Formulation of Calculus on Fractal Curves in R^n and Physical Application
Abstract
It is well known that fractals pervade nature. For example obsereved pathof a quantum mechanical particle or Brownian and Fractional Browniantrajectories are known to be fractals and are continuous butnon-differentiable. I will present a new calculus on fractal curves suchas the von-Koch curve. This will include a Riemann-like integral along afractal curve called the F^alpha-integral, where alpha denotes thedimension of the fractal curve. Also, a corresponding derivative, calledF^alpha-derivative will be introduced. I will discuss various aspects andresults of this Calculus. The second part of the talk will consist of theapplication of this Calculus to Fokker-Planck equation on Fractal Curves.I will start with a Chapmann-Kolmogorov equation on fractal curves andtalk about fractal diffusion and drift coefficients for a suitabletransition probability. In this process, a diffusion equation on fractalcurves will be obtained and finally I will show, how the probabilitydistribution shows deviation from ordinary diffusive behaviour due tounderlying fractal space.
