Initial value problem

In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Modeling a system in physics or other sciences frequently amounts to solving an initial value problem. In that context, the differential initial value is an equation which specifies how the system evolves with time given the initial conditions of the problem. Definition An initial value problem is a differential equation :y'(t) = f(t, y(t)) with f\colon \Omega \subset \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n where \Omega is an open set of \mathbb{R} \times \mathbb{R}^n, together with a point in the domain of f :(t_0, y_0) \in \Omega, called the initial condition. A solution to an initial value problem is a function y that is a solution to the differential equation

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Techniques et théories de base pour les équations aux dérivées partielles d'évolution. Etude d'exemples fondamentaux: équations du premier ordre, équation des ondes, équation de la chaleur. Théorème de Cauchy-Kowalevsky, problèmes abstraits d'évolution.

Asymmetric dark matter and baryogenesis from pseudoscalar inflation

Yann David Cado, Eray Sabancilar

We show that both the baryon asymmetry of the Universe and the dark matter abundance can be explained within a single framework that makes use of maximally helical hypermagnetic fields produced during pseudoscalar inflation and the chiral anomaly in the Standard Model. We consider a minimal asymmetric dark matter model free from anomalies and constraints. We find that the observed baryon and the dark matter abundances are achieved for a wide range of inflationary parameters, and the dark matter mass ranges between 7-15 GeV. The novelty of our mechanism stems from the fact that the same source of CP violation occurring during inflation explains both baryonic and dark matter in the Universe with two inflationary parameters, hence addressing all the initial condition problems in an economical way.

Iop Publishing Ltd2017

Methodology for streams definition and graphical representation in Total Site Analysis

Laurent Gabriel Félix Alexandre Stéphane Bungener, François Maréchal, Elfie Marie Méchaussie, Greta Martha Van Eetvelde

The current regulatory framework in Europe regarding industrial energy consumption puts emphasis on regular energy assessments and increasing energy efficiency. Among the available strategies to identify energy savings opportunities, Total Site Analysis (TSA) can be a powerful method to generate utility savings in industrial sites or clusters, targeting for heat recovery and cogeneration potential. The grey box representation of the energy requirements focuses on process/utility heat exchanges when defining hot and cold streams. This representation is usually most suitable when carrying out a TSA on large industrial systems, as direct heat recovery schemes are rarely viable. Since its initial problem definition and solving in the 1990s, a high number of theoretical developments and practical applications have expanded the TSA knowledge. Still an important body of work on TSA techniques and case studies only addresses general aspects and issues that are encountered, and no in-depth explanation yet is found on how to consider specific types of heat consumers and producers. This paper presents a methodology for data collection and streams definition in TSA. It provides a step-by- step approach for defining the process and utility requirements of the main types of heat flows typically found in large industrial systems, including their graphical representations. Using the proposed definitions, it is possible to systematically build composite curves for the total site profiles.

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Integral Convexity And Parabolic Systems

Bernard Dacorogna

In this work we give optimal, i.e., necessary and sufficient, conditions for integrals of the calculus of variations to guarantee the existence of solutions-both weak and variational solutions-to the associated L-2-gradient flow. The initial values are merely L-2 functions with possibly infinite energy. In this context, the notion of integral convexity, i.e., the convexity of the variational integral and not of the integrand, plays the crucial role; surprisingly, this type of convexity is weaker than the convexity of the integrand. We demonstrate this by means of certain quasi-convex and nonconvex integrands.

SIAM PUBLICATIONS2020

Differential equation

In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the

Ordinary differential equation

In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable. As with other DE, its unknown(s) consists of one (or more) f

Dynamical system

In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space, such as in a parametric curve. Examples include the mathematical mode