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The recently introduced technique, namely, the extended complex method, is used to explore exact solutions for the generalized fifth-order KdV equation. Appropriately, the rational, periodic, and elliptic function solutions are obtained by this technique. The 3D graphs explain the different physical phenomena to the exact solutions of this equation. This idea specifies that the extended complex method can acquire exact solutions of several differential equations in engineering. These results reveal that the extended complex method can be directly and easily used to solve further higher-order nonlinear partial differential equations (NLPDEs). All computer simulations are constructed by maple packages.

In the 20 century, nonlinear science (NLS) plays a significant role in special inventions, for example, the invention of the radio, the discovery of DNA structure for biology, the development of quantum theory for theoretical physics and chemistry, and the invention of transister for computer engineering. It is well known that NLS belongs to the NLPDEs which are introduced in several areas such as fluid thermodynamics, plasma diffusion, biology, physics, geometry, and population dynamics.

Lots of studies are focused on the differential equations [

In the present work, our main purpose is to calculate the generalized fifth-order KdV equation by the extended complex method based on the concept of Yuan et al. [

Let us consider the general form of NLPDE

A transformation

The

Let the meromorphic solutions

Putting the indeterminate forms

The meromorphic solutions are got with the arbitrary pole. Substitute inverse transformation

In this section, we would like to find the exact analytical solutions of a generalized fifth-order KdV equation by extended complex approach. Substitute

Putting (

By assuming that the coefficients of same powers concerning

By solving number of these equations, we obtain

where

By assuming that the coefficients of the same powers concerning

Solve the numbers of these equations, then attain

By the weak

By assuming that the coefficients of the same powers concerning

Solve these equation; then, we have

Applying the additional formula to the

By the above approach, so, we obtain the meromorphic solutions of Eq. (

Here, we display the exact solutions for

Perspective view of 3D graph of

Perspective view of 3D graph of

Perspective view of 3D graph of

Perspective view of 3D graph of

Perspective view of 3D graph of

Perspective view of 3D graph of

Perspective view of 3D graph of

Perspective view of 3D graph of

Figures

Figures

Figures

Figures

Khan et al. [

We employed the extended complex technique to explore the exact analytical solutions of the generalized fifth-order KdV equation. The graphical phenomena are showed by setting the values of arbitrary parameters, and the graphical representations are revealed the mechanism of wave behavior, for example, Figures

The data used to support the finding of this study are mentioned in the article.

The authors mentioned here that they have no conflict of interests.

This work is supported by the NSFC (11901111) and Visiting Scholar Program of Chern Institute of Mathematics.