View source for Analysis on fractals

'''Analysis on fractals''' or '''calculus on fractals''' is a generalization of [[Differentiable manifold|calculus on smooth manifolds]] to [[calculus]] on [[fractals]].

The theory describes dynamical phenomena which occur on objects modelled by fractals.
It studies questions such as "how does heat diffuse in a fractal?" and "How does a fractal vibrate?"

In the smooth case the operator that occurs most often in the equations modelling these questions is the [[Laplacian]], so the starting point for the theory of analysis on fractals is to define a Laplacian on fractals. This turns out not to be a full [[differential operator]] in the usual sense but has many of the desired properties. There are a number of approaches to defining the Laplacian: probabilistic, analytical or measure theoretic.

==See also==

* [[Time scale calculus]] for dynamic equations on a [[cantor set]].
*[[Differential geometry]]
*[[Discrete differential geometry]]

==References==
* {{cite book|author1=Christoph Bandt |author2=Siegfried Graf |author3=Martina Zähle | title = Fractal Geometry and Stochastics II | publisher = Birkhäuser| year = 2000 | isbn = 978-3-7643-6215-7}}
* {{cite book| author=Jun Kigami| title = Analysis on Fractals | publisher = Cambridge University Press| year = 2001 | isbn = 978-0-521-79321-6}}
* {{cite book| author=Robert S. Strichartz| title = Differential Equations on Fractals | publisher = Princeton| year = 2006 | isbn = 978-0-691-12542-8}}
* {{cite book| author=Pavel Exner| author2=Jonathan P. Keating| author3=Peter Kuchment| author4=Toshikazu Sunada| author4-link=Toshikazu Sunada| author5= Alexander Teplyaev| name-list-style=amp| title = Analysis on graphs and its applications: Isaac Newton Institute for Mathematical Sciences, Cambridge, UK, January 8-June 29, 2007 | publisher = AMS Bookstore| year = 2008 | isbn = 978-0-8218-4471-7}}

==External links==
* [http://www.ams.org/notices/199910/fea-strichartz.pdf Analysis on Fractals], Robert S. Strichartz - Article in Notices of the AMS
* [http://www.math.uconn.edu/~teplyaev/fractals/ University of Connecticut - Analysis on fractals Research projects]
* [http://adsabs.harvard.edu/abs/2003math.ph..10047P, Calculus on fractal subsets of real line - I: formulation]

[[Category:Fractals]]


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