## THERMAL SCIENCE

International Scientific Journal

### Thermal Science - Online First

online first only
### Variational principle of the Whitham-Broer-Kaup equation in shallow water wave with fractal derivatives

**ABSTRACT**

The Whitham-Broer-Kaup equation exists widely in shallow water waves, but unsmooth boundary seriously affects the properties of solitary waves and has certain deviations in scientific research. The aim of this paper is to introduce its modification with fractal derivatives in a fractal space and to establish a fractal variational formulation by the semi-inverse method. The obtained fractal variational principle shows conservation laws in an energy form in the fractal space and also hints its possible solution structure.

**KEYWORDS**

PAPER SUBMITTED: 2020-03-01

PAPER REVISED: 2020-06-15

PAPER ACCEPTED: 2020-06-15

PUBLISHED ONLINE: 2021-01-31

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