ar X iv : m at h - ph / 0 31 00 47 v 1 2 3 O ct 2 00 3 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 α, 0 < α ≤ 1, called F α-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 F α-derivative, is defined, which enables us to differentiate functions, like the Cantor staircase, " changing " only on a fractal set. The F α-derivative is local unlike the classical fractional derivative. The F α-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 γ-dimension. F α-differential equations are equations involving F α-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.