Application of Kudryashov Method to Some Equations Used in Physics Science

Year 2019, , 1485 - 1492, 31.12.2019

Guldem Yıldız

https://doi.org/10.18185/erzifbed.566013

Abstract

In this study,
Kudryashov Method is used to find the wave solutions of the Gardner equation,
fifth order Caudrey-Dodd-Gibbon equation and Sawada-Kotera equation, which are
non-linear partial differential equations used as a mathematical model in the
physics science field and engineering applications. The exact solutions
obtained are compared with the results in the literature and hyperbolic type
and soliton solutions are obtained.

Keywords

Caudrey-Dodd-Gibbon equation, Gardner equation, , Kudryashov Method, Sawada-Kotera equation

Fizik Biliminde Kullanılan Bazı Denklemlere Kudryashov Metodun Uygulanması

Abstract

Bu çalışmada fizik bilim alanında ve mühendislik uygulamalarında
matematiksel model olarak kullanılan lineer olmayan kısmi türevli diferansiyel denklemlerden Gardner denklemi, beşinci
mertebeden Caudrey-Dodd-Gibbon
denklemi ve Sawada-Kotera denkleminin dalga çözümlerini bulmak için Kudryashov Metot kullanılmıştır. Elde edilen tam çözümler
literatürde bulunan sonuçlarla karşılaştırılmış ve hiperbolik tip ve soliton
çözümler elde edilmiştir.

Keywords

Caudrey-Dodd-Gibbon denklemi, , Gardner denklemi, , Kudryashov Metot

References

• Alam, M.N., Akbar, M.A., 2014. “Traveling wave solutions for the mKdV equation and the Gardner equation”, Journal of the Egyptian Mathematical Society, 22, 402–406.
• Betchewe, G., Victor, K.K., Thomas, B.B., Crepin, K.T., 2013. “New solutions of the Gardner equation: Analytical and numerical analysis of its dynamical understanding”, Appl. Math. Comput., 223, 377–388.
• Bildik, N., Konuralp, A., Bek, F.O. and Küçükarslan, S., 2006. “Solution of Different Type of the Partial Differential Equation by Differential Transform Method and Adomian’s Decomposition Method”, Applied Mathematics and Computation, 172(1), 551-567.
• Biswas, A., 2008. “Soliton Perturbation Theory for the Gardner Equation”, Adv. Stud. Theor. Phys. 2(16), 787–794.
• Dağhan, D., Dönmez, O., 2016. “Exact Solutions of the Gardner Equation and their Applications to the Different Physical Plasmas”, General and Applied Physiscs, 46, 321–333.
• He, JH., 2000. “A coupling method of homotopy technique and perturbation technique for nonlinear problems”, Int J Nonlinear Mech, 35, 37-43.
• Jiang, B., Bi, Q., 2010. “A study on the bilinear Caudrey_Dodd_Gibbon equation”, Nonlinear Analysis, 72, 4530-4533.
• Kabir, M.M., Khajeh, A., Abdi Aghdam, E., Koma, Y., 2011. “Modified Kudryashov method for finding exact solitary wave solutions of higher-order nonlinear equations”, Math. Meth. Appl. Sci., 34, 213–219.
• Kaplan, M., Bekir, A. ve Akbulut, A., 2016. “A generalized Kudryashov method to some nonline are evolution equations in mathematical physics”, Nonlinear Dyn, 85, 2843–2850.
• Kamchatnov, A.M., Kuo, Y.-H., Lin, T.-C. , Horng, T.-L., Gou, S.-C., Clift, R., El, G.A., Grimshaw, R.H.J., 2012. “Undular bore theory for the Gardner equation”, Phys. Rev. E 86, 036605.
• Karaagac, B., 2019. “A Numerical Approach to Caudrey_Dodd_Gibbon Equation Via Collocation Method Using Quintic B-Spline”, TWMS J. App. and Eng. Math. 9, 1-8.
• Kudryashov, N. A., 2012. “One method for finding exact solition of nonlinear differential equations”, Commun. Nonlinear Sci. Numer. Simulat., 17, 2248–2253.
• Mirzazadeh, M., Eslami, M., Biswas, A., 2014. “Dispersive optical solitons by Kudryashov's method”, Optik, 125, 6874–6880.
• Ryabov, P. N., Sinelshchikov, D. I., Kochanov, M. B., 2011. “Application of the Kudryashov method for finding exact solutions of the high order nonlinear evolution equations”, Applied Mathematics and Computation, 218, 3965–3972.
• Salas, A.H., 2008. “Exact solutions for the general fifth KdV equation by the exp function method”, Applied Mathematics and Computation, 205, 291-297.
• Shakeel, M., Mohyud-din, S.T., 2015. “Solution of Fifth Order Caudrey-Dodd-Gibbon-Sawada-Kotera Equation by the Alternative (G′/G)-Expansion Method with Generalized Riccati Equation”, Walailak Journal of Science and Technology, 12(10):949-960.
• Zuntao, F., Shida, L., Liu, S., 2004. “New kinds of solutions to Gardner equation”, Chaos, Solitons & Fract. 20, 301–309.
• Wazwaz, A.-M., 2006. “Analytic study of the fifth order integrable nonlinear evolution equations by using the tanh method”, Appl. Math. Comput., 174, 289–299.
• Wazwaz, A.-M., 2007. “New solitons and kink solutions for the Gardner equation”, Commun. Nonlinear Sci. Numer. Simul. 12, 1395–1404.
• Wazwaz, A.-M., 2008. “Multiple-soliton solutions for the fifth order Caudrey–Dodd–Gibbon (CDG) equation”, Applied Mathematics and Computation 197, 719–724.
• Wazwaz, A.-.M., 2011. “Multiple soliton solutions for (2+1)-dimensional Sawada–Kotera and Caudrey–Dodd–Gibbon equations”, Mathematical Methods in the Applied Science, DOI: 10.1002/mma.1460.