Communications in Nonlinear Science and Numerical Simulation
Special Issue on Recent Advances in Fractal-Based Analysis and Application to Science and Engineering
• 大类 : 物理 - 2区
• 小类 : 应用数学 - 1区
• 小类 : 数学跨学科应用 - 2区
• 小类 : 力学 - 2区
• 小类 : 物理：流体与等离子体 - 2区
• 小类 : 物理：数学物理 - 1区
With the expression “Fractal-Based Analysis” we consider the mathematics associated with two fundamental ideas, namely, (1) self-similarity and (2) contractivity.
Self-similarity.This is the “fractals” part of “fractal-based analysis.” Let u ∈ X denote some mathematical object of interest, e.g., a set, a function, an image or a measure, in an appropriate space X. The ﬁrst step is to construct a number of “shrunken” and “distorted” copies of u. Then combine these fractal components of u in some way to make a new object v ∈ X. T maps u to v a fractal transform.
Contractivity.Under suitable conditions on the parameters which characterize the shrinking and distortion mentioned above, the fractal transform T will be contractive in the metric space X. From Banach’s Fixed Point Theorem, T has a unique and attractive ﬁxed point u*.
Given the nature of fractal transforms T, their ﬁxed points u* will exhibit some kind of self-similarity, making them generally “fractal” in nature. Iterated Function Systems (IFSs) are particular fractal transforms, where T is deﬁned by a set of contractive mappings in Rn. The seminal works by Mandelbrot (1982), Hutchinson (1981), Barnsley and Demko (1985), and Barnsley (1989) showed how construct fractals, self-similar sets and measures supported on such sets. After these pioneering papers, there have been further advancements in IFS theory and wide development of applications of IFS theory in a variety of fields, including in image analysis, biology, economics, engineering, medicine, and others.
The focus of this special issue is to collect recent papers in the area of Fractal-based Analysis, and their application to different disciplines. Papers relevant to the scope of the special issue should include, but are not limited to, the following areas: Analysis on Fractals, Fractal Geometry, Fractal-Based Image Analysis, Inverse Problems using Fractal-Based Methods, Numerical methods and Fractal-Based Methods, Application to Biology, Economics, Engineering, Finance, Physics, and Social Sciences.
This special issue will be dedicated to Prof. Edward Robert Vrscay of the Department of Applied Mathematics of the University of Waterloo, who is a recognized scholar in these areas and well-known in the international community for, in particular, his career contributions to the development of the Theory of Generalized Fractal Transforms and their application to Mathematical Imaging.