Équation différentielle homogène

L'expression équation différentielle homogène a deux significations totalement distinctes et indépendantes. Équation différentielle du premier ordre, homogène de degré n Une équation différentielle du premier ordre mais non nécessairement linéaire est dite homogène de degré n si elle peut s'écrire sous la forme : \frac{dy}{dx}=F(x,y) où F est une fonction homogène de degré n, c'est-à-dire vérifiant : F(tx,ty)=t^nF(x,y),. Autrement dit (en posant h(u)=F(1,u)), c'est une équation qui s'écrit :\frac{dy}{dx}=x^nh\left(\frac y x\right). Le cas n = 0 Le cas le plus étudié est celui où le degré d'homogénéité est 0, à tel point que dans ce cas on ne mentionne même pas le degré. La résolution d'une telle équation se fait par séparation des variables : grâce à la substitution u(x)=\frac{y(x)} x, l'équation homogène :\frac{dy}{dx}=h\left(\frac y x\right). se transforme en une équation à variables séparées

Dynamical System Modeling of Self-Regulated Systems Undergoing Multiple Excitations: First Order Differential Equation Approach

Julien Gilles Edgar Gateau

This article proposes a dynamical system modeling approach for the analysis of longitudinal data of self-regulated homeostatic systems experiencing multiple excitations. It focuses on the evolution of a signal (e.g., heart rate) before, during, and after excitations taking the system out of its equilibrium (e.g., physical effort during cardiac stress testing). Such approach can be applied to a broad range of outcomes such as physiological processes in medicine and psychosocial processes in social sciences, and it allows to extract simple characteristics of the signal studied. The model is based on a first order linear differential equation with constant coefficients defined by three main parameters corresponding to the initial equilibrium value, the dynamic characteristic time, and the reaction to the excitation. Assuming the presence of interindividual variability (random effects) on these three parameters, we propose a two-step procedure to estimate them. We then compare the results of this analysis to several other estimation procedures in a simulation study that clarifies under which conditions parameters are accurately estimated. Finally, applications of this model are illustrated using cardiology data recorded during effort tests.

ROUTLEDGE JOURNALS, TAYLOR & FRANCIS LTD2020

On the stability of blowup solutions for the critical corotational wave-map problem

Joachim Krieger, Shuang Miao

We show that the finite time blow up solutions for the co-rotational Wave Maps problem constructed in [7,15] are stable under suitably small perturbations within the co-rotational class, provided the scaling parameter

λ(t)=t−1−ν

is sufficiently close to

t−1

, i. e. the constant

ν

is sufficiently small and positive. The method of proof is inspired by [3,12], but takes advantage of geometric structures of the Wave Maps problem already used in [1,21] to simplify the analysis. In particular, we heavily exploit that the resonance at zero satisfies a natural first order differential equation.

2020

MATHICSE Technical Report : Numerical approximation of parametrized problems in cardiac electrophysiology by a local reduced basis method

Andrea Manzoni, Stefano Pagani, Alfio Quarteroni

The efficient solution of coupled PDEs/ODEs problems arising in cardiac electrophysiology is of key importance whenever interested to study the electrical behavior of the tissue for several instances of relevant physical and/or geometrical parameters. This poses significant challenges to reduced order modeling (ROM) techniques – such as the reduced basis method – traditionally employed when dealing with the repeated solution of parameter dependent differential equations. Indeed, the nonlinear nature of the problem, the presence of moving fronts in the solution, and the high sensitivity of this latter to parameter variations, make the application of standard ROM techniques very problematic. In this paper we propose a local ROM built through a k-means clustering in the state space of the snapshots for both the solution and the nonlinear term. Several comparisons among alternative local ROMs on a benchmark test case show the effectivity of the proposed approach. Finally, the application to a parametrized problem set on an idealized leftventricle geometry shows the capability of the proposed ROM to face complex problems.

MATHICSE2017

On the stability of blowup solutions for the critical corotational wave-map problem

Joachim Krieger, Shuang Miao

λ(t)=t−1−ν

is sufficiently close to

t−1

, i. e. the constant

ν

2020