Analysis on fractals
Analysis on fractals or calculus on fractals is a generalization of 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
- Abstract differential geometry
References
- Christoph Bandt; Siegfried Graf; Martina Zähle (2000). Fractal Geometry and Stochastics II. Birkhäuser. ISBN 978-3-7643-6215-7.
- Jun Kigami (2001). Analysis on Fractals. Cambridge University Press. ISBN 978-0-521-79321-6.
- Robert S. Strichartz (2006). Differential Equations on Fractals. Princeton. ISBN 978-0-691-12542-8.
- Pavel Exner, Jonathan P. Keating, Peter Kuchment, Toshikazu Sunada, and Alexander Teplyaev (2008). Analysis on graphs and its applications: Isaac Newton Institute for Mathematical Sciences, Cambridge, UK, January 8-June 29, 2007. AMS Bookstore. ISBN 978-0-8218-4471-7.
External links
- Analysis on Fractals, Robert S. Strichartz - Article in Notices of the AMS
- University of Connecticut - Analysis on fractals Research projects
- Calculus on fractal subsets of real line - I: formulation
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