Adequate soliton solutions to the time fractional Zakharov-Kuznetsov equation and the space-time fractional Zakharov-Kuznetsov-Benjamin-Bona-Mahony equation

• Published in: Arab journal of basic and applied sciences 2021, Vol.28 (1), p.370-385
• Main Authors: Islam, Nurul, Parvin, Rehana, Pervin, Rashidah, Akbar, Ali
• Format: Article
• Language: English
• Subjects: Auxiliary equation method; complex transformation; nonlinear fractional differential equation; the ZK equation; the ZKBBM equation
• ISSN: 1815-3852; 2576-5299; 2576-5299
• DOI: 10.1080/25765299.2021.1969740

The time fractional (2, 2, 2) Zakharov-Kuznetsov (ZK) equation and the space-time fractional Zakharov-Kuznetsov-Benjamin-Bona-Mahony (ZKBBM) equation demonstrate the characteristic of shallow water waves, turbulent motion, waves of electro-hydro-dynamics in the local electric field, sound wave, waves of driving flow of fluid, ion acoustic waves in plasmas, traffic flow, financial mathematics, etc. The time-fractional (2, 2, 2) ZK equation is the particular case of the general time-fractional (λ, μ,δ ) ZK equation, where λ, μ represent the space coordinate and δ represents the temporal coordinate. Hereinto to evade the complexity and to ascertain soliton solutions of this model, we accept λ=2, μ=2, δ=2 and in this case, the general ZK equation is called the time-fractional (2, 2, 2) ZK equation. In this article by making use of the concept of fractional complex transformation, the auxiliary equation method is put in use to search the closed form soliton solutions to the above indicated fractional nonlinear equations (FNLEs).The ascertained solutions are in the form of exponential, rational, hyperbolic and trigonometry functions with significant precision. We illustrate the soliton solutions relating to physical concern by setting the definite values of the free parameters through depicting diagram and interpreted the physical phenomena. The developed solutions assert that the method is effective, able to measure NLEEs, influential, powerful and offer vast amount of travelling wave solutions of nonlinear evolution equations in the area of mathematical sciences and engineering.