Systèmes d'équations différentielles linéaires singulièrement perturbées et développements asymptotiques combinés

Abstract : In this thesis we prove a theorem of uniform simplification for second order and singularly perturbed differential equations in a full neighborhood of a degenerate point, called a turning point. This is an analytic version of a formal result due to Hanson and Russell, which generalizes a well known theorem of Sibuya. To solve this problem we use the Gevrey composite asymptotic expansions introduced by Fruchard and Schäfke. In the first part we recall the main definitions and theorems of this recent theory. We establish three general results used in the second part of this thesis to prove the main theorem of analytic reduction. Finally we consider ordinary differential equations of order greater than two, which are singularly perturbed and have a turning point, and we prove a theorem of analytic reduction.
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Charlotte Hulek. Systèmes d'équations différentielles linéaires singulièrement perturbées et développements asymptotiques combinés. Mathématiques générales [math.GM]. Université de Strasbourg, 2014. Français. ⟨NNT : 2014STRAD020⟩. ⟨tel-01021178v2⟩

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