On new analytical solutions of fractional systems in shallow water dynamics





Nonlinear fractional partial differential equation, shallow water waves, analytical wave solution, conformable fractional derivative


Recently, fractional calculus has got considerable attention from researchers since many problems in natural sciences and engineering are modelled with differential equations having fractional order. The nonlinear coupled time-fractional Boussinesq-Burger (B-B) equation, the nonlinear time-fractional long water wave (ALW) equation, and the nonlinear (2+1)-dimensional space-time fractional generalized Nizhnik-Novikov-Veselov (GNNV) equation are used to express the structure of shallow water waves (SWWs) with different distributions. The analytical solutions of these equations play a substantial role in explaining the properties of complex phenomena in applied sciences. In the current work, we utilize the exponential rational function (ERF) method with the definition of fractional derivative in the conformable sense to achieve new exact traveling wave solutions of these fractional systems. The correctness, validity, and graphics of the new traveling wave solutions are achieved with the aid of Mathematica. Results demonstrate the effectiveness and strength of this technique to solve the system of fractional differential equations (FDEs).


K. Chunga and T. Toulkeridis, First evidence of paleo-tsunami deposits of a major historic event in ecuador, Sci. Tsunami Hazards 33 (2014) 55–69.

A.A. Imani, D.D. Ganji, H.B. Rokni, H. Latifizadeh, E. Hesameddini and M.H. Rafiee, Approximate traveling wave solution for shallow water wave equation, Appl. Math. Model 36 (2012) 1550, https://doi.org/10.1016/j.apm.2011.09.030.

H.O. Roshid, M.M. Roshid, N. Rahman and M.R. Pervin, New solitary wave in shallow water, plasma and ion acoustic plasma via the GZK-BBM equation and the RLW equation, Propuls. Power Res. 6 (2017) 49, https://doi.org/10.1016/j.jppr.2017.02.002.

W. Hereman, Shallow water waves and solitary waves, Mathematics of Complexity and Dynamical Systems, (Springer, New York, 2012), pp. 1520–1532.

C.B. Vreugdenhil, Numerical methods for shallow-water flow, (Springer Science & Business Media, 2013).

R. Hilfer, Applications of fractional calculus in physics, (World Scientific, Singapore, 2000).

A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and applications of fractional differential equations, (Elsevier Science Limited, 2006).

J. Sabatier, O.P. Agrawal and J.T. Machado, Advances in fractional calculus, (Springer, Dordrecht, 2007). 9. M. Tahir and A.U. Awan, Optical singular and dark solitons with Biswas-Arshed model by modified simple equation method, Optik 202 (2020), https://doi.org/10.1016/j.ijleo.2019.163523.

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

N. Islam, K. Khan and M.H. Islam, Travelling wave solution of Dodd-Bullough-Mikhailov equation: a comparative study between generalized Kudryashov and improved F-expansion methods, J. Phys. Commun. 3 (2019), https://doi.org/10.1088/2399-6528/ab1a47.

M. Senol, New analytical solutions of fractional symmetric regularized-long-wave equation, Rev. Mex. Fis. 66 (2020) 297, https://doi.org/10.31349/RevMexFis.66.297.

H. Durur, Different types analytic solutions of the (1 + 1)- dimensional resonant nonlinear Schrödinger’s equation using (G 0 /G)-expansion method, Mod. Phys. Lett. B 34 (2020), https://doi.org/10.1142/S0217984920500360.

S.T. Mohyud-Din and S. Bibi, Exact solutions for nonlinear fractional differential equations using exponential rational function method, Opt Quantum Electron 49 (2017) 64, https://doi.org/10.1007/s11082-017-0895-9.

H. Yepez-Martínez, J.F. Gómez-Aguilar and A. Atangana, First integral method for non-linear differential equations with conformable derivative, Math Model Nat Phenom 13 (2018) 14, https://doi.org/10.1051/mmnp/2018012.

M. Odabasi, Traveling wave solutions of conformable timefractional Zakharov-Kuznetsov and Zoomeron equations, Chin. J. Phys. 64 (2020) 194, https://doi.org/10.1016/j.cjph.2019.11.003.

B. Ghanbari and J.F. Gómez-Aguilar, The generalized exponential rational function method for Radhakrishnan-KunduLakshmanan equation with β-conformable time derivative, Rev. Mex. Fis., 65 (2019) 503, https://doi.org/10.31349/RevMexFis.65.503.

Z. Wen, The generalized bifurcation method for deriving exact solutions of nonlinear space-time fractional partial differential equations, Appl. Math. Comput. 366 (2020), https://doi.org/10.1016/j.amc.2019.124735.

