Year 2021, Volume 8 , Issue 1, Pages 56 - 64 2021-06-30

(1/G')-Açılım Yöntemi ile Benney-Luke Denkleminin Tam Çözümleri
Exact solutions of the Benney–Luke equation via (1/G')-expansion method

Hülya DURUR [1] , Asıf YOKUŞ [2]


Bu çalışmada, Benney-Luke (BL) denklemini çözmek için (1/ G') -açılım yöntemi uygulanmıştır. BL denkleminin tam çözümleri bu yöntem ile elde edilmektedir. BL denkleminden elde edilen çözümler hiperbolik formdadır. Elde edilen çözümlerin 3 boyutlu, 2 boyutlu ve kontur grafikleri sunulmaktadır. Sonuçlar, (1/ G') - açılım yönteminin doğrusal olmayan evrim denklemlerinin çözümlerini bulmak için etkili ve basit bir matematiksel enstrüman olduğu gösterilmiştir.

In this study, the (1/G') -expansion method was implemented to solve the Benney–Luke (BL) equation. Exact solutions of the BL equation were obtained via this method. The solutions obtained from the BL equation were in hyperbolic form. 3D, 2D and contour graphs of obtained solutions are presented. Results show that the (1/G') -expansion method provides an efficient and straightforward mathematical instrument for finding solutions of nonlinear evolution equations (NLEEs). 

