür. Çözümlerin fiziksel özelliklerinin gösterilmesi için 3D, 2D ve kontur çizimleri verilmiştir. Bu yöntemin temel amacı, bu denklemlerin kesin çözümlerini açıklığa kavuşturmaktır. Ayrıca yöntemin etkinliği, bu makalede sunulan bulgularla gösterilmektedir.

Anahtar Kelimeler

• Kesirli türev
• WBMM denklemleri
• Tam çözüm

Kaynakça

1. Abdeljawad, T. (2015). On conformable fractional calculus. J Comput Appl Math, 279, 57-66.
2. Aktürk, T. & Kubal, Ç. (2022). The behavior of plasma and space-charge waves represented by Nonlinear mathematical models. Journal of Ocean Engineering and Science, 23, 50, https://doi.org/10.1016/j.joes.2022.06.031.
3. Ala, V., Demirbilek, U. & Mamedov, Kh. R. (2021). On the exact solutions to conformable equal width wave equation by improved Bernoulli sub-equation function method. Bulletin of the South Ural State University Ser. Mathematics. Mechanics. Physics, 13(3), 5–13.
4. Atangana, A., Baleanu, D. & Alsaedi, A. (2015). New properties of conformable derivative. Open Math, 13, 1-10.
5. Atas, S.S., Ali, K.K., Sulaiman, T.A. & Bulut, H. (2022). Optical solitons to the Fokas system equation in monomode optical fibers. Opt Quant Electron, 54(11), 1-13.
6. Baskonus, H. M. & Bulut, H. (2015). On the complex structures of Kundu-Eckhaus equation via improved Bernoulli sub-equation function method. Waves in Random and Complex Media, 66, 720-728.
7. Baskonus, H.M., Bulut, H. & Sulaiman, T.A. (2019). New complex hyperbolic structures to the Lonngrenwave equation by using sine-gordon expansion method. Applied Mathematics and Nonlinear Sciences, 4, 129-138.
8. Benjamin, T., Bona, J. & Mahony, J. (1972). Model equations for long waves in nonlinear dispersive systems. Philos Trans. R. Soc. London, Ser A 272(1220), 47.

9. Ekici, M. & Ünal, M. (2022). Application of the rational (G′/G)-expansion method for solving some coupled and combined wave equations. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 71(1), 116-132.
10. Fu, Y. & Li, J. (2017). Exact stationary -wave solutions in the standart model of the Kerr-nonlinear optical fiber with the Bragggrating. Journal of Applied Analysis and Computation, 7, 1177-1184.
11. Khalil, R., Al Horani, A., Yousef, A. & Sabadheh, M. (2014). A new definition of fractional derivative. J. Comput. Appl. Math., 264, 65-70.
12. Ma, W.X. (2011). Generalized bilinear differential equations. Stud. Nonlinear Sci., 2, 140–144.
13. Mamun, A.A., An, T., Shahen, N. H. M., Ananna, S. N., Hossain, M. F., & Muazu, T. (2020). Exact and explicit travelling-wave solutions to the family of new 3D fractional WBBM equations in mathematical physics. Results in Physics, 19, 103517.
14. Mamun, A.A., Ananna, S. N., An, T., Asaduzzaman, M., & Miah, M. M. (2022a). Solitary wave structures of a family of 3D fractional WBBM equation via the tanh–coth approach. Partial Differential Equations in Applied Mathematics, 5, 100237.
15. Mamun, A.A., Ananna, S.N., An, T., Asaduzzaman, Md. & Rana, M.S. (2022b.) Sine-Gordon expansion method to construct the solitary wave solutions of a family of 3D fractional WBBM equations, Results in Physics, 40, 105845.
16. Mamun, A.A., Ananna, S.N., Gharami, PP., An, T. & Md. Asaduzzaman. (2022c). The improved modified extended tanh-function method to develop the exact travelling wave solutions of a family of 3D fractional WBBM equations, Results in Physics, 41, 105969.
17. Roshid, H.O., Akbar, M.A., Alam Md. N., Hoque, Md. F.& Rahman, N. (2014). New extended 〖(G〗^'/G)-expansion method to solve nonlinear evolution equation: The (3 + 1)-dimensional potential-YTSF equation. SpringerPlus, 3(122), 6.
18. Ünal, M. & Ekici, M. (2021). The double (G'/G,1/G)-expansion method and its applications for some nonlinear partial differential equations. Journal of the Institute of Science and Technology, 1(11), 599-608.
19. Wazwaz, A.M. (2008). The Hirota’s direct method and the tanh-coth method for multiple-soliton solutions of the Sawada-Kotera-Ito seventh-order equation. Appl Math Comput., 199,133-8.
20. Wazwaz, A.M. (2017). Exact soliton and kink solutions for new (3+1)-dimensional nonlinear modifed equations of wave propagation. Open Eng., 7,169–174.
21. Yusuf, A., Inc, M., Aliyu, A. & Baleanu, D. (2019). Optical solitons possessing beta derivative of the Chen Lee-Liu equation in optical fiber. Front Phys., 7(34), doi:10.3389/fphy.2019.00034.