G. Bakıcıerler, S. Alfaqeih and E. Mısırlı, Analytic solutions of a (2 + 1)-dimensional nonlinear Heisenberg ferromagnetic spin chain equation, Phys. A: Stat. Mech. Appl. 582 (2021), https://doi.org/10.1016/j.physa.2021.126255.

H. Yepez-Martínez and J.F. Gómez-Aguilar, Fractional sub-equation method for Hirota-Satsuma-coupled KdV equation and coupled mKdV equation using the Atangana’s conformable derivative, Waves Random Complex Media 29 (2019) 678, https://doi.org/10.1080/17455030.2018.1464233.

K.S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, (John-Wily and Sons, New York, 1993).

I. Podlubny, Fractional differential equations, (Academic Press, London, 1999).

G. Jumarie, Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for non-differentiable functions, Appl. Math. Lett. 22 (2009) 378, https://doi.org/10.1016/j.aml.2008.06.003.

M. Dalir and M. Bashour, Applications of fractional calculus, Appl. Math. Sci. 4 (2010) 1021-1032.

T.D. Leta, W. Liu and J. Ding, Existence of periodic, solitary and compacton travelling wave solutions of a (3 + 1)- dimensional time-fractional nonlinear evolution equations with applications, Anal. Math. Phys. 11 (2021) 1, https://doi.org/10.1007/s13324-020-00458-0.

E. Atilgan, M. Senol, A. Kurt and O. Tasbozan, New wave solutions of time-fractional coupled Boussinesq-WhithamBroer-Kaup equation as a model of water waves, China Ocean Eng. 33 (2019) 477, https://doi.org/10.1007/s13344-019-0045-1.

R. Khalil, M. Al 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.

T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math. 279 (2015) 57, https://doi.org/10.1016/j.cam.2014.10.016.

D. Zhao and M. Luo, General conformable fractional derivative and its physical interpretation, Calcolo 54 (2017) 903, https://doi.org/10.1007/s10092-017-0213-8.

R. Khalil, M. Al Horani and M.A. Hammad, Geometric meaning of conformable derivative via fractional cords, J. Math. Comput. Sci. 19 (2019) 241, https://doi.org/10.22436/jmcs.019.04.03.

R. Almeida, M. Guzowska and T. Odzijewicz, A remark on local fractional calculus and ordinary derivatives, Open Math. 14 (2016) 1122, https://doi.org/10.1515/math-2016-0104.

B. Ghanbari and D. Baleanu, New optical solutions of the fractional Gerdjikov-Ivanov equation with conformable derivative, Front. Phys. 8 (2020) 167, https://doi.org/10.3389/fphy.2020.00167.

F. Gao and C. Chi, Improvement on conformable fractional derivative and its applications in fractional differential equations, J. Funct. Spaces 2020 (2020) 1, https://doi.org/10.1155/2020/5852414.

A. Atangana, D. Baleanu and A. Alsaedi, Analysis of timefractional Hunter-Saxton equation: a model of neumatic liquid crystal, Open Phys. 14 (2016) 145, https://doi.org/10.1515/phys-2016-0010.

M.M. Khater and D. Kumar, New exact solutions for the time fractional coupled Boussinesq-Burger equation and approximate long water wave equation in shallow water, J. Ocean Eng. Sci. 2 (2017) 223, https://doi.org/10.1016/j.joes.2017.07.001.

S. Saha Ray, A novel method for travelling wave solutions of fractional Whitham-Broer-Kaup, fractional modified Boussinesq and fractional approximate long wave equations in shallow water, Math. Methods Appl. Sci. 38 (2015) 1352, https://doi.org/10.1002/mma.3151.

M. Mirzazadeh, M. Ekici, A. Sonmezoglu, S. Ortakaya, M. Eslami and A. Biswas, Soliton solutions to a few fractional nonlinear evolution equations in shallow water wave dynamics, Eur. Phys. J. Plus 131 (2016) 1, https://doi.org/10.1140/epjp/i2016-16166-7.

G. Bakıcıerler, S. Alfaqeih and E. Mısırlı, Application of the modified simple equation method for solving two nonlinear time-fractional long water wave equations, Rev. Mex. Fis. 67 (2021) 1, https://doi.org/10.31349/RevMexFis.67.060701.

O. Guner, New travelling wave solutions for fractional regularized long-wave equation and fractional coupled NizhnikNovikov-Veselov equation, Int. J. Optim. Control: Theor. 8 (2018) 63, https://doi.org/10.11121/ijocta.01.2018.00417.

