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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.
List of contents
Four Introductory Examples.- Composite Asymptotic Expansions:
General Study.- Composite Asymptotic Expansions: Gevrey Theory.- A Theorem of Ramis-Sibuya Type.- Composite Expansions and Singularly Perturbed Differential Equations.- Applications.- Historical Remarks.- References.- Index.
Summary
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.
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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)
“This book focuses on the theory of composite asymptotic expansions for functions of two variables when functions of one variable and functions of the quotient of these two variables are used at the same time. … The book addresses graduate students and researchers in asymptotic analysis and applications.” (Vladimir Sobolev, zbMATH, Vol. 1269, 2013)
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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)
"This book focuses on the theory of composite asymptotic expansions for functions of two variables when functions of one variable and functions of the quotient of these two variables are used at the same time. ... The book addresses graduate students and researchers in asymptotic analysis and applications." (Vladimir Sobolev, zbMATH, Vol. 1269, 2013)
