Lecture

Advanced Analysis II: Homogeneous ODEs and Banach Spaces

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In course

MATH-105(a): Advanced analysis II - vector analysis

Etudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles de plusieurs variables. ...

Description

This lecture covers the resolution of homogeneous ordinary differential equations (ODEs) with separated variables, as well as the existence and uniqueness of solutions in Banach spaces. It also delves into the Banach fixed-point theorem and its applications, emphasizing the completeness of spaces and the contraction mapping principle.

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Instructors (2)

Fabio Nobile

Teaching & PhD PhD Students Veronica Montanaro, Matteo Raviola, Francesca Bettinelli, Eliott Van Dieren, Riccardo Saporiti, Giacomo Mossinelli, Noé Stauffer, Fabio Zoccolan Past EPFL PhD Students Diane Sylvie Guignard, Francesco Tesei, Eleonora Musharbash, Matthieu Claude Martin, Davide Pradovera, Sundar Subramaniam Ganesh, Eva Vidlicková, Juan Pablo Madrigal Cianci, Andrea Zanoni, Thomas Simon Spencer Trigo Trindade Past EPFL PhD Students as codirector Michele Pisaroni Courses Advanced analysis II - ve

Boris Buffoni

Research activityResearch ResearchCurrent Research Fields https://sma.epfl.ch/~buffoni/home.html Teaching & PhD Courses Analysis II MATH-106(b) The course studies fundamental concepts of analysis and the calculus of functions of several variables. Evolutional partial differential equations MATH-421 Fundamental techniques and theories for partial differential equations of evolution. Application to basic examples: first-order equations, wave equation, heat equation. The Cauchy

Official source

Ontological neighbourhood

Mathematics

Analysis: Differential anaysis

Related concepts (30)

Fixed-point theorem

In mathematics, a fixed-point theorem is a result saying that a function F will have at least one fixed point (a point x for which F(x) = x), under some conditions on F that can be stated in general terms. The Banach fixed-point theorem (1922) gives a general criterion guaranteeing that, if it is satisfied, the procedure of iterating a function yields a fixed point.

Brouwer fixed-point theorem

Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function mapping a nonempty compact convex set to itself there is a point such that . The simplest forms of Brouwer's theorem are for continuous functions from a closed interval in the real numbers to itself or from a closed disk to itself. A more general form than the latter is for continuous functions from a nonempty convex compact subset of Euclidean space to itself.

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) function(s) and involves the derivatives of those functions. The term "ordinary" is used in contrast with partial differential equations which may be with respect to one independent variable. A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form where a_0(x), .

Banach fixed-point theorem

In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem or Banach-Caccioppoli theorem) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces, and provides a constructive method to find those fixed points. It can be understood as an abstract formulation of Picard's method of successive approximations. The theorem is named after Stefan Banach (1892–1945) who first stated it in 1922.

Homogeneous differential equation

A differential equation can be homogeneous in either of two respects. A first order differential equation is said to be homogeneous if it may be written where f and g are homogeneous functions of the same degree of x and y. In this case, the change of variable y = ux leads to an equation of the form which is easy to solve by integration of the two members. Otherwise, a differential equation is homogeneous if it is a homogeneous function of the unknown function and its derivatives.

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