## UE Ordinary differential equations

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Degrees incorporating this pedagocial element :

### Description

The aim of the course is to strengthen knowledge of ordinary differential equations (ODEs) by focusing on dynamic issues, but also to provide new analytical tools and to introduce some questions related to equations of partial derivatives.

1. Theorems of local existence for the Cauchy problem: Integral formulation of an EDO, Cauchy-Lipschitz theorem (finite dimension and in any Banach space) and Cauchy-Peano theorem.
2. Gronwall's Lemma: Uniqueness in the Cauchy-Lipschitz theorem. Maximum solutions.
3. Global Existence and Explosion: theorem of ends.
4. Qualitative study in dimension 1: Comparison principle, global existence criteria, examples of finite-time explosion, asymptotic behaviour of solutions.
5. Flow associated with an EDO: Dependence on the initial data; theorem of straightening of the flow.
6. Large-time behaviour of autonomous EDO solutions: Stability and asymptotic stability of equilibrium points; linear wells; Lyapunov theorems.
7. EDO of order 2: Theorems of oscillation and comparison, Sturm-Liouville problem.
8. Hierbertian theory of the Fourier series in one variable. Use of Fourier series for solving the heat equation on a segment.
9. Convolution and Fourier transformation in Rd: Convolution of two functions in L1(Rd). Fourier Transformation on L1(Rd). Fourier transformation of a convolution product. Fourier transformation of the density of a centred Gaussian. An inversion formula for an integrable function having an integrable Fourier transformation. Plancherel Formula. Definition of the Fourier transform of a function of L2(Rd). Convolution of two finite positive measures. Case where one of the two measurements has a density.

### Prerequisites

The course will use the following concepts of the third year Bachelor's program.

• Banach spaces, contraction fixed point theorem, Ascoli theorem
• Hilbert spaces, projection theorem, Hilbert bases
• Theory of Lebesgue integration, dominated convergence theorem.
• Fourier series in one variable: definition, quadratic and point convergence in the continuous piece or C1 by pieces

### Bibliography

Background documentation

• Sylvie Benzoni-Gavage, Calcul différentiel et équations différentielles : cours et exercices corrigés, Dunod, 2014. (EDO)
• Jean-Michel Bony, Cours d'analyse : théorie des distributions et analyse de Fourier, Éditions de l'École Polytechnique, 2001. (Séries et transformation de Fourier)
• Jean-Michel Bony, Méthodes mathématiques pour les sciences physiques, Éditions de l'École Polytechnique, 2001. (Séries et transformation de Fourier)
• Marc Briane et Gilles Pagès, Théorie de l'intégration, Vuibert, 2000. (Intégration, convolution)
• Antoine Chambert-Loir et Stéfane Fermigier, Analyse 1 : exercices de mathématiques pour l'agrégation, Masson, 1997. (Séries de Fourier)
• Jean-Pierre Demailly, Analyse numérique et équations différentielles, EDP Sciences, 2016. (EDO)
• Xavier Gourdon, Analyse, Ellipses, 2008. (EDO, Sturm)
• François Laudenbach, Calcul différentiel et intégral, éditions de l'École Polytechnique, 2011. (EDO)
• Walter Rudin, Analyse réelle et complexe : cours et exercices, Dunod, 2009. (Séries et transformation de Fourier)
• Claude Wagschal, Dérivation et intégration, Hermann, 2009. (Séries et transformation de Fourier)