Two Reliable Techniques for Solving Conformable Space-Time Fractional PHI-4 Model Arising in Nuclear Physics via β-derivative

Authors

  • Berfin Elma Ege University
  • Emine Mısırlı Ege University

DOI:

https://doi.org/10.31349/RevMexFis.67.050707

Keywords:

The functional variable method, the first integral method, fractional Phi-4 equation, traveling wave solutions, beta-derivative.

Abstract

Nowadays,  nonlinear fractional partial differential equations have been highly using for modelling of physical phenomena. Therefore, it is very important to achieve exact solutions of fractional differential equations for understanding complex phenomena in mathematical physics. In this study,  new exact traveling wave solutions are reached of space-time fractional Phi-4 equation indicated by Atangana’s conformable derivative using two powerful different techniques. These are the functional variable method and the first integral method. Obtaining new solutions of this equation show that method is effective to understanding other nonlinear complex problems in particle and nuclear physics.

References

I. Podlubny, Fractional Differential Equations: an Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Elsevier, (1998).

Caputo, M., Fabrizio, M., A new definition of fractional derivative without singular kernel. Progr. Fract. Differ. Appl, 1(2) (2015) 1-13. https://doi.org/10.1016/j.aml.2019.02.033

Li, C., Guo, Q., Zhao, M., On the solutions of (2 + 1)-dimensional time-fractional Schrödinger equation. Appl. Math. Lett., 94 (2019) 238–243.

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. DOI: 10.2298/TSCI160111018A.

K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley, New York, (1993).

M. Senol, O. Tasbozan, A. Kurt, Numerical Solutions of Fractional Burgers’Type Equations with Conformable Derivative. Chinese Journal of Physics, 58 (2019) 75-84. DOI: 10.1016/j.cjph.2019.01.001.

S. Sarwar, S. Iqbal, Stability analysis, dynamical behavior and analytical solutions of nonlinear fractional differential system arising in chemical reaction. Chinese Journal of Physics, 56 (2018) 374-384. https://doi.org/10.1016/j.cjph.2017.11.009.

R.L. Magin, Fractional Calculus in Bioengineering, BegellHouse Publisher, 125 West Redding. Conn, (2006).

V.F. Morales-Delgado, M.A. Taneco-Hernández, J.F. Gómez-Aguilar, On the solutions of fractional order of evolution equations. The European Physical Journal Plus, 132 (2017) 1-14. https://doi.org/10.1140/epjp/i2017-11341-0.

A. Atangana, J.F. Gómez-Aguilar, Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena. The European Physical Journal Plus, 133 (2018) 1-22. https://doi.org/10.1140/epjp/i2018-12021-3.

Matinfar, M., Eslami, M. Kordy, M., The functional variable method for solving the fractional Korteweg–de Vries equations and the coupled Korteweg–de Vries equations. Pramana, 85(4) (2014) 583-592. DOI: 10.1007/s12043-014-0912-5.

B. Lu, The first integral method for some time fractional differential equations. J. Math. Anal. Appl., 395 (2012) 684. https://doi.org/10.1016/j.jmaa.2012.05.066.

A.A. Gaber, A.F. Aljohani, A. Ebaid and J. Tenreiro Machado, The generalized Kudryashov method for nonlinear space-time fractional partial differential equations of Burgers type. Nonlinear Dyn., 95 (2019) 361. https://doi.org/10.1007/s11071-018-4568-4.

Y. Pandir, Y. Gurefe and E. Misirli, The extended trial equation method for some time-fractional differential equations. Discrete Dyn. Nat. Soc., 2013 (2013) 491359. https://doi. org/10.1155/2013/491359 14.

Y. Pandir, Y. Gurefe and E. Misirli, New exact solutions of the time-fractional Nonlinear dispersive KdV equation. Int. J. Model. Opt., 3 (2013) 349. DOI: 10.7763/IJMO.2013. V3.296.

Ali Kurt, Orkun Tasbozan, and Yucel Cenesiz, Homotopy analysis method for conformable burgers–korteweg-de vries equation. Bull. Math. Sci. Appl, 17(17) (2016) 23. DOI: 10.18052/www.scipress.com/BMSA.17.17.

Asim Zafar, Rational exponential solutions of conformable space-time fractional equal-width equations. Nonlinear Engineering, (2018). https://doi.org/10.1515/nleng-2018-0076.

D. Kumar, K. Hosseini, and F. Samadani, The sine-gordon expansion method to look for the traveling wave solutions of the tzitzéica type equations in nonlinear optics. Optik - International Journal for Light and Electron Optics, 149(439) (2017) 446. https://doi.org/10.1016/j.ijleo.2017.09.066.

S. Zhang, Q.A. Zhong, Fractional sub-equation method and its applications to nonlinear PDE’s. Physics Letters A, 375 (2011), 1069-1073. DOI:10.1016/j.physleta.2011.01.02.

Guner, O., Bekir, A.,Bilgil, H., A note on exp-function method combined with complex transform method applied to fractional differential equations. Advances in Nonlinear Analysis, 4(3) (2015) 201-208. DOI:10.1515/anona-2015-0019.

K. Kamruzzaman, M.A. Akbar, Exact solutions of the (2+1)-dimensional cubic Klein-Gordon equation and the (3+1)-dimensional Zakharov200 Kuznetsov equation using the modified simple equation method. J. Assoc. Arab Univ. Basic Appl., 15 (2014) 74-81. https://doi.org/10.1016/j.jaubas.2013.05.001.

K. Hosseini, A. Bekir A, R. Ansari, Exact solutions of nonlinear conformable time-fractional Boussinesq equations using the exp(−φ(ε)) - expansion method. Opt Quant Electron. 49 (2017) 13. https://doi.org/10.1007/s11082-017-0968-9.

Jiang, Y., Ma, J., High-order finite element methods for time-fractional partial differential equations. J. Comput. Appl. Math., 235 (2011) 3285–3290. https://doi.org/10.1016/j.cam.2011.01.011.

Duan, J.-S., Chaolu, T., Rach, R., Solutions of the initial value problem for nonlinear fractional ordinary differential equations by the Rach–Adomian–Meyers modified decomposition method. Appl. Math. Comput. 218 (2012) 8370–8392. DOI: 10.1016/j.amc.2012.01.063.

Khalil, R., Al Horani, M., Yousef, A., Sababheh, M., A new definition of fractional derivative. Journal of Computational and Applied Mathematics, 264 (2014) 65-70. DOI: 10.1016/j.cam.2014.01.002.

Gurefe, Y., The generalized Kudryashov method for the nonlinear fractional partial differential equations with the beta-derivative. Revista Mexicana de Física, 66 (2020) 771-781. DOI: https://doi.org/10.31349/RevMexFis.66.771.

Khan, U., Ellahi, R., Khan, R., Mohyud-Din, S. T., Extracting new solitary wave solutions of Benny–Luke equation and Phi-4 equation of fractional order by using (G′/G)-expansion method. Optical and Quantum Electronics, 49(11) (2017) 362. DOI: 10.1016/j.amc.2012.01.063.

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Published

2021-09-01

Issue

Section

07 Gravitation, Mathematical Physics and Field Theory