S. Kumar, A. Kumar, and D. Baleanu, Two analytical methods for time-fractional nonlinear coupled Boussinesq-Burger’s equations arise in propagation of shallow water waves, Nonlinear Dyn. 85 (2016) 699, https://doi.org/10.1007/s11071-016-2716-2.

S. Javeed, S. Saif, A. Waheed and D. Baleanu, Exact solutions of fractional mBBM equation and coupled system of fractional Boussinesq-Burgers, Results Phys. 9 (2018) 1275, https://doi.org/10.1016/j.rinp.2018.04.026.

A.A. Al-Shawba, F.A. Abdullah, A. Azmi, and M.A. Akbar, Reliable methods to study some nonlinear conformable systems in shallow water, Adv Differ Equ 2020 (2020) 1, https://doi.org/10.1186/s13662-020-02686-x.

K. Fan and C. Zhou, Mechanical solving a few fractional partial differential equations and discussing the effects of the fractional order, Adv. Math. Phys. 2020 (2020) 3758353, https://doi.org/10.1155/2020/3758353.

H.C. Yaslan, New analytic solutions of the space-time fractional Broer-Kaup and approximate long water wave equations, J. Ocean Eng. Sci. 3 (2018) 295, https://doi.org/10.1016/j.joes.2018.10.004.

M. Kaplan and A. Akbulut, Application of two different algorithms to the approximate long water wave equation with conformable fractional derivative, Arab. J. Basic Appl. Sci. 25 (2018) 77, https://doi.org/10.1080/25765299.2018.1449348.

D. Kumar, M. Kaplan, M. Haque, M.S. Osman, and D. Baleanu, A variety of novel exact solutions for different models with the conformable derivative in shallow water, Front. Phys. 8 (2020) 177, https://doi.org/10.3389/fphy.2020.00177.

M.A. Arefin, M.A. Khatun, M.H. Uddin, and M. Inc, Investigation of adequate closed form travelling wave solution to the space-time fractional non-linear evolution equations, J. Ocean Eng. Sci. (2021), https://doi.org/10.1016/j.joes.2021.08.011.

Q. Feng, A new analytical method for seeking traveling wave solutions of space-time fractional partial differential equations arising in mathematical physics, Optik 130 (2017) 310, https://doi.org/10.1016/j.ijleo.2016.10.106.

B. Zheng, Exact solutions for some fractional partial differential equations by the (G 0 /G) method, Math. Probl. Eng. 2013 (2013) 1, http://dx.doi.org/10.1155/2013/826369.

Y. Liu and L. Yan, Solutions of fractional KonopelchenkoDubrovsky and Nizhnik-Novikov-Veselov equations using a generalized fractional subequation method, Abstr. Appl. 2013 (2013) 1, http://dx.doi.org/10.1155/2013/839613.

H. Demiray, A travelling wave solution to the KdV-Burgers equation, Appl. Math. Comput. 154 (2004) 665, https://doi.org/10.1016/S0096-3003(03)00741-0.

A. Bekir and M. Kaplan, Exponential rational function method for solving nonlinear equations arising in various physical models, Chin. J. Phys. 54 (2016) 365, https://doi.org/10.1016/j.cjph.2016.04.020.

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

P. Veeresha, D.G. Prakasha, M.A. Qurashi and D. Baleanu, A reliable technique for fractional modified Boussinesq and approximate long wave equations, Adv. Differ. Equ. (2019), https://doi.org/10.1186/s13662-019-2185-2.

M.E. Elbrolosy and A.A. Elmandouh, Bifurcation and new traveling wave solutions for (2 + 1)-dimensional nonlinear Nizhnik-Novikov-Veselov dynamical equation, Eur. Phys. J. Plus 135 (2020) 533, https://doi.org/10.1140/epjp/s13360-020-00546-x.

L. Debnath and K. Basu, Nonlinear water waves and nonlinear evolution equations with applications, Encyclopedia of Complexity and Systems Science (2014) 1–59, https://doi.org/10.1007/978-3-642-27737-5_609-1.

K.K. Ali, H. Dutta, R. Yilmazer and S. Noeiaghdam, On the new wave behaviors of the Gilson-Pickering equation, Front. Phys. 8 (2020) 54, https://doi.org/10.3389/fphy.2020.00054.

E.W. Weisstein, Concise Encyclopedia of Mathematics, 2nd edition, (CRC Press, New York, 2002).




How to Cite

G. Bakıcıerler and E. Mısırlı, “On new analytical solutions of fractional systems in shallow water dynamics”, Rev. Mex. Fís., vol. 68, no. 5 Sep-Oct, pp. 050701 1–, Aug. 2022.



07 Gravitation, Mathematical Physics and Field Theory