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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.

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 & 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.

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