Variational principles for two kinds of non-linear geophysical KdV equation with fractal derivatives

Abstract

It is an important and difficult inverse problem to construct variational principles from complex models directly, because their variational formulations are theoretical bases for many methods to solve or analyze the non-linear problems. At first, this paper extends two kinds of non-linear geophysical KdV equations in continuum mechanics to their fractional partners in fractal porous media or with irregular boundaries. Then, by designing skillfully, the trial-Lagrange functional, variational principles are successfully established for the non-linear geophysical KdV equation with Coriolis term, and the high-order extended KdV equation with fractal derivatives, respectively. Furthermore, the obtained variational principles are proved to be correct by minimizing the functionals with the calculus of variations.