Analytical approach for travelling wave solution of non-linear fifth-order time-fractional Korteweg–De Vries equation

Abstract

In this paper, we have studied analytical travelling wave solution of a non-linear fifth-order time-fractional Korteweg–De Vries (KdV) equation under the conformal fractional derivative. This equation is very important as it has many applications in various fields such as fluid dynamics, plasma physics, shallow water waves etc. Also, most importantly, the considered fractional-order derivative plays a vital role as it can be varied to obtain different waves. Here, a new form of exact travelling wave solution is derived using the powerful sine–cosine method. To understand the physical phenomena, some visual representations of the solution by varying different parameters are given. Accordingly, it has been observed that the obtained wave solutions are solitons in nature. Also, from the results, one can conclude that the corresponding wave of the solution will translate from left to right by increasing the fractional order $\alpha$. Furthermore, extending the range of x it can be noticed that there is a reduction in the heights of the waves. Also similar observations have been made for a particular time interval and by increasing the values of x. Also, it can be observed that the present method is straightforward as well as computationally efficient compared to the existing methods. The obtained solution has been verified using Maple software and the results are validated.