CALCULUS ON FRACTAL SUBSETS OF REAL LINE - I: FORMULATION

A new calculus based on fractal subsets of the real line is formulated. In this calculus, an integral of order alpha, 0 < alpha <= 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 <= 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 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 gamma-dimension. Spaces of F-alpha-differentiable and F-alpha-integrable functions are analyzed. Analogues of Sobolev Spaces are constructed on F and F-alpha-differentiability is generalized using Sobolev-like construction. 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 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.