Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Existence Unicité

In courses (2)

MATH-250: Advanced numerical analysis I

Construction and analysis of numerical methods for the solution of problems from linear algebra, integration, approximation, and differentiation. ...

MATH-251(c): Numerical analysis

Le cours présente des méthodes numériques pour la résolution de problèmes mathématiques comme des systèmes d'équations linéaires ou non linéaires, l'approximation de fonctions, l'intégration et la dérivation, et les équations différentielles. ...

Description

This lecture discusses the Lipschitz condition for functions, showing that if a function satisfies this property, then the Cauchy problem has a unique solution belonging to C1. By examining examples, the instructor demonstrates the application of this concept to ensure a unique and global solution for all times.

Instructors (3)

Simone Deparis

Daniel Kressner

Annalisa Buffa

Official source

Related concepts (21)

Lipschitz continuity

In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exists a real number such that, for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is not greater than this real number; the smallest such bound is called the Lipschitz constant of the function (and is related to the modulus of uniform continuity).

Trigonometric functions

In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis.

Jacobi elliptic functions

In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions. They are found in the description of the motion of a pendulum (see also pendulum (mathematics)), as well as in the design of electronic elliptic filters. While trigonometric functions are defined with reference to a circle, the Jacobi elliptic functions are a generalization which refer to other conic sections, the ellipse in particular. The relation to trigonometric functions is contained in the notation, for example, by the matching notation for .

Even and odd functions

In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power series and Fourier series. They are named for the parity of the powers of the power functions which satisfy each condition: the function is an even function if n is an even integer, and it is an odd function if n is an odd integer.

Hölder condition

In mathematics, a real or complex-valued function f on d-dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are real constants C ≥ 0, α > 0, such that for all x and y in the domain of f. More generally, the condition can be formulated for functions between any two metric spaces. The number α is called the exponent of the Hölder condition. A function on an interval satisfying the condition with α > 1 is constant. If α = 1, then the function satisfies a Lipschitz condition.

Related lectures (42)

Functional Derivatives

Covers the concept of functional derivatives and their calculation process with examples.

Bernoulli's Hospital Rule MOOC: Analysis I MOOC: Analysis I (part 6) : Study of functions, limited developments

Covers the statement of the Bernoulli's Hospital Rule and its application.

Derivability and Continuity

Explores derivability and continuity of functions, showcasing examples of functions with differentiability properties.

Analysis I Exam Solutions

Provides solutions to an Analysis I exam, covering various topics.

Derivatives and Tangent Lines: Understanding Functions and Limits