Savitribai Phule Pune University | All Models Are False, Some Are Useful |
Title | Calculus on Fractal Curves in R^n |
Author/s |
Abhay Parvate
Centre for Modeling and Simulation and Department of Physics, Savitribai Phule Pune University, Pune 411 007 India Seema Satin and A. D. Gangal Department of Physics, Savitribai Phule Pune University, Pune 411 007 India |
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 using Sobolev like constructions. Finally we touch upon an example of diffusion equation on fractal curves, to illustrate the utility of the framework. |
Keywords | |
Download | arXiv |
Citing This Document | Abhay Parvate, Seema Satin, and A. D. Gangal , Calculus on Fractal Curves in R^n . Technical Report CMS-TR-20090722 of the Centre for Modeling and Simulation, Savitribai Phule Pune University, Pune 411007, India (2009); available at http://cms.unipune.ac.in/reports/. |
Notes, Published Reference, Etc. | |
Contact |
abhay AT cms.unipune.ac.in
satin AT physics dot unipune.ac.in adg AT physics dot unipune.ac.in |
Supplementary Material |