Composite Asymptotic Expansions

✍ Scribed by Augustin Fruchard, Reinhard Schäfke (auth.)

• Publisher: Springer-Verlag Berlin Heidelberg
• Year: 2013
• Tongue: English
• Leaves: 168
• Series: Lecture Notes in Mathematics 2066
• Edition: 1
• Category: Library

✦ Synopsis

The purpose of these lecture notes is to develop a theory of asymptotic expansions for functions involving two variables, while at the same time using functions involving one variable and functions of the quotient of these two variables. Such composite asymptotic expansions (CAsEs) are particularly well-suited to describing solutions of singularly perturbed ordinary differential equations near turning points. CAsEs imply inner and outer expansions near turning points. Thus our approach is closely related to the method of matched asymptotic expansions. CAsEs offer two unique advantages, however. First, they provide uniform expansions near a turning point and away from it. Second, a Gevrey version of CAsEs is available and detailed in the lecture notes. Three problems are presented in which CAsEs are useful. The first application concerns canard solutions near a multiple turning point. The second application concerns so-called non-smooth or angular canard solutions. Finally an Ackerberg-O’Malley resonance problem is solved.

✦ Table of Contents

Front Matter....Pages i-x
Four Introductory Examples....Pages 1-15
Composite Asymptotic Expansions: General Study....Pages 17-41
Composite Asymptotic Expansions: Gevrey Theory....Pages 43-61
A Theorem of Ramis–Sibuya Type....Pages 63-80
Composite Expansions and Singularly Perturbed Differential Equations....Pages 81-118
Applications....Pages 119-150
Historical Remarks....Pages 151-153
Back Matter....Pages 155-161

✦ Subjects

Approximations and Expansions; Ordinary Differential Equations; Sequences, Series, Summability