A new calculus based on fractal subsets of the real line is formulated. In
this calculus, an integral of order $\alpha, 0 < \alpha \leq 1$, called
$F^\alpha$-integral, is defined, which is suitable to integrate functions with
fractal support $F$ of dimension $\alpha$. Further, a derivative of order
$\alpha, 0 < \alpha \leq 1$, 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.