  • Kheiri, H., Alipour, N. and Dehghani, R., (2011). Homotopy analysis and Homotopy-Pade methods for the modified Burgers-Korteweg-de-Vries and the Newell Whitehead equation. Mathematical Sciences, 5(1), 33-50.
  • Durur, H., (2020). Different Types Analytic Solutions of the (1+1)-Dimensional Resonant Nonlinear Schrödinger’s Equation Using (G′/G)-Expansion Method. Modern Physics Letters B, 34(03), 2050036.
  • Ahmad H., Rafiq, M., Cesarano, C. and Durur, H., 2020. Variational Iteration Algorithm-I with an Auxiliary Parameter for Solving Boundary Value Problems. Earthline Journal of Mathematical Sciences (ISSN: 2581-8147), 3(2), 229-247.
  • Duran, S., (2020). Solitary Wave Solutions of the Coupled Konno-Oono Equation by using the Functional Variable Method and the Two Variables (G'/G, 1/G)-Expansion Method. Adıyaman Üniversitesi Fen Bilimleri Dergisi, 10(2), 585-594.
  • Yavuz, M. and Özdemır, N., (2018). An Integral Transform Solution for Fractional Advection-Diffusion Problem. Mathematical Studies and Applications 2018 4-6 October, 442.
  • Yokus, A., Durur, H., Ahmad, H., Thounthong, P., & Zhang, Y. F. (2020). Construction of exact traveling wave solutions of the Bogoyavlenskii equation by (G′/G, 1/G)-expansion and (1/G′)-expansion techniques. Results in Physics, 19, 103409.
  • Yokus, A., Durur, H., & Ahmad, H. (2020). Hyperbolic type solutions for the couple Boiti-Leon-Pempinelli system. Facta Universitatis, Series: Mathematics and Informatics, 35(2), 523-531.
  • Durur, H., & Yokuş, A. (2020) Vakhnenko-Parkes Denkleminin Hiperbolik Tipte Yürüyen Dalga Çözümü. Erzincan Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 13(2), 550-556.
  • Yokus, A., Durur, H., Ahmad, H., & Yao, S. W. (2020). Construction of Different Types Analytic Solutions for the Zhiber-Shabat Equation. Mathematics, 8(6), 908.
  • Su-Ping, Q. and Li-Xin, T., (2007). Modification of the Clarkson–Kruskal Direct Method for a Coupled System. Chinese Physics Letters, 24(10), 2720.
  • Yokuş, A., & Kaya, D. (2020). Comparison exact and numerical simulation of the traveling wave solution in nonlinear dynamics. International Journal of Modern Physics B, 34(29), 2050282.
  • Yavuz, M. and Özdemir, N., (2018). A Quantitative Approach to Fractional Option Pricing Problems with Decomposition Series. Konuralp Journal of Mathematics, 6(1), 102-109.
  • Rady, A. A., Osman, E. S. and 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.
  • Darvishi, M., Arbabi, S., Najafi, M. and Wazwaz, A., (2016). Traveling Wave Solutions of a (2+1)-Dimensional Zakharov-Like Equation by the First Integral Method and the Tanh Method. Optik, 127(16), 6312-6321.
  • Durur, H., Şenol, M., Kurt, A. and Taşbozan, O., (2019). Approximate Solutions of the Time-Fractional Kadomtsev-Petviashvili Equation with Conformable Derivative. Erzincan University Journal of the Institute of Science and Technology, 12(2), 796-806.
  • Aziz, I. and Šarler, B., (2010). The Numerical Solution of Second-Order Boundary-Value Problems by Collocation Method with the Haar Wavelets. Mathematical and Computer Modelling, 52(9-10), 1577-1590.
  • Kumar, D., Seadawy, A. R. and Joardar, A. K., (2018). Modified Kudryashov Method Via New Exact Solutions for Some Conformable Fractional Differential Equations Arising in Mathematical Biology. Chinese journal of physics, 56(1), 75-85.
  • Baskonus, H. M., Bulut, H. and Sulaiman, T. A., (2019). New Complex Hyperbolic Structures to the Lonngren-Wave Equation by Using Sine-Gordon Expansion Method. Applied Mathematics and Nonlinear Sciences, 4(1), 129-138.
  • Eskitaşçıoğlu, E. İ., Aktaş, M. B. and Baskonus, H. M., (2019). New Complex and Hyperbolic Forms for Ablowitz–Kaup–Newell–Segur Wave Equation with Fourth Order. Applied Mathematics and Nonlinear Sciences, 4(1), 105-112.
  • Dusunceli, F., Celik, E., Askin, M. and Bulut, H., (2020). New Exact Solutions for the Doubly Dispersive Equation Using the Improved Bernoulli Sub-Equation Function Method. Indian Journal of Physics, 1-6.
  • Kaya, D., Yokuş, A. and Demiroğlu, U., (2020). Comparison of Exact and Numerical Solutions for the Sharma–Tasso–Olver Equation. In Numerical Solutions of Realistic Nonlinear Phenomena (pp. 53-65). Springer, Cham.
  • Durur, H., Kurt, A., & Tasbozan, O. (2020). New Travelling Wave Solutions for KdV6 Equation Using Sub Equation Method. Applied Mathematics and Nonlinear Sciences, 5(1), 455-460.
  • Durur, H., Tasbozan, O., & Kurt, A. (2020). New Analytical Solutions of Conformable Time Fractional Bad and Good Modified Boussinesq Equations. Applied Mathematics and Nonlinear Sciences, 5(1), 447-454.
