Calculus on Fractal Curves in R^n

TL;DR

This paper develops a new calculus framework for fractal curves in R^n, defining integrals and derivatives that are conjugate to ordinary calculus, enabling analysis and modeling on fractal geometries.

Contribution

It introduces the F-alpha-calculus with integrals and derivatives on fractal curves, establishing conjugacy with standard calculus and constructing Sobolev spaces on fractals.

Findings

F-alpha-integral and derivative are conjugate to Riemann integral and derivative.

An algorithm for calculating the mass function on fractals is presented.

The framework allows modeling phenomena like absorption along fractal paths.

Abstract

A new calculus on fractal curves, such as the von Koch curve, is formulated. We define a Riemann-like integral along a fractal curve F, called F-alpha-integral, where alpha is the dimension of F. A derivative along the fractal curve called F-alpha-derivative, is also defined. The mass function, a measure-like algorithmic quantity on the curves, plays a central role in the formulation. An appropriate algorithm to calculate the mass function is presented to emphasize algorithmic aspect. Several aspects of this calculus retain much of the simplicity of ordinary calculus. We establish a conjugacy between this calculus and ordinary calculus on the real line. The F-alpha-integral and F-alpha-derivative are shown to be conjugate to the Riemann integral and ordinary derivative respectively. In fact they can thus be evaluated using the corresponding operators in ordinary calculus and conjugacy. Sobolev Spaces are constructed on F, and F-alpha- differentiability is generalized. Finally we touch upon an example of absorption along fractal path to illustrate the utility of the framework in model making.