Composite Asymptotic Expansions

Beschreibung

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.
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.
Presents a comprehensive theory of infinite composite asymptotic expansions (CAsEs), an alternative to the method of matched asymptotic expansions Generalizes the classical theory of Gevrey asymptotic expansions to such CAsEs, thus establishing a new powerful tool for the study of turning points of singularly perturbed ODEs Using CAsEs, especially their versions of Gevrey type, to obtain new results for three classical problems in the theory of singularly perturbed ODEs Includes supplementary material: sn.pub/extras

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Augustin Fruchard

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Gevrey expansions composite asymptotic expansion ordinary differential equation singular perturbation turning point ordinary differential equations

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From the reviews:

“This memoir develops the theory of Composite Asymptotic Expansions … . The book is very technical, but written in a clear and precise style. The notions are well motivated, and many examples are given. … this book will be of great interest to people studying asymptotics for singularly perturbed differential equations.” (Jorge Mozo Fernández, Mathematical Reviews, December, 2013)

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