Year: 2016
Type: article
Abstract: Modeling phenomena from experimental data always begins with a choice of hypothesis on the observed dynamics such as determinism, randomness, and differentiability. Depending on these choices, different behaviors can be observed. The natural question associated to the modeling problem is the following: “With a finite set of data concerning a phenomenon, can we recover its underlying nature? From this problem, we introduce in this paper the definition of multi-scale functions, scale calculus, and scale dynamics based on the time scale calculus [see Bohner, M. and Peterson, A., Dynamic Equations on Time Scales: An Introduction with Applications (Springer Science & Business Media, 2001)] which is used to introduce the notion of scale equations. These definitions will be illustrated on the multi-scale Okamoto’s functions. Scale equations are analysed using scale regimes and the notion of asymptotic model for a scale equation under a particular scale regime. The introduced formalism explains why a single scale equation can produce distinct continuous models even if the equation is scale invariant. Typical examples of such equations are given by the scale Euler-Lagrange equation. We illustrate our results using the scale Newton’s equation which gives rise to a non-linear diffusion equation or a non-linear Schrödinger equation as asymptotic continuous models depending on the particular fractional scale regime which is considered.
Source: Journal of Mathematical Physics
DOI: 10.1063/1.4948745
Authors: Jacky Cresson, Frédéric Pierret
Institutions: Centre National de la Recherche Scientifique, Université de Pau et des Pays de l'Adour, Université Paris Sciences et Lettres, Observatoire de Paris, Sorbonne Université, Laboratoire de Mathématiques et de leurs Applications, Systèmes de Référence Temps-Espace, Laboratoire Temps Espace, Université de Lille, Université Pierre-et-Marie-Curie
Language: English
Subfield: Modeling and Simulation
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2221829072