Multiscale functions, Scale dynamics and Applications to partial differential equations

TL;DR

This paper introduces multiscale functions, scale calculus, and scale dynamics based on time-scale calculus to understand how different continuous models emerge from data at various scales, especially in PDEs.

Contribution

It develops a formalism connecting scale regimes with different PDE models, including deriving classical and non-linear equations from scale Newton's equations.

Findings

Derived non-linear diffusion and Schrödinger equations from scale Newton's equations.

Explained the existence of multiple models for scale-invariant equations.

Revealed how classical equations are recovered under specific scale assumptions.

Abstract

Modeling phenomena from experimental data, always begin with a \emph{choice of hypothesis} on the observed dynamics such as \emph{determinism}, \emph{randomness}, \emph{derivability} etc. Depending on these choices, different behaviors can be observed. The natural question associated to the modeling problem is the following : \emph{"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 \emph{multi-scale functions}, \emph{scale calculus} and \emph{scale dynamics} based on the \emph{time-scale calculus} (see \cite{bohn}). These definitions will be illustrated on the \emph{multi-scale Okamoto's functions}. The introduced formalism explains why there exists different continuous models associated to an equation with different \emph{scale regimes} whereas the equation is \emph{scale invariant}. A typical example of such an equation, is the \emph{Euler-Lagrange equation} and particularly the \emph{Newton's equation} which will be discussed. Notably, we obtain a \emph{non-linear diffusion equation} via the \emph{scale Newton's equation} and also the \emph{non-linear Schr\"odinger equation} via the \emph{scale Newton's equation}. Under special assumptions, we recover the classical \emph{diffusion} equation and the \emph{Schr\"odinger equation}.