Modified exponential function method for nonlinear mathematical models with Atangana conformable derivative

Authors

DOI:

https://doi.org/10.31349.RevMexFis.67.040704

Keywords:

The modified exponential function method, The space-time fractional modified Benjamin-Bona-Mahony equation, Fractional Sharma-Tasso-Olver equation, Atangana conformable derivative, contour surfaces

Abstract

In this study, we investigate the analytical solutions of the modified Benjamin Bona Mahony and Sharma-Tosso-Olver equations, which are defined with Atangana conformable fractional derivative, using the modified exponential function method. Analytical solutions of the modified Benjamin Bona Mahony and Sharma-Tosso-Olver equations were obtained by using the modified exponential function method. Two, three-dimensional and contour graphics are used to understand the physical interpretations of the resulting analytical solutions to the mathematical model. When all these results and graphs are analzyed, it has been shown that the modified exponential function method is an effective method for obtaining analytical solutions for all other nonlinear fractional partial differential equations containing conformable fractional derivatives of Atangana.

References

C. S. Liu, Trial equation method and its applications to nonlinear evolution equations, Acta Physica Sinica., 54(6), 2505-2509, 2005.

Y. Chen, & Z. Yan, New exact solutions of (2+ 1)-dimensional Gardner equation via the new sine-Gordon equation expansion method, Chaos, Solitons & Fractals, 26(2), 399-406, 2005.

N. A. Kudryashov, One method for finding exact solutions of nonlinear differential equations, Communications in Nonlinear Science and Numerical Simulation, 17, 2248-2253, 2012.

Y. Pandir, Y. Gurefe, U. Kadak & E. Misirli, Classification of exact solutions for some nonlinear partial differential equations with generalized evolution, Abstract and Applied Analysis, 2012(16) 2012.

G. Shen, Y. Sun & Y. Xiong, New travelling-wave solutions for Dodd-Bullough equation, Journal of Applied Mathematics, 2013 (2013).

H. Bulut, T. Aktürk, Y. Gurefe, Traveling wave solutions of the (N+ 1)-dimensional sine-cosine-Gordon equation, AIP Conference Proceedings, American Institute of Physics, 1637(1) (2014) 145-149.

T. Akturk, H. Bulut & Y. Gurefe, New function method to the (n+1)-dimensional nonlinear problems, An International Journal of Optimization and Control: Theories & Applications, 7(3) (2017) 234-239.

S. Yunchuan, New travelling wave solutions for Sine-Gordon equation, Journal of Applied Mathematics, 2014 (2014).

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

D. Kumar, J. Singh and D. Baleanu, A hybrid computational approach for Klein-Gordon equations on Cantor sets. Nonlinear Dyn., 87 (2017) 511-517.

K.M. Owolabi and A. Atangana, Numerical simulation of noninteger order system in subdiffusive, diffusive, and superdiffusive scenarios. J. Comput. Nonlinear Dyn., 12 (2017) 1-10.

R. Metzler and J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339 (2000) 1-77.

H.M. Srivastava, D. Kumar and J. Singh, An efficient analytical technique for fractional model of vibration equation. Appl. Math. Model., 45 (2017) 192-204.

H. Jafari and H.K. Jassim, Numerical solutions of telegraph and Laplace equations on cantor sets using local fractional Laplace decomposition method. Int. J. Adv. Appl. Math. Mech., 2 (2015) 144-151.

K.M. Owolabi and A. Atangana, Numerical simulations of chaotic and complex spatiotemporal patterns in fractional reaction diffusion systems. Comput. Appl. Math., 1 (2017) 1-24.

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) 763-769.

B. Lu, The first integral method for some time fractional differential equations. J. Math. Anal. Appl., 395 (2012) 684-693.

X. J. Yang and Y.D. Zhang, A new Adomian decomposition procedure scheme for solving local fractional Volterra integral equation. Adv. Inf. Tech. Manag., 1(4) (2012) 158-161.

W. Deng, Finite element method for the space and time fractional Fokker-Planck equation. SIAM J. Numer. Anal., 47(1) (2008) 204-226.

M.S. Hu, R.P. Agarwal and X.J. Yang, Local fractional Fourier series with application to wave equation in fractal vibrating string. Abstract. Appl. Anal., 2012 (2012) 567401.

G.H. Gao, Z.Z. Sun and Y.N. Zhang, A finite difference scheme for fractional sub-diffusion equations on an unbounded domain using artificial boundary conditions. J. Comput. Phys., 231(7) (2012) 2865-2879.

Y. Gurefe, The generalized Kudryashov method for the nonlinear fractional partial differential equations with the beta-derivative, Revista Mexicana de Física, (66)6 (2020) 771-781.

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

G. Yel and H. M, Baskonus, Solitons in conformable time-fractional Wu–Zhang system arising in coastal design, Pramana, (93)4 (2019) 1-10.

R. Khalil, M. Al Horani, A. Yousef and M. Sababheh, A new definition of fractional derivative. J. Comput. Appl. Math., 264 (2014) 65-70.

A. Atangana, D. Baleanu and A. Alsaedi, New properties of conformable derivative. Open Math., 13 (2015) 1-10.

H. Yépez-Martínez, J.F. Gómez-Aguilar and A. Atangana, First integral method for non-linear differential equations with comformable derivative, Math. Model. Nat. Phenom., 13 (2018) 1-22.

M. A. Rahman, The exp (− Φ (η))-expansion method with application in the (1+ 1)-dimensional classical Boussinesq equations, Results in Physics, 4 (2014) 150-155.

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Published

2021-07-02

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Section

07 Gravitation, Mathematical Physics and Field Theory