Solitary Wave Solutions for Some Fractional Evolution Equations via New Kudryashov Approach

Authors

  • Serife Muge Ege Ege University

DOI:

https://doi.org/10.31349/RevMexFis.68.010703

Abstract

In this work, we construct solitary wave solutions of a nonlinear evolution equation in the physical phenomena of waves;
namely the time-fractional fifth-order Sawada-Kotera equation and the (4+1)-dimensional space-time fractional Fokas
equation by Kudryashov method with a new function. As a result, new types of exact analytical solutions are obtained.
Here the fractional derivative is described in beta sense.

 

References

I. Podlubny, Fractional Differential Equations. (Academic Press, 1999)

S.S. Ray, Nonlinear Differential Equations in Physics. (Springer, 2020)

S.S. Ray and S. Sahoo, A novel analytical method with fractional complex transform for new exact solutions of time-fractional fifth-order Sawada-Kotera equation. Rep. Math. Phys. 75 (2015) 63. https://doi.org/10.1016/S0034-4877(15)60024-6.

S.S. Ray and S. Sahoo, Two efficient reliable methods for solving fractional fifth order modified Sawada?Kotera equation appearing in mathematical physics. J. Ocean Eng. Sci. 3 (2016) 219. htts://doi.org/10.1016/j.joes.2016.06.002.

O.S. Iyiola, A numerical study of Ito equation and Sawada-Kotera equation both of time-fractional type. Adv. Math. Sci.

Journal 2 (2013) 71.

M.Senol, L. Akinyemi, A. Ata and O.S. Iyiola Approximate and generalized solutions of conformable Type Coudrey-Dodd-Gibbon-Sawada-Kotera equation. Int. J. Mod. Phys. B 35 (2021) 2150021. https://doi.org/10.1142/S0217979221500211.

P.Veeresha, H.M. Baskonus and W. Gao Strong Interacting Internal Waves in Rotating Ocean: Novel Fractional Approach.

Axinoms 10 (2021) 123. https://doi.org/10.3390/axioms10020123.

S. Zhang and H. Zhang, Fractional sub-equation method and its applications to nonlinear fractional PDEs. Phys. Lett. A 375 (2011) 1069. https://doi.org/10.1016/j.physleta.2011.01.029.

H. Y´epes-Mart´ınez, J.F. G´omez-Aguilar and D. Baleanu, Betaderivative and sub-equation method applied to the optical solitons

in medium with parabolic law nonlinearity and higher order dispersion. Optik 155 (2018) 357. https://doi.org/10.1016/j.ijleo.2017.10.104.

H.M. Baskonus and H. Bulut, Exponential prototype structures for (2+1)-dimensional Boiti-Leon-Pempinelli systems in mathematical physics. Wave in Complex and Random Media 26 (2016) 189. https://doi.org/10.1080/17455030.2015.1132860.

R.M. Jena, S. Chakraverty, S.K. Jena and H.M. Sedighi On the wave solutions of time-fractional Sawada-Kotera-Ito equation

arising in shallow water. Math. Meth. Appl. Sci. 44 (2021) 583. https://doi/10.1002/mma.6763.

S. Sarwar, K.M. Furati and M. Arshad, Abundant wave solutions of conformable space-timefractional order Fokas wave model arising inphysical sciences. Alex. Eng. J. 60 (2021) 2687. https://doi.org/10.1016/j.aej.2021.01.001g.

Y. Zhao and Y. He, The extended fractionalD® » G=G-expansion method and its applications to a space-time fractional Fokas

equation. Math. Probl. Eng. 2017 (2017) 8251653. https://doi.org/10.1155/2017/8251653.

A. Dascioglu, S.C. Unal and D.V. Bayram, New Analytical Solutions for Space and Time Fractional Phi-4 Equation. MTU J. Eng. Nat. Sci. 1 (2020) 30. https://dergipark.org.tr/en/pub/naturengs/issue/54615/697242.

H. M. Baskonus, Complex surfaces to the fractional (2+1)-dimensional Boussinesq dynamical model with the local Mderivative. Eur. Phys. J. Plus 134 (2019) 322, https://doi.org/10.1140/epjp/i2019-12680-4.

W. Gao, G. Yel, H. M. Baskonus and C. Cattani, Complex solitons in the conformable (2+1)-dimensional Ablowitz-Kaup-Newell-Segur equation. AIMS Mathematics 5 (2019) 507. https://doi.org/10.3934/math.2020034.

