Wave Structures for Nonlinear Schrodinger Types Fractional Partial Differential Equations Arise in Physical Sciences
DOI:
https://doi.org/10.18034/ei.v9i2.560Keywords:
The rational ( )-expansion method, wave variable transformation, nonlinear fractional partial differential equation, analytic solution, solitonAbstract
Nonlinear partial differential equations are mostly renowned for depicting the underlying behavior of nonlinear phenomena relating to the nature of the real world. In this paper, we discuss analytic solutions of fractional-order nonlinear Schrodinger types equations such as the space-time fractional nonlinear Schrodinger equation and the (2+1)-dimensional time-fractional Schrodinger equation. The considered equations are converted into ordinary differential equations with the help of wave variable transformation and then the recently established rational ( )-expansion method is employed to construct the exact solutions. The obtained solutions have appeared in the forms of a trigonometric function, hyperbolic function, and rational function which are compared with those of literature and claimed to be different. The graphical representations of the solutions are finally brought out for their physical appearances. The applied method is seemed to be efficient, concise, and productive which might be used for further research.
Mathematics Subject Classifications: 35C08, 35R11
Downloads
References
Abdel-Salam, E., Yousif, E., El-Aasser, M. (2016). Analytical solution of the space-time fractional nonlinear Schrodinger equation. Rep. Math. Phys., 77, 19-34.
Alam, M. & Li, X. (2019). Exact traveling wave solutions to higher order nonlinear equations. J. Ocean Eng. Sci., 4, 276-288.
Alam, M. (2015). Exact solutions to the foam drainage equation by using the new generalized 〖(G〗^'/G-expansion method. Res. Phys., 5, 168-177.
Alam, M., Akbar, M., Hoque, M. (2014). Exact traveling wave solutions of the (3+1)-dimensional mKdV-ZK equation and the (1+1)-dimensional compound KdVb equation using new approach of the generalized 〖(G〗^'/G)-expansion method. Pramana J. Phys., 83, 317-329.
Bazyar, M. H. & Song, C. (2017). Analysis of transient wave scattering and its applications to site response analysis using the scaled boundary finite-element method. Soil Dyn. Earth quake Engg., 98, 191-205.
Bekir, A., Guner, O. A. (2013). Exact solutions of nonlinear fractional differential equations by 〖(G〗^'/G)-expansion method. Chin. Phys. B, 103, 404-9.
Bhrawy, A. H., Ahmed, E. A., & Baleanu, D. (2014a). The modified simple equation method for nonlinear fractional differential equation. Proc. Romanian Acad. A, 15, 322.
Bhrawy, A. H., Al-Zahrani, A. A., Alhamed, Y. A., Baleanu, D. (2014b).The modified simple equation method for nonlinear fractional differential equation. Rom. J. Phys. 59, 646.
Biswas, A., Ebadi, G., Triki, H., Yildirim, A., Yousefzadeh, N. (2013a). Topological soliton and other exact solutions to KdV-Caudrey-Dodd-Gibbon equation. Results Math, 63, 687-703.
Biswas, A., Kumar, S., Krishnan, E. V., Ahmed, B., Strong, A., Johnson, S., Yildirim, A. (2013b). Topological solitons and other solutions to potential Korteweg-de-Vries equation. Rom. Rep. Phys., 65, 1125-37.
Doha, E. H., Baleanu, D., Bhrawy, A. H., Hafez, R. M. (2014). The modified simple equation method for nonlinear fractional differential equation. Proc. Romanian Acad. A. 15, 130.
Eslami, M., Vajargah, F., Mirzazadeh, M., & Biswas, A. (2004). Application of first integral method to fractional partial differential equations. Indian J. Phys., 88, 177-184.
Ghanbari, B., Baleanu, D., Qurashi, M. A. (2019). New exact solutions of the generalized Benjamin Bonamahony equation. Symmerty 11, 20.
Golmankhaneh, A. K., Baleanu, D. (2011). Homotopy perturbation method for solving a system of Schrodinger-Korteweg-de Vries equation. Romanian Rep. Phys., 63, 609-623.
Guo, P. (2019). The Adomian decomposition method for a type of fractional differential equations. J. Apple. Math. Phys., 7, 2459-2466.
Islam, M. T. & Akter, M. A. (2021a). Distinct solutions of nonlinear space-time fractional evolution equations appearing in mathematical physics via a new technique. Partial Diff. Eq. Appl. Math., 3.
Islam, M. T. & Akter, M. A. (2021b). Exact analytic wave solutions to some nonlinear fractional differential equations for the shallow water wave arise in physics and engineering. J. Research Engg. Appl. Sci., 6(1), 11-18.
