New Analytical Solutions of Fractional Symmetric Regularized-Long-Wave Equation

Authors

DOI:

https://doi.org/10.31349/RevMexFis.66.297

Keywords:

Fractional partial differential equations, Conformable fractional derivative, Symmetric regularized-long-wave (SRLW) equation, New extended direct algebraic method

Abstract

In this study, new extended direct algebraic method is successfully implemented to acquire new exact wave solution sets for symmetric regularized-long-wave (SRLW) equation which arise in long water flow models. By the help of Mathematica symbolic calculation package, the method produced a great number of analytical solutions. We also presented a few graphical illustrations for some surfaces. The fractional derivatives are considered in the conformable sense. All of the solutions were checked by substitution to ensure the reliability of the method. Obtained results confirm that the method is straightforward, powerful and effective method to attain exact solutions for nonlinear fractional differential equations. Therefore, the method is a good candidate to take part in the existing literature.

Author Biography

Mehmet Senol, Nevsehir Haci Bektas Veli University

Assistant Professor Doctor in Department of Mathematics

References

Zhaosheng, Y., & Jianzhong, L. (1998). Numerical research on the coherent structure in the viscoelastic second-order mixing layers. Applied Mathematics and Mechanics, 19(8), 717-723.

Senol, B., Ates, A., Alagoz, B. B., & Yeroglu, C. (2014). A numerical investigation for robust stability of fractional-order uncertain systems. ISA transactions, 53(2), 189-198.

Oldham, K. B. (2010). Fractional differential equations in electrochemistry. Advances in Engineering Software, 41(1), 9-12.

Heaviside, O. (2008). Electromagnetic theory (Vol. 3). Cosimo, Inc.

Carpinteri, A., & Mainardi, F. (Eds.). (2014). Fractals and fractional calculus in continuum mechanics (Vol. 378). Springer.

Podlubny, I. (1998). Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications (Vol. 198). Elsevier.

Yakar, A., & Koksal, M. E. (2012). Existence results for solutions of nonlinear fractional differential equations. In Abstract and Applied Analysis (Vol. 2012). Hindawi.

Baleanu, D., Ugurlu, Y., & Kilic, B. (2015). Improved (G0=G)-Expansion Method for the Time-Fractional Biological Population Model and Cahn-Hilliard Equation. Journal of Computational and Nonlinear Dynamics, 10(5), 051016.

Kurt, A., Tasbozan, O., & Baleanu, D. (2017). New solutions for conformable fractional Nizhnik-Novikov-Veselov system via G0=G expansion method and homotopy analysis methods. Optical and Quantum Electronics, 49(10), 333.

Tasbozan, O., Senol, M., Kurt, A., & Ozkan, O. (2018). New solutions of fractional Drinfeld-Sokolov-Wilson system in shallow water waves. Ocean Engineering, 161, 62-68.

Inc, M., Aliyu, A. I., Yusuf, A., & Baleanu, D. (2018). Optical solitons for Biswas-Milovic Model in nonlinear optics by Sine-Gordon equation method. Optik, 157, 267-274.

Ghanbari, B., & Gomez, J. F. (2019). The generalized exponential rational function method for Radhakrishnan-Kundu-Lakshmanan equation with conformable time derivative. Revista Mexicana de Fisica, 65(5 Sept-Oct), 503-518.

Ghanbari, B., & Aguilar, J. F. G. (2019). Optical soliton solutions of the Ginzburg-Landau equation with conformable derivative and Kerr law nonlinearity. Revista Mexicana de Fisica, 65(1), 73-81.

Cenesiz, Y., Baleanu, D., Kurt, A., & Tasbozan, O. (2017). New exact solutions of Burgers' type equations with conformable derivative. Waves in Random and Complex Media, 27(1), 103-116.

Eslami, M., & Rezazadeh, H. (2016). The first integral method for Wu-Zhang system with conformable time-fractional derivative. Calcolo, 53(3), 475-485.

Hosseini, K., Mayeli, P., & Ansari, R. (2017). Modifed Kudryashov method for solving the conformable time-fractional Klein-Gordon equations with quadratic and cubic nonlinearities. Optik-International Journal for Light and Electron Optics, 130, 737-742.

Kumar, D., Seadawy, A. R., & Joardar, A. K. (2018). Modi_ed Kudryashov method via new exact solutions for some conformable fractional differential equations arising in mathematical biology. Chinese Journal of Physics, 56(1), 75-85.

Hashemi, M. S., & Bayram, M. (2019). Symmetry properties and exact solutions of the time fractional Kolmogorov-Petrovskii-Piskunov equation. Revista Mexicana de Fisica, 65(5 Sept-Oct), 529-535.

Morales-Delgado, J. G. A. V., & Taneco-Hernandez, M. A. (2019). Analytical solution of the time fractional diffusion equation and fractional convection-diffusion equation. Rev. Mex. Fisica, 65, 8288.