  • Akter, J. and Akbar, M. A., (2015). Exact Solutions to the Benney–Luke Equation and the Phi-4 Equations by Using Modified Simple Equation Method. Results in Physics, 5, 125-130.
  • Quintero, J. R. and Grajales, J. C. M., (2008). Instability of Solitary Waves for a Generalized Benney–Luke Equation. Nonlinear Analysis: Theory, Methods and Applications, 68(10), 3009-3033.
  • Islam, S. R., Khan, K. and Woadud, K. A. A., (2018). Analytical Studies on the Benney–Luke Equation in Mathematical Physics. Waves in Random and Complex Media, 28(2), 300-309.
  • Islam, Z., Hossain, M. M. and Sheikh, M. A. N., (2017). Exact Traveling Wave Solutions to Benney-Luke Equation. GANIT: Journal of Bangladesh Mathematical Society, 37, 1-14.
  • Ibrahim, I. A., Taha, W. M. and Noorani, M. S. M., (2019). Homogenous Balance Method for Solving Exact Solutions of the Nonlinear Benny-Luke Equation and Vakhnenko-Parkes Equation. ZANCO Journal of Pure and Applied Sciences, 31(s4), 52-56.
  • Triki, H., Yildirim, A., Hayat, T., Aldossary, O. M. and Biswas, A., (2012). Shock wave solution of Benney-Luke equation. Romanian Journal of Physics, 57(7-8), 1029-1034.
  • Yavuz, M., & Sene, N. (2020). Approximate solutions of the model describing fluid flow using generalized ρ-laplace transform method and heat balance integral method. Axioms, 9(4), 123.
  • Kumar, D., Paul, G. C., Biswas, T., Seadawy, A. R., Baowali, R., Kamal, M., & Rezazadeh, H. (2020). Optical solutions to the Kundu-Mukherjee-Naskar equation: mathematical and graphical analysis with oblique wave propagation. Physica Scripta, 96(2), 025218.
  • Yavuz, M. (2020). European option pricing models described by fractional operators with classical and generalized Mittag‐Leffler kernels. Numerical Methods for Partial Differential Equations.
  • Gao, W., Rezazadeh, H., Pinar, Z., Baskonus, H. M., Sarwar, S., & Yel, G. (2020). Novel explicit solutions for the nonlinear Zoomeron equation by using newly extended direct algebraic technique. Optical and Quantum Electronics, 52(1), 1-13.
  • Yavuz, M., & Abdeljawad, T. (2020). Nonlinear regularized long-wave models with a new integral transformation applied to the fractional derivative with power and Mittag-Leffler kernel. Advances in Difference Equations, 2020(1), 1-18.
  • Modanli, M. (2019). On the numerical solution for third order fractional partial differential equation by difference scheme method. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 9(3), 1-5.
  • Yokus, A., & Yavuz, M. (2018). Novel comparison of numerical and analytical methods for fractional Burger–Fisher equation. Discrete & Continuous Dynamical Systems-S, 0.
  • Haq, F., Aziz, I., & Islam, S. U. (2010). A Haar wavelets based numerical method for eight-order boundary problems. International Journal of Applied Mathematics and Computer Science, 6, 25-31.
  • Çelik, N., Seadawy, A. R., Özkan, Y. S., & Yaşar, E. (2021). A model of solitary waves in a nonlinear elastic circular rod: Abundant different type exact solutions and conservation laws. Chaos, Solitons & Fractals, 143, 110486.
  • Yavuz, M., & Yokus, A. (2020). Analytical and numerical approaches to nerve impulse model of fractional‐order. Numerical Methods for Partial Differential Equations, 36(6), 1348-1368.
  • Uddin, M. F., Hafez, M. G., Hammouch, Z., Rezazadeh, H., & Baleanu, D. (2021). Traveling wave with beta derivative spatial-temporal evolution for describing the nonlinear directional couplers with metamaterials via two distinct methods. Alexandria Engineering Journal, 60(1), 1055-1065.
  • Modanli, M., Abdulazeez, S. T., & Husien, A. M. (2020). A residual power series method for solving pseudo hyperbolic partial differential equations with nonlocal conditions. Numerical Methods for Partial Differential Equations.
  • Özkan, Y. S., Seadawy, A. R., & Yaşar, E. (2020). On the optical solitons and local conservation laws of Chen–Lee–Liu dynamical wave equation. Optik, 165392.
  • Duran, S., (2021). Breaking theory of solitary waves for the Riemann wave equation in fluid dynamics. International Journal of Modern Physics B, (In press).
Primary Language en
Subjects Basic Sciences
Journal Section Articles
Authors

Orcid: 0000-0002-9297-6873
Author: Hülya DURUR (Primary Author)
Institution: Ardahan University
Country: Turkey


Orcid: 0000-0002-1460-8573
Author: Asıf YOKUŞ
Institution: FIRAT UNIVERSITY
Country: Turkey


Supporting Institution yok
Dates

Application Date : November 29, 2020
Acceptance Date : February 18, 2021
Publication Date : June 30, 2021

APA Durur, H , Yokuş, A . (2021). Exact solutions of the Benney–Luke equation via (1/G')-expansion method . Bilecik Şeyh Edebali Üniversitesi Fen Bilimleri Dergisi , 8 (1) , 56-64 . DOI: 10.35193/bseufbd.833244