Exact Solutions of n+1 -Dimensional Space-Time Fractional Zakharov-Kuznetsov Equation

Year 2018, Volume: 5 Issue: 3, 179 - 183, 30.09.2018

Muhammad Nasir Ali Syed Muhammad Husnine Sana Noor Adnan Tuna

https://doi.org/10.17350/HJSE19030000088

Cited By: 1

Abstract

In this article, we study the n+1 -dimensional space time fractional Zakharov-Kuznetsov equation for calculating the exact solutions. For this purpose fractional derivative is used in the form of modified Riemann-Liouville derivatives. Complex fractional transformation is applied for transforming the nonlinear partial differential equation into another nonlinear ordinary differential equation. Exact solutions are obtained by using modified simple equation method and 1⁄G' -expansion method. Obtained solutions are new and may be of significant importance in the field of plasma physics to investigate the waves in the magnetized plasma and in the dust plasma.

Keywords

Modified Riemann-Liouville derivatives, Complex fractional transformation, Modified simple equation method, 1⁄G' - expansion method, Zakharov-Kuznetsov equation.

References

• 1. Gepreel KA, Omran S. Exact solutions of nonlinear fractional partial differential equation. Chinese Physics B 21(11) (2017) 110204.
• 2. Guner O, Aksoy E, Bekir A, Cevikel AC. Different methods for (3+1)-dimensional space-time fractional modified KdVZakharov-Kuznetsov equation. Computers & Mathematics with Applications 71(6) (2016) 1259-1269.
• 3. Jumarie G. Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for non-differentiable functions. Applied Mathematics Letters 22(3) (2009) 378-385.
• 4. Khan K, Akbar MA. Exact solutions of the (2+1)-dimensional cubic Klein–Gordon equation and the (3+1)-dimensional Zakharov-Kuznetsov equation using the modified simple equation method. Journal of Association of Arab Universities for Basic and Applied Sciences 15 (2014) 74–81.
• 5. Lu B. The first integral method for some time fractional differential equation. Journal of Mathematical Analysis and Applications 395(2) (2012) 684-693.
• 6. Miller KS, Ross B. An Introduction to Fractional Calculus and Fractional Differential Equations, John Wiley, New York, 1993.
• 7. Bekir A, Guner O. Exact solutions of nonlinear fractional differential equations by (G^'⁄G)-expansion method. Chinese Physics B 22(11) (2013) 110202.
• 8. Chen C, Jiang Y-L. Lie group analysis for two classes of fractional differential equations. Communications in Nonlinear Science and Numerical Simulation 26(1-3) (2015) 24-35.
• 9. Pandir Y, Gurefe Y. New exact solutions of the generalized fractional Zakharov-Kuznetsov equations. Life Science Journal 10(2) (2013) 2701-2705.
• 10. Zheng B. Exp-function method for solving fractional partial differential equations. The Scientific World Journal 2013 (2013) 465723.
• 11. Zakharov VE, Kuznetsov EA. On three-dimensional solitons. Soviet Physics 39 (1974) 285-288.
• 12. Daghan D, Donmez O. Exact solutions of Gardner equation and their application to the different physical plasma. Brazilian Journal of Physics 46(3) (2016) 321-333.