Savitribai Phule Pune University | All Models Are False, Some Are Useful |
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 F^{α}-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 F^{α}-derivative, is defined,
which enables us to differentiate functions, like the Cantor staircase, ``changing'' only on a fractal set.
The F^{α}-derivative is local unlike the classical fractional derivative.
The F^{α}-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 F^{α}-differentiable and F^{α}-integrable functions are analyzed. Analogues of Sobolev Spaces are constructed on F and F^{α}-differentiability is generalized using Sobolev-like construction. F^{α}-differential equations are equations involving F^{α}-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. |
Keywords | |
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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/. |
Notes, Published Reference, Etc. |
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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 |
Supplementary Material |