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2013 | Buch

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Composite Asymptotic Expansions

verfasst von: Augustin Fruchard, Reinhard Schäfke

Verlag: Springer Berlin Heidelberg

Buchreihe : Lecture Notes in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

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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Inhaltsverzeichnis

Frontmatter

Chapter 1. Four Introductory Examples

Abstract

Here we present simple examples, showing that solutions of singularly perturbed differential equations naturally have composite asymptotic expansions (ca se s) near turning points. A theory of ca se s might thus help to understand them.All examples are linear equations of first order. The first example is among the simplest ones having a turning point. The second one contains a control parameter for “duck hunting” or “canard hunting”. The third example also contains a control parameter, but the turning point is no longer simple; this implies that the canard solutions are no longer overstable solutions in the sense of Guy Wallet. Finally the fourth example relates to “fake ducks” or “fake canard solutions”: the slow curve is first repelling and then attracting. In this situation, any solution with bounded initial condition at the turning point is defined and bounded on an interval containing this turning point, but this solution can have a ca se only if the initial condition has an asymptotic expansion. We will see that this necessary condition is also sufficient.

Augustin Fruchard, Reinhard Schäfke

Chapter 2. Composite Asymptotic Expansions: General Study

Abstract

In this chapter, we present the general theory of ca se s: their definition and their behavior with respect to the basic operations of addition, multiplication, division, differentiation, integration, composition and analytic continuation. We also link our ca se s to the inner and outer expansions of the classical method of matching. Using these inner and outer expansions is also a good method for determining the coefficients of a composite expansion in practice, provided one can show the existence of a composite expansion independently.

Augustin Fruchard, Reinhard Schäfke

Chapter 3. Composite Asymptotic Expansions: Gevrey Theory

Abstract

In the following chapters (Chaps. 5 and 6) we will apply ca se s to singularly perturbed differential equations. We will see at this occasion that the notions of composite formal series and composite expansions of Gevrey kind play a key-role.The notion of Gevrey asymptotics has played a key-role in the classical theory of asymptotic expansions uniform in a full neighborhood of a turning point. In the present chapter, we generalize this concept to composite expansions and thus to expansions valid on quasi-sectors with vertex at a turning point.

Augustin Fruchard, Reinhard Schäfke

Chapter 4. A Theorem of Ramis–Sibuya Type

Abstract

In this chapter, we present a key-result of our work: a family of functions of two variables defined and analytic on modified polysectors forming a consistent good covering and such that their differences on the intersections of their domains are exponentially small have ca se s of Gevrey order \(\frac{1} {p}\) and the associated formal series \(\sum \limits _{n}{a}_{n}(x){\eta }^{n}\) and \(\sum \limits _{n}\widehat{{g}}_{n}(X){\eta }^{n}\) are the same.Differently from the classical theory, the differences on the intersections of the x-domains are \(\mathcal{O}(\exp \left (-B\left \vert {x}^{p}/{\eta }^{p}\right \vert \right ))\) whereas the differences on the intersections of the η-domains are \(\mathcal{O}(\exp \left (-A\left \vert {\eta }^{p}\right \vert \right ))\), A, B > 0. In the neighborhood of the origin, we therefore do necessarily not have a classical Gevrey expansion.

Augustin Fruchard, Reinhard Schäfke

Chapter 5. Composite Expansions and Singularly Perturbed Differential Equations

Abstract

We assume Φ analytic with respect to x and y in a domain \(\mathcal{D}\subset {\mathbb{C}}^{2}\) and of Gevrey order 1 with respect to \(\epsilon \) in S; this also allows to treat equations containing a control parameter. This is useful for equations where canards solutions might occur for certain values of the parameter.

Augustin Fruchard, Reinhard Schäfke

Chapter 6. Applications

Abstract

We present three problems in which cases are useful. Our first application concerns canard solutions near a multiple turning point for an equation analytic, not only with respect to the variable x, but also with respect to the small parameter \(\epsilon \). On the one hand the problem is to find a necessary and sufficient condition, that can be tested on the coefficients of the equation, for the existence of a solution close to the slow curve on some open interval containing the turning point. Such a solution will be called a local canard. On the other hand we show that, if there is such a local canard, then there is also a global canard. In other words, there is no phenomenon of buffer points for this kind of equation.

Augustin Fruchard, Reinhard Schäfke

Chapter 7. Historical Remarks

Abstract

The literature on matching, i.e. the method of matched asymptotic expansions is abundant. Already in the fifties and sixties this method was common, see e.g. the work of Kaplun/Lagerstrom [40], Erdélyi [18]. The matching is the main subject of Eckhaus’ book [17] and the Chaps. VII and VIII of Wasow’s book [62]. In the latter book, the method is presented for linear systems of singularly perturbed differential equations.

Augustin Fruchard, Reinhard Schäfke

Backmatter

Titel: Composite Asymptotic Expansions
verfasst von: Augustin Fruchard
Reinhard Schäfke
Copyright-Jahr: 2013
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-642-34035-2
Print ISBN: 978-3-642-34034-5
DOI: https://doi.org/10.1007/978-3-642-34035-2

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