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 α, 0 < α ≤ 1, called Fα-integral, is defined, which is suitable to integrate functions with fractal support F of dimension α. Further, a derivative of order α, 0 < α ≤ 1, called Fα-derivative, is defined, which enables us to differentiate functions, like the Cantor staircase, "changing" only on a fractal set. The Fα-derivative is local unlike the classical fractional derivative. The Fα-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 generalization 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 γ-dimension.
Spaces of Fα-differentiable and Fα-integrable functions are analyzed. Analogues of Sobolev Spaces are constructed on F and Fα-differentiability is generalized using Sobolev-like construction.
Fα-differential equations are equations involving Fα-derivatives. They can be used to model sublinear dynamical systems and fractal time processes, since sublinear behaviors 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.
