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Abstract
The space-time fractional-coupled Burgers equations and the space-time fractional foam drainage equation are important as an electro-hydro-dynamical model to progress the local electric field and ion acoustic waves in plasma, the shallow water wave problems, and also fluid flow of liquid through foam arisen by gravity and capillarity. In this article, we determine new and further general exact wave solutions to the above-mentioned space-time fractional equations using the generalized (G′/G)-expansion method with the assistance of the fractional complex transformation. It is shown that the method is further effective, convenient, and can be used to establish new solutions for other kind non-linear fractional differential equations arising in mathematical physics. Finally, we depict the 3D and 2D figures of the obtained wave solutions in order to interpret them in geometrical sense.
Public Interest Statement
Fractional differential equations have gained much importance and popularity to the researchers. In order for better understanding the complex phenomena, exact solutions play a vital role. It has recently proved to be a valuable tool to the modeling in various fields of mathematical physics, biology, population dynamics, engineering, fluid-dynamic traffic model, etc. The results obtained from the fractional system are of a more general nature. In the present article, we use the generalized (G′/G)-expansion method to investigate closed form wave solutions of the space-time fractional-coupled Burgers equations and the space-time fractional foam drainage equation. Consequently, we obtain abundant closed form wave solutions of these two equations among them some are new solutions. We expect that the new closed form solutions will be helpful to explain the associated phenomena. Therefore, diverse group of researchers developed and extended different methods for investigating closed form solutions to NLEEs.
Additional information
Notes on contributors
M. Nurul Islam
M. Nurul Islam is a lecturer at the Department Mathematics, Islamic University, Kushtia, Bangladesh. He received his BSc (Honors) and MSc degrees from the Department of Mathematics, Jahangirnagar University, Savar, Dhaka, Bangladesh. He is interested in the research field of non-linear differential equations, fluid mechanics, and also in fractional calculus. He has published more than seven research articles. At present, he is working as a PhD student of his fields of interest.
M. Ali Akbar
M. Ali Akbar is a professor at the Department of Applied Mathematics, University of Rajshahi, Bangladesh. He received his PhD in Mathematics from the Department of Mathematics, University of Rajshahi, Bangladesh. He is actively involved in research in the field of non-linear differential equations and fractional calculus. He has published more than 170 research articles of which 65 articles are published in ISI (Thomson Reuter)-indexed journals and other 15 articles published in Scopus-indexed journals.