(1+1)-Boyutlu Benjamin-Bona-Mahony (BBM) Denkleminin Modifiye Edilmiş Kudryashov Metodu ile Soliton Çözümleri

Year 2024, , 316 - 324, 29.11.2024

Sait San , Zeynep Aydın

https://doi.org/10.35193/bseufbd.1387390

Abstract

Bu çalışma, (1+1)-boyutlu Benjamin-Bona-Mahony (BBM) denkleminin analitik soliton çözümlerinin modifiye edilmiş modifiye Kudryashov metodu ile elde edilmesine yöneliktir. Birinci aşamada, doğrusal olmayan kısmi türevli diferansiyel denklem formuna sahip olan model, uygun dalga dönüşümü ile doğrusal olmayan adi diferansiyel denkleme indirgenmektedir. İkinci aşamada ise, homojen denge prensibi ve Riccati yardımcı diferansiyel denklemi kullanılarak doğrusal cebirsel denklem sistemi elde edilerek bu sistemin çözümünden incelenen modelin bilinmeyen parametreleri belirlenmektedir. Elde edilen farklı çözüm setlerine bağlı olarak analitik soliton çözümleri elde edilerek ana denklemi sağlama kontrolü yapılmaktadır. Son aşamada ise çözümlerin fiziksel olarak yorumlanmasını kolaylaştırmak amacıyla kontur ve üç boyutlu grafik sunumları yapılmaktadır.

Keywords

Dalga dönüşümü, Analitik Çözüm, Kısmi Diferansiyel Denklemler, Soliton Çözümü

The Soliton Solutions of the (1+1)-Dimensional Benjamin-Bona-Mahony (BBM) Equation Via the Modified New Kudryashov Method

Abstract

This study is aimed at obtaining analytical soliton solutions of the (1+1)-dimensional Benjamin-Bona-Mahony (BBM) equation with the new modified Kudryashov method. In the first stage, the model, which has the form of a nonlinear partial differential equation, is reduced to a nonlinear ordinary differential equation with the appropriate wave transformation. In the second stage, a system of linear algebraic equations is obtained by using the homogeneous equilibrium principle and the Riccati auxiliary differential equation, and the unknown parameters of the model examined are determined from the solution of this system. Depending on the different solution sets obtained, analytical soliton solutions are obtained and the main equation is checked. In the final stage, contour and three-dimensional graphic presentations are made to facilitate the physical interpretation of the solutions.

Keywords

Wave Transformation, Analytic Solution, Partial Differential Equations, Soliton Solution

