Analysis on fractals or calculus on fractals is a generalization of
calculus on smooth manifolds to
calculus
Calculus is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.
Originally called infinitesimal calculus or "the ...
on
fractals
In mathematics, a fractal is a Shape, geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scale ...
.
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
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is th ...
, 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 mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retur ...
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
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Time scale calculus
In mathematics, time-scale calculus is a unification of the theory of difference equations with that of differential equations, unifying integral and differential calculus with the calculus of finite differences, offering a formalism for studying ...
for dynamic equations on a
cantor set
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and mentioned by German mathematician Georg Cantor in 1883.
Throu ...
.
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Differential geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...
*
Discrete differential geometry
References
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External links
Analysis on Fractals Robert S. Strichartz - Article in Notices of the AMS
University of Connecticut - Analysis on fractals Research projectsCalculus on fractal subsets of real line - I: formulation
Fractals
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