Variational principle of the fractal Rangwala-Rao equation in optics and quantum system

In general, it is very hard to establish the variational principle (VP) for the nonlinear partial differential equations (NPDEs) in mathematics and physics. In this paper, a fractal nonlinear Rangwala-Rao equation is proposed with the fractal derivative and its fractal VP is explored for the first time. With the aid of the semi-inverse method (SIM), a novel trial Lagrange function (TLF) is introduced and then updated step by step to construct the fractal VP. The developed fractal VP is also confirmed via computing the functional stationary conditions. Especially, the fractal VP becomes the VP of the classic Rangwala-Rao equation when the fractal order [Formula: see text]. The obtained fractal VP can offers some new insights into the fractal variatioanl and numerical methods in physics.