Composite Asymptotic Expansions
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| Author | : Augustin Fruchard |
| Publisher | : Springer |
| Total Pages | : 169 |
| Release | : 2012-12-15 |
| Genre | : Mathematics |
| ISBN | : 3642340350 |
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.
| Author | : Vladimir A. Smirnov |
| Publisher | : Springer |
| Total Pages | : 270 |
| Release | : 2003-07-01 |
| Genre | : Science |
| ISBN | : 3540445749 |
'The sturgeon they sent was second grade fresh,' said the barman. 'Really, what nonsense/' 'Why nonsense?' '"Second grade fresh" that's what I call nonsense/ There's only one degree of freshness the first, and it's the last) (M. A. Bulgakov, The Master and Margarita) The goal of this book is to describe in detail how Feynman integrals can be expanded in suitable parameters, when various momenta or masses are small or large. In a narrow sense, this problem is connected with practical calcula tions. In a situation where a given Feynman integral depends on parameters of very different scales, a natural idea is to replace it by a sufficiently large number of terms of an expansion of it in ratios of small and large scales. It will be explained how this problem of expansion can be systematically solved, by formulating universal prescriptions that express terms of the expansion by using the original Feynman integral with its integrand expanded into a Taylor series in appropriate momenta and masses. It turns out that knowledge of the structure of the asymptotic expansion at the diagrammatic level is a key point in understanding how to perform expansions at the operator level. There are various examples of these ex pansions: the operator product expansion, the large mass expansion, Heavy Quark Effective Theory, Non Relativistic QCD, etc. Each of them serves as a realization of the factorization of contributions of different scales.
| Author | : Shaofan Li |
| Publisher | : CRC Press |
| Total Pages | : 1256 |
| Release | : 2016-04-19 |
| Genre | : Science |
| ISBN | : 9814411248 |
This book presents the latest developments and applications of micromechanics and nanomechanics. It particularly focuses on some recent applications and impact areas of micromechanics and nanomechanics that have not been discussed in traditional micromechanics and nanomechanics books on metamaterials, micromechanics of ferroelectric/piezoelectric,
| Author | : Michael Ruzhansky |
| Publisher | : John Wiley & Sons |
| Total Pages | : 767 |
| Release | : 2018-04-05 |
| Genre | : Mathematics |
| ISBN | : 111941430X |
An authoritative text that presents the current problems, theories, and applications of mathematical analysis research Mathematical Analysis and Applications: Selected Topics offers the theories, methods, and applications of a variety of targeted topics including: operator theory, approximation theory, fixed point theory, stability theory, minimization problems, many-body wave scattering problems, Basel problem, Corona problem, inequalities, generalized normed spaces, variations of functions and sequences, analytic generalizations of the Catalan, Fuss, and Fuss–Catalan Numbers, asymptotically developable functions, convex functions, Gaussian processes, image analysis, and spectral analysis and spectral synthesis. The authors—a noted team of international researchers in the field— highlight the basic developments for each topic presented and explore the most recent advances made in their area of study. The text is presented in such a way that enables the reader to follow subsequent studies in a burgeoning field of research. This important text: Presents a wide-range of important topics having current research importance and interdisciplinary applications such as game theory, image processing, creation of materials with a desired refraction coefficient, etc. Contains chapters written by a group of esteemed researchers in mathematical analysis Includes problems and research questions in order to enhance understanding of the information provided Offers references that help readers advance to further study Written for researchers, graduate students, educators, and practitioners with an interest in mathematical analysis, Mathematical Analysis and Applications: Selected Topics includes the most recent