Khalil, R., Al Horani, M., Yousef, A., & Sababheh, M. (2014). A new definition of fractional derivative. Journal of Computational and Applied Mathematics, 264, 65-70.

Atangana, A., Baleanu, D., & Alsaedi, A. (2015). New properties of conformable derivative. Open Mathematics, 13(1).

Tasbozan, O., Cenesiz, Y., & Kurt, A. (2016). New solutions for conformable fractional Boussinesq and combined KdV-mKdV equations using Jacobi elliptic function expansion method. The European Physical Journal Plus, 131(7), 244.

Abdeljawad, T. (2015). On conformable fractional calculus. Journal of computational and Applied Mathematics, 279, 57-66.

Ahmadian, S., & Darvishi, M. T. (2016). New exact traveling wave solutions for space-time fractional (1+ 1)-dimensional SRLW equation. Optik-International Journal for Light and Electron Optics, 127(22), 10697-10704.

Akbulut, A., Kaplan, M., & Bekir, A. (2016). Auxiliary Equation Method for Fractional Differential Equations with Modifed Riemann-Liouville Derivative. International Journal of Nonlinear Sciences and Numerical Simulation, 17(7-8), 413-420.

Ali, K. K., Nuruddeen, R. I., & Raslan, K. R. (2018). New structures for the space-time fractional simplified MCH and SRLW equations. Chaos, Solitons & Fractals, 106, 304-309.

Alzaidy, J. F. (2013). The fractional sub-equation method and exact analytical solutions for some nonlinear fractional PDEs. Am. J. Math. Anal, 1(1), 14-19.

Ege, S. M., & Misirli, E. (2016). Travelling wave solutions of some fractional differential equations. Romanian Journal of Mathematics and Computer Science, 6(1), 106-115.

Korkmaz, A., Hepson, O. E., Hosseini, K., Rezazadeh, H., & Eslami, M. (2018). Sine-Gordon expansion method for exact solutions to conformable time fractional equations in RLW-class. Journal of King Saud University-Science.

Mohyud-Din, S. T., & Bibi, S. (2017). Exact solutions for nonlinear fractional differential equations using exponential rational function method. Optical and Quantum Electronics, 49(2), 64.

Shakeel, M., & Mohyud-Din, M. (2015). A Novel (G'/G)-Expansion Method and its Application to the Space-Time Fractional Symmetric Regularized Long Wave (SRLW) Equation. Adv. Trends Math., 2, 1-16.

Sonmezoglu, A. (2015). Exact solutions for some fractional differential equations. Advances in Mathematical Physics, 2015.

Zahran, E. H. (2015). Exact traveling wave solution for nonlinear fractional partial differential equation

arising in soliton using the exp()-expansion method. International Journal of Computer Applications, 109(13), 12-17.

Xu, F. (2008). Application of Exp-function method to symmetric regularized long wave (SRLW) equation. Physics Letters A, 372(3), 252-257.

Zayed, E. M., Amer, Y. A., & Shohib, R. M. (2014). Exact traveling wave solutions for nonlinear fractional partial differential equations using the improved (G0=G)-expansion method. International Journal of Engineering, 4(7), 8269.

Zayed, E. M., Amer, Y. A., & Shohib, R. M. (2015). The fractional (D_G=G)-expansion method and its

applications for solving four nonlinear space-time fractional PDEs in Mathematical Physics. Ital. J. Pure Appl. Math, 34, 463-482.

Rezazadeh, H., Tariq, H., Eslami, M., Mirzazadeh, M., & Zhou, Q. (2018). New exact solutions of nonlinear conformable time-fractional Phi-4 equation. Chinese Journal of Physics, 56(6), 2805-2816.

Rezazadeh, H., Mirhosseini-Alizamini, S. M., Eslami, M., Rezazadeh, M., Mirzazadeh, M., & Abbagari, S. (2018). New optical solitons of nonlinear conformable fractional Schrodinger-Hirota equation. Optik, 172, 545-553.

Rezazadeh, H. (2018). New solitons solutions of the complex Ginzburg-Landau equation with Kerr law nonlinearity. Optik, 167, 218-227.

Maliet, W. (1992). Solitary wave solutions of nonlinear wave equations. American Journal of Physics, 60(7), 650-654.

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Published

2020-05-01

How to Cite

[1]
M. Senol, “New Analytical Solutions of Fractional Symmetric Regularized-Long-Wave Equation”, Rev. Mex. Fís., vol. 66, no. 3 May-Jun, pp. 297–307, May 2020.

Issue

Vol. 66 No. 3 May-Jun (2020): Revista Mexicana de Fìsica

Section

07 Gravitation, Mathematical Physics and Field Theory

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Authors retain copyright and grant the Revista Mexicana de Física right of first publication with the work simultaneously licensed under a CC BY-NC-ND 4.0 that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.