Calculus on Fractal Subsets of Real Line — Ii: Conjugacy With Ordinary Calculus

Calculus on fractals, or F α -calculus, developed in a previous paper, is a calculus based fractals F ⊂ R, and involves F α -integral and F α -derivative of orders α, 0 < α ≤ 1, where α is the dimension of F. The F α -integral is suitable for integrating functions with fractal support of dimension α, while the F α -derivative enables us to differentiate functions like the Cantor staircase. Several results in F α -calculus are analogous to corresponding results in ordinary calculus, such as the Leibniz rule, fundamental theorems, etc. The functions like the Cantor staircase function occur naturally as solutions of F α -differential equations. Hence the latter can be used to model processes involving fractal space or time, which in particular include a class of dynamical systems exhibiting sublinear behaviour. In this paper we show that, as operators, the F α -integral and F α -derivative are conjugate to the Riemann integral and ordinary derivative respectively. This is accomplished by constructing a map ψ which takes F α -integrable functions to Riemann integrable functions, such that the corresponding integrals on appropriate intervals have equal values. Under suitable conditions, a restriction of ψ also takes F α -differentiable functions to ordinarily differentiable functions such that their values at appropriate points are equal. Further, this conjugacy is generalized to one between Sobolev spaces in ordinary calculus and F α -calculus. This conjugacy is useful, among other things, to find solutions to F α -differential equations: they can be mapped to ordinary differential equations, and the solutions of the latter can be transformed back to get those of the former. This is illustrated with a few examples.