Analysis on fractals
Encyclopedia
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.
Differentiable manifold
A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...
to calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
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
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, accepting a function and returning another .This article considers only linear operators,...
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 calculusTime scale calculusIn 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 hybrid discrete–continuous dynamical systems...
for dynamic equations on a cantor setCantor setIn mathematics, the Cantor set is a set of points lying on a single line segment that has a number of remarkable and deep properties. It was discovered in 1875 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883....
.
- Differential geometry
- Discrete differential geometryDiscrete differential geometryDiscrete differential geometry is the study of discrete counterparts of notions in differential geometry. Instead of smooth curves and surfaces, there are polygons, meshes, and simplicial complexes...
- Abstract differential geometryAbstract differential geometryThe adjective abstract has often been applied to differential geometry before, but the abstract differential geometry of this article is a form of differential geometry without the calculus notion of smoothness, developed by Anastasios Mallios and others from 1998 onwards.Instead of calculus, an...
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