TY - JOUR
TI - Variational principles for two kinds of non-linear geophysical KdV equation with fractal derivatives
AU - Cao Xiao-Qun
AU - Liu Bai-Nian
AU - Liu Meng-Zhu
AU - Peng Ke-Cheng
AU - Tian Wen-Long
JN - Thermal Science
PY - 2022
VL - 26
IS - 3
SP - 2505
EP - 2515
PT - Article
AB - 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.