Variational principle of the 2-D steady-state convection-diffusion equation with fractal derivatives

Abstract

The convection-diffusion equation describes a convection and diffusion process, which is the cornerstone of electrochemistry. The process always takes place in a porous medium or on an uneven boundary, and an abnormal diffusion occurs, which will lead to deviations in prediction of the convection-diffusion process. To overcome the problem, a fractal modification is suggested to deal with the ?abnormal? problem, and a 2-D steady-state convection-diffusion equation with fractal derivatives in the fractal space is established. Furthermore, its fractal variational principle is obtained by the semi-inverse method. The fractal variational formula can not only provide the conservation law in the fractal space in the form of energy, but also give the possible solution structure of the equation.