New soliton wave structures of fractional Gilson-Pickering equation using tanh-coth method and their applications

Abstract The present article discovers the new solitary wave solutions of the well–known Gilson-Pickering model by considering the fractional derivative. The solitary wave solutions are established by considering the proficient technique named as a tanh-coth method with aid of the computational program. The equation is reformulated to a fractional order derivative by using the conformable derivative operator. As a result, a set of new soliton solutions of fractional Gilson-Pickering are efficaciously assembled. Moreover, different types of soliton solutions in the form of 3D-plots, contour plots, and 2D-plots by considering the several different values of parameters are presented. The attained outcomes are generally new for the considered model equation, and the results demonstrate that the used method is competent, direct, and concise which can be used in more complex phenomena.

[1]  Melike Kaplan,et al.  Applications of two reliable methods for solving a nonlinear conformable time-fractional equation , 2017 .

[2]  Arzu Akbulut,et al.  Auxiliary equation method for time-fractional differential equations with conformable derivative , 2017, Comput. Math. Appl..

[3]  Xinghua Fan,et al.  Travelling Wave Solutions for the Gilson-Pickering Equation by Using the Simplified ' / -expansion Method , 2009 .

[4]  Vasily E. Tarasov,et al.  Fractional Vector Calculus , 2010 .

[5]  Dumitru Baleanu,et al.  An Efficient Non-standard Finite Difference Scheme for a Class of Fractional Chaotic Systems , 2018 .

[6]  Tao Liu,et al.  Exact Solutions to Time-Fractional Fifth Order KdV Equation by Trial Equation Method Based on Symmetry , 2019, Symmetry.

[7]  M. S. Hashemi,et al.  New exact solution of the conformable Gilson–Pickering equation using the new modified Kudryashov’s method , 2020 .

[8]  W. Malfliet.  Solitary wave solutions of nonlinear wave equations , 1992 .

[9]  H. Rezazadeh,et al.  Dynamical Behavior of Traveling Wave Solutions for a (2+1)-Dimensional Bogoyavlenskii Coupled System , 2021 .

[10]  M. Eslami,et al.  New solitary wave solutions for the conformable Klein-Gordon equation with quantic nonlinearity , 2020, AIMS Mathematics.

[11]  Billy Sperlich,et al.  Instant Biofeedback Provided by Wearable Sensor Technology Can Help to Optimize Exercise and Prevent Injury and Overuse , 2017, Front. Physiol..

[12]  Wentao Huang,et al.  Bifurcations of travelling wave solutions for the Gilson–Pickering equation , 2009 .

[13]  Abdul-Majid Wazwaz,et al.  Exact and explicit travelling wave solutions for the nonlinear Drinfeld–Sokolov system , 2006 .

[14]  Aniqa Zulfiqar,et al.  Soliton solutions of fractional modified unstable Schrödinger equation using Exp-function method , 2020 .

[15]  Aniqa Zulfiqar,et al.  Exact solitary wave solutions of fractional modified Camassa-Holm equation using an efficient method , 2020 .

[16]  Elsayed M. E. Zayed,et al.  The fractional complex transformation for nonlinear fractional partial differential equations in the mathematical physics , 2016 .

[17]  S. Momani,et al.  Numerical computations of coupled fractional resonant Schrödinger equations arising in quantum mechanics under conformable fractional derivative sense , 2020, Physica Scripta.

[18]  D. Kaya,et al.  Role of Gilson–Pickering equation for the different types of soliton solutions: a nonlinear analysis , 2020, The European Physical Journal Plus.

[19]  Aniqa Zulfiqar,et al.  Comparative study of two techniques on some nonlinear problems based ussing conformable derivative , 2020, Nonlinear Engineering.

[20]  M. Sababheh,et al.  A new definition of fractional derivative , 2014, J. Comput. Appl. Math..

[21]  Aniqa Zulfiqar,et al.  Computational Solutions of Fractional (2 + 1)-Dimensional Ablowitz–Kaup–Newell–Segur Equation Using an Analytic Method and Application , 2021, Arabian Journal for Science and Engineering.

[22]  Anjan Biswas,et al.  Soliton solutions and conservation laws of the Gilson–Pickering equation , 2011 .

[23]  D. Baleanu,et al.  New Optical Solutions of the Fractional Gerdjikov-Ivanov Equation With Conformable Derivative , 2020, Frontiers in Physics.

[24]  M. S. Hashemi,et al.  New wave surfaces and bifurcation of nonlinear periodic waves for Gilson-Pickering equation , 2021 .

[25]  H. M. Baskonus,et al.  1-Soliton solutions of the (2 + 1)-dimensional Heisenberg ferromagnetic spin chain model with the beta time derivative , 2021, Optical and Quantum Electronics.

[26]  Lixin Tian,et al.  Solitons, peakons and periodic cusp wave solutions for the Fornberg-Whitham equation , 2010 .

[27]  F. Zabihi,et al.  A not-a-knot meshless method with radial basis functions for numerical solutions of Gilson–Pickering equation , 2017, Engineering with Computers.

[28]  Adem C. Cevikel,et al.  Study on Fractional Differential Equations with Modified Riemann–Liouville Derivative via Kudryashov Method , 2019, International Journal of Nonlinear Sciences and Numerical Simulation.

[29]  Ahmet Bekir,et al.  The exp-function method for some time-fractional differential equations , 2017, IEEE/CAA Journal of Automatica Sinica.

[30]  Yinnian He,et al.  A generalized fractional sub-equation method for fractional differential equations with variable coefficients☆ , 2012 .

[31]  Aniqa Zulfiqar,et al.  New optical solutions of conformable fractional perturbed Gerdjikov-Ivanov equation in mathematical nonlinear optics , 2021 .

[32]  Rami Ahmad El-Nabulsi,et al.  Path Integral Formulation of Fractionally Perturbed Lagrangian Oscillators on Fractal , 2018, Journal of Statistical Physics.

[33]  N. Ali,et al.  Multiple closed form solutions to some fractional order nonlinear evolution equations in physics and plasma physics , 2019, AIMS Mathematics.

[34]  Asit Saha,et al.  Performance of a hybrid computational scheme on traveling waves and its dynamic transition for Gilson–Pickering equation , 2019, International Journal of Modern Physics C.

[35]  R. El-Nabulsi.  Emergence of quasiperiodic quantum wave functions in Hausdorff dimensional crystals and improved intrinsic Carrier concentrations , 2019, Journal of Physics and Chemistry of Solids.

[36]  Abdul-Majid Wazwaz,et al.  Two reliable methods for solving variants of the KdV equation with compact and noncompact structures , 2006 .

[37]  Marko D. Petkovic,et al.  Soliton solutions of a few nonlinear wave equations , 2010, Appl. Math. Comput..