Using Fractal Calculus to Express Electric Potential and Electric Field in Terms of Staircase and Characteristic Functions

Abstract

The Dirac Delta function is usually used to express the discrete distribution of electric charges in electrostatic problems. The integration of the product of the Dirac Delta function and the Green functions can calculate the electric potential and the electric field. Using fractal calculus, characteristic function, ${\chi }_{C_n}(x)$, as an alternative for dirac delta function is used to describe Cantor set charge distribution which is typical example of a discrete set. In these cases we deal with $F^{\alpha }$-integration and $F^{\alpha }$-derivative of the product of characteristic function and function of staircase function, namely $f(S_{C_n}^{\alpha }(x))$, which lead to calculation of electric potential and electric field. Recently, a calculus based fractals, called F${}^{\alpha }$-calculus, has been developed which involve F${}^{\alpha }$-integral and F${}^{\alpha }$-derivative, of orders $\alpha$, $0<\alpha <1$, where $\alpha$ is dimension of $F$. In F${}^{\alpha }$-calculus the staircase function and characteristic function have special roles. Finally, using COMSOL Multiphysics software we solve ordinary Laplace's equation (not fractional) in the fractal region with Koch snowflake boundary which is non-differentiable fractal, and give their graphs for the three first iterations.