Jannah, M., Islam, M. T., Akter, M. A. (2021). Explicit travelling wave solutions to nonlinear partial differential equations arise in mathematical physics and engineering. J. Eng. Advan., 2, 58-63.
Kaplan, M., Unsal, O., Bekir. A. (2016). Exact solutions of nonlinear Schrodinger equation by using symbolic computation. Math. Meth. Appl. Sci., 39.
Karaagac, B. (2019). New exact solutions for some fractional order differential equations via improved sub-equation method. Discrete Contin. Dyn. Syst., 12, 447-454.
Khalil, R., Horani, M. A., Yousef, A., Sababheh, M. A. M. (2014). A new definition of fractional derivative. J. Comput. Appl. Math., 264, 65–70.
Kudryashov, N. A. (2012). One method for finding exact solutions of nonlinear differential equations. Commun. Nonlin. Sci. Numer. Simul., 17, 2248-53.
Laskin, N. (2002) Fractional Schrodinger equation. Phys. Rev. E., 66, 056108.
Li, C., Guo, Q., Zhao, M. (2019). On the solutions of (2+1)-dimensional time-fractional Schrodinger equation. Appl. Math. Lett., 94, 238-243.
Lu, D., Seadawy, A., Arshad, M. (2017). Application of extend simple equation method on unstable Schrodinger equations. Optic-Int. J. Light Elect. Opt., 140, 136-144.
Mohyud-Din, S. T. (2012). A meshless numerical solution of the family of generalized fifth-order Korteweg-de Vries equations. Int. J. Numer. Meth. Heat Fluid Flow, 2, 641-658.
Nuruddeen, R. I. & Nass, A. M. (2017). Exact solutions of wave-type equations by the aboodh decomposition method. Stochastic Model Appl., 21, 23-30.
Nuruddeen, R. I. (2017). Elzaki decomposition method and its applications in solving linear and nonlinear Schrodinger equations. Sohag J. Math., 4, 1-5.
Omar, A. A. (2019a). Fitted fractional reproducing kernel algorithm for the numerical solutions of abc-fractional Volterra integro-differential equations. Chaos Solit. Fract., 126, 394-402.
Omar, A. A. (2019b). Modulation of reproducing kernel hilbert space method for numerical solutions of Riccati and Bernoulli equations in the Atangana-Baleanu fractional sense. Chaos Solit. Fract., 125, 163-170.
Rizvi, S. T. R., Ali, K., Bashir, S., Younis, M., Ashraf, R., Ahmad, M. O. (2017). Exact solution of (2+1)-dimensional fractional Schrodinger equation. Superlattices Microstruct., 107, 234-239.
Saha, R. S. (2016). New analytical exact solutions of time fractional KdV-KZK equation by Kudryashov methods. Chin. Phys. B, 25, 040204.
Saxena, R., & Kalla, S. (2010). Solution of space-time fractional Schrodinger equation occurring in quantum mechanics. Fract. Calc. Appl. Anal., 13, 177-190.
Shah, R., Khan, H., Arif, M., Kumam, P. (2019). Application of Laplace Adomian decomposition method for the analytical solution of third-order dispersive fractional partial differential equations. Entropy, 21, 335.
Sheikholeslami, M. (2017). Cuo-water nanofluid free convection in a porous cavity considering Darcy law. Eur. Phys. J. Plus, 132.
Younis, M., Rehman, H., Rizvi, S., & Mahmood, S. (2017). Dark and singular optical solitons perturbation with fractional temporal evolution. Superlattices Microstruct, 104, 525-531.
Zayed, E. M. & Alurrfi, K. A. (2016). The (G^'/G,1/G)-expansion method and its applications to two nonlinear Schrodinger equations describing the propagation of femtosecond pulses in nonlinear optical fibers. Optik-International Journal for Light and Electron Optics, 127, 1581-1589.
Zayed, E. M. E. & Abdelaziz, M. A. M. (2012). The two-variable G^'/G,1/G-expansion method for solving the nonlinear KdV-mKdV equation. Math. Prob. Engg., 14, 725061.
Zayed, E. M., Shahoot, A. M., Alurrfi, K.A. (2018). The 〖(G〗^'/G,1/G)-expansion method and its applications for constructing many new exact solutions of the higher-order nonlinear Schrodinger equation and the quantum Zakharov Kuznetsov equation. Optical and Quantum Electronics, 50, 96.
--0--
Downloads
Published
How to Cite
Issue
Section
License
Engineering International is an Open Access journal. Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal the right of first publication with the work simultaneously licensed under a CC BY-NC 4.0 International License that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of their work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgment of its initial publication in this journal. We require authors to inform us of any instances of re-publication.
