Calculus on fractal subsets of real line - I: formulation

Abstract

A new calculus based on fractal subsets of the real line is formulated. In this calculus, an integral of order $\alpha ,0<\alpha \le 1$, called $F^{\alpha }$-integral, is defined, which is suitable to integrate functions with fractal support $F$ of dimension $\alpha$. Further, a derivative of order $\alpha ,0<\alpha \le 1$, called $F^{\alpha }$-derivative, is defined, which enables us to differentiate functions, like the Cantor staircase, ``changing'' only on a fractal set. The $F^{\alpha }$-derivative is local unlike the classical fractional derivative. The $F^{\alpha }$-calculus retains much of the simplicity of ordinary calculus. Several results including analogues of fundamental theorems of calculus are proved. The integral staircase function, which is a generalisation of the functions like the Cantor staircase function, plays a key role in this formulation. Further, it gives rise to a new definition of dimension, the $\gamma$-dimension. $F^{\alpha }$-differential equations are equations involving $F^{\alpha }$-derivatives. They can be used to model sublinear dynamical systems and fractal time processes, since sublinear behaviours are associated with staircase-like functions which occur naturally as their solutions. As examples, we discuss a fractal-time diffusion equation, and one dimensional motion of a particle undergoing friction in a fractal medium.

32 pages, 1 figure, to be submitted to Nonlinearity