H.M. Baskonus, New acoustic wave behaviors to the Davey-Stewartson equation with power-law nonlinearity arising in fluid dynamic. Nonlinear Dyn. 86 (2016) 177, https://doi.org/10.1007/s11071-016-2880-4.

F. Tascan, M. Kaplan and A. Bekir, Exponential rational function method for space-time fractional differential equations. Wave Random Complex, 26 (2016) 142, https://doi.org/10.1080/17455030.2015.1125037.

F. Meng, A new approach for solving factional partial differential equations. J. Appl. Math., 2013 (2013) 1, https://dx.doi.org/10.1155/2013/256823.

Y. Gurefe, The generalized Kudryashov method for the nonlinear fractional partial differential equations with the betaderivative. Rev. Mex. Fis. 66 (2020) 771. htts://doi.org/10.31349/RevMexFis.66.771.

S.T. Demiray, Y. Pandir and H. Bulut, Generalized Kudryashov method for time-Fractional differential equations. Abstr. Appl.

Anal. 2014 (2010) 901540. https://doi.org/10.1155/2014/901540.

M. Bagheri and A. Khani, Analytical Method for Solving the Fractional Order Generalized KdV Equation by a Beta-Fractional Derivative. Adv. Math. Phys. 2020 (2020) 8819183. https://doi.org/10.1155/2020/8819183.

S.M. Ege and E. Misirli, Extended Kudryashov method for fractional nonlinear differential equations. Math. Sci. Appl. Enotes, 6 (2018) 19. https://doi.org/10.36753/mathenot.421751g.

E.M.E. Zayed et al., Cubic-quartic optical solitons with Kudryashov’s arbitrary form of nonlinear refractive index. Optik 238 (2021) 166747, https://doi.org/10.1016/j.ijleo.2021.166747.

Y. Yildirim et al., Highly dispersive optical solitons and conservation laws with Kudryashov’s sextic power-law of nonlinear

refractive index. Optik, 240 (2021) 166915. https://doi.org/10.1016/j.ijleo.2021.166915.

D. Kumar, A.R. Seadawy and A.K. Joardar, Modified Kudryashov method via new exact solutions for some conformable fractional differential equations arising in mathematical biology. Chin. J. Phys., 56 (2018) 75. https://doi.org/10.1016/j.cjph.2017.11.020.

Y. Yildirim et al., Cubic-quartic optical soliton perturbation with Kudryashov’s law of refractive index having quadrupledpower law and dual form of generalized nonlocal nonlinearity by sine-Gordon equation approach.J. Opt. 50 (2021) 593. https://doi.org/10.1007/s12596-021-00686-y.

E.M.E. Zayed, K.A. Gepreel, R.M.A. Shohib and M.E.M. Alngar, Solitons in magneto-optics waveguides for the nonlinear Biswas-Milovic equation with Kudryashov’s law of refractive index using the unified auxiliary equation method. Optik, 235 (2021) 166602. https://doi.org/10.1016/j.ijleo.2021.166602.

S. Sain, A.G. Choudhuryb and S. Garai, Solitary wave solutions for the KdV-type equations in plasma: a new approach with the Kudryashov function. Eur. Phys. J. Plus, 136 (2021) 226. https://doi.org/10.1140/epjp/s13360-021-01217-1.

N. A. Kudryashov, Method for finding highly dispersive optical solitons of nonlinear differential equations. Optik 206 (2020)

, https://doi.org/10.1016/j.ijleo.2019.163550.

N. A. Kudryashov, Solitary waves of the non-local Schrodinger equation with arbitrary refractive index. Optik 231 (2021)

, https://doi.org/10.1016/j.ijleo.2021.166443.

R. Khalil, M. Horani, A. Yousef and M. Sababheh, A new definition of fractional derivative. J. Comput. Appl. Math. 264 (2014) 65, https://doi.org/10.1016/j.cam.2014.01.002.

A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: theory and application to

heat transfer model. Therm. Sci., 20 (2016) 763, https://doi.org/10.2298/TSCI160111018A.

A. Atangana and R.T. Alqahtan, Modelling the Spread of River Blindness Disease via the Caputo Fractional Derivative and the

Beta-derivative. Entropy, 18 (2016) 40, https://doi.org/10.3390/e18020040.

Downloads

Published

2022-01-01

How to Cite

[1]
S. M. Ege, “Solitary Wave Solutions for Some Fractional Evolution Equations via New Kudryashov Approach”, Rev. Mex. Fís., vol. 68, no. 1 Jan-Feb, pp. 010703 1–, Jan. 2022.

Issue

Section

07 Gravitation, Mathematical Physics and Field Theory