References

• Alsayyed, O., Jaradat, H. M., Jaradat, M. M. M., & Mustafa, Z. (2016). Multi-soliton solutions of the BBM equation arisen in shallow water. http://dx.doi.org/10.22436/jnsa.009.04.35
• Altun, S., Ozisik, M., Secer, A., & Bayram, M. (2022). Optical solitons for Biswas–Milovic equation using the new Kudryashov’s scheme. Optik, 270, 170045. https://doi.org/10.1016/j.ijleo.2022.170045.
• An, J.Y., & Zhang, W. G. (2006). Exact periodic solutions to generalized BBM equation and relevant conclusions. Acta Mathematicae Applicatae Sinica, 22(3), 509-516. http://dx.doi.org/10.1007/s10255-006-0326-3
• Cinar, M., Secer, A., & Bayram, M. (2022). Analytical solutions of (2+ 1)-dimensional Calogero-Bogoyavlenskii-Schiff equation in fluid mechanics/plasma physics using the New Kudryashov method. Physica Scripta, 97(9), 094002. https://doi.org/10.1088/1402-4896/ac883f.
• Esen, H., Secer, A., Ozisik, M., & Bayram, M. (2022). Soliton solutions to the nonlinear higher dimensional Kadomtsev-Petviashvili equation through the new Kudryashov’s technique. Physica Scripta, 97(11), 115104. https://doi.org/10.1088/1402-4896/ac98e4.
• Estévez, P. G., Kuru, Ş., Negro, J., & Nieto, L. M. (2009). Travelling wave solutions of the generalized Benjamin–Bona–Mahony equation. Chaos, Solitons & Fractals, 40(4), 2031-2040. https://doi.org/10.1016/j.chaos.2007.09.080
• Fan, E., Zhang, H., (1998). A note on the homogeneous balance method, Phys. Lett. A 246 403–406. https://doi.org/10.1016/S0375-9601(98)00547-7.
• Feng, Z. (2002). The first-integral method to study the Burgers–Korteweg–de Vries equation. Journal of Physics A: Mathematical and General, 35(2), 343. https://doi.org/10.1088/0305-4470/35/2/312.
• Frias, B. A., Salas, A. H., & (2010). New periodic and soliton solutions for the Generalized BBM and Burgers–BBM equations. Applied Mathematics and Computation, 217(4), 1430-1434. https://doi.org/10.1016/j.amc.2009.05.068.
• Gomez, C. A., & Salas, A. H. (2010). Exact solutions for the generalized BBM equation with variable coefficients. Mathematical Problems in Engineering, 2010. http://dx.doi.org/10.1155/2010/498249.
• Guan-Ting, L., Tian-You F., (2005). New applications of developed Jacobi elliptic function expansion methods, Physics Letters A, 345.1-3: 161-166. https://doi.org/10.1016/j.physleta.2005.07.034.
• He, J. H. (1998). Approximate solution of nonlinear differential equations with convolution product nonlinearities. Computer methods in applied mechanics and engineering, 167(1-2), 69-73. https://doi.org/10.1016/S0045-7825(98)00109-1.
• He, J.H., Wu, X.H., (2006). Exp-function method for nonlinear wave equations, Chaos, Solitons & Fractals, vol. 30, no. 3, pp. 700-708, Nov. https://doi.org/10.1016/j.chaos.2006.03.020.
• Kudryashov, N. A. (2011). One method for finding exact solutions of nonlinear differential equations. Communications in Nonlinear Science and Numerical Simulation, 17(6), 2248-2253. https://doi.org/10.1016/j.cnsns.2011.10.016.
• Kudryashov, N. A. (2015). On nonlinear differential equation with exact solutions having various pole orders. Chaos, Solitons & Fractals, 75, 173-177. https://doi.org/10.1016/j.chaos.2015.02.016.
• Kudryashov, N. A. (2020). Method for finding highly dispersive optical solitons of nonlinear differential equations. Optik, 206, 163550. https://doi.org/10.1016/j.ijleo.2019.163550.
• Kudryashov, N. A. (2020). Method for finding highly dispersive optical solitons of nonlinear differential equations. Optik, 206, 163550. https://doi.org/10.1016/j.ijleo.2019.163550.
• Noor, M. A., Noor, K. I., Waheed, A., & Al-Said, E. A. (2011). Some new solitonary solutions of the modified Benjamin–Bona–Mahony equation. Computers & Mathematics with Applications, 62(4), 2126-2131. http://dx.doi.org/10.1016/j.camwa.2011.06.060.
• Ozisik, M., Secer, A., & Bayram, M. (2022). The bell-shaped perturbed dispersive optical solitons of Biswas–Arshed equation using the new Kudryashov’s approach. Optik, 267, 169650. https://doi.org/10.1016/j.ijleo.2022.169650.
• Ozisik, M., Secer, A., Bayram, M., & Aydin, H. (2022). An encyclopedia of Kudryashov’s integrability approaches applicable to optoelectronic devices. Optik, 265, 169499. https://doi.org/10.1016/j.ijleo.2022.169499.
• Önder, İ., Özışık, M., & Seçer, A. (2022). The soliton solutions of (2+ 1)-dimensional nonlinear two-coupled Maccari equation with complex structure via new Kudryashov scheme. New Trends in Mathematical Sciences, 10(1). http://dx.doi.org/10.20852/ntmsci.2022.468.
• Rady, A. A., Osman, E. S., & Khalfallah, M. (2010). The homogeneous balance method and its application to the Benjamin–Bona–Mahoney (BBM) equation. Applied Mathematics and Computation, 217(4), 1385-1390. https://doi.org/10.1016/j.amc.2009.05.027
• Tang, Y., Xu, W., Gao, L., & Shen, J. (2007). An algebraic method with computerized symbolic computation for the one-dimensional generalized BBM equation of any order. Chaos, Solitons & Fractals, 32(5), 1846-1852. http://dx.doi.org/10.1007/s11071-007-9282-6
• Wang, M., Li, X., & Zhang, J. (2008). The (G′ G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Physics Letters A, 372(4), 417-423. https://doi.org/10.1016/j.physleta.2007.07.051.
• Wang, M.L., (1995). Solitary wave solutions for variant Boussinesq equations, Phys. Lett. A 199 169–172. https://doi.org/10.1016/0375-9601(95)00092-H.
• Wazwaz, A. M. (2007). The tanh–coth method for solitons and kink solutions for nonlinear parabolic equations. Applied Mathematics and Computation, 188(2), 1467-1475. https://doi.org/10.1016/j.amc.2006.11.013
• Yan, C. (1996). A simple transformation for nonlinear waves. Physics Letters A, 224(1-2), 77-84. https://doi.org/10.1016/S0375-9601(96)00770-0
• Benjamin Thomas Brooke , Bona J. L. and Mahony J. J. (1972) Model equations for long waves in nonlinear dispersive systems, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences 27247–78. https://doi.org/10.1098/rsta.1972.0032