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A NEW PERSPECTIVE ON THE STUDY OF THE FRACTAL COUPLED BOUSSINESQ–BURGER EQUATION IN SHALLOW WATER

KANG-JIA WANG

School of Physics and Electronic Information Engineering, Henan Polytechnic University, Jiaozuo 454003, P. R. China

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GUO-DONG WANG

School of Physics and Electronic Information Engineering, Henan Polytechnic University, Jiaozuo 454003, P. R. China

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, and

HONG-WEI ZHU

https://orcid.org/0000-0002-1573-0220

School of Physics and Electronic Information Engineering, Henan Polytechnic University, Jiaozuo 454003, P. R. China

E-mail Address: zhw0011@outlook.com

Corresponding author.

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https://doi.org/10.1142/S0218348X2150122XCited by:33 (Source: Crossref)

Abstract

The well-known coupled Boussinesq–Burger equation can be used to describe the flow of the shallow water of the harbor. But when the boundary is nonsmooth, it becomes powerless. So, the fractal calculus is needed to be applied to it and the fractal coupled Boussinesq–Burger equation (FCBBe) is presented for the first time in this paper. By using the semiinverse method, we have successfully established the fractal variational formulation of the FCBBe, which can not only provide the conservation laws in an energy form in the fractal space but also reveal the possible solution structures of the equation. Then a novel variational approach based on He’s variational method and the two-scale transform are used to seek its periodic wave solutions. The main advantage of variational approach is that it can reduce the order of differential equation and make the equation more simple. Finally, the numerical results have been shown through graphs to discuss the effect of different fractal orders on the wave motion. The obtained results in this work are expected to shed a bright light on the study of fractal nonlinear partial differential equations in the fractal space.

Keywords:

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