Centre for Modeling & Simulation
Savitribai Phule Pune University

Technical Report CMS-TR-20080908

Title Calculus on Fractal Subsets of Real Line - I : Formulation
Author/s Abhay Parvate
Centre for Modeling and Simulation and Department of Physics,
Savitribai Phule Pune University, Pune 411 007 India

A. D. Gangal
Department of Physics, Savitribai Phule Pune University, Pune 411 007 India
Abstract A new calculus based on fractal subsets of the real line is formulated. In this calculus, an integral of order α, 0 < α ≤ 1, called Fα-integral, is defined, which is suitable to integrate functions with fractal support F of dimension α. Further, a derivative of order α, 0 < α ≤ 1, called Fα-derivative, is defined, which enables us to differentiate functions, like the Cantor staircase, ``changing'' only on a fractal set. The Fα-derivative is local unlike the classical fractional derivative. The Fα-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 γ-dimension.

Spaces of Fα-differentiable and Fα-integrable functions are analyzed. Analogues of Sobolev Spaces are constructed on F and Fα-differentiability is generalized using Sobolev-like construction.

Fα-differential equations are equations involving Fα-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.
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Citing This Document Abhay Parvate, and A. D. Gangal , Calculus on Fractal Subsets of Real Line - I : Formulation . Technical Report CMS-TR-20080908 of the Centre for Modeling and Simulation, Savitribai Phule Pune University, Pune 411007, India (2008); available at http://cms.unipune.ac.in/reports/.
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Publisher Acknowledgement: Electronic version of an article published as
Fractals 17(1) 53--81 (2009) [DOI 10.1142/S0218348X09004181]
© World Scientific Publishing Company
Contact abhay AT cms.unipune.ac.in
adg AT physics dot unipune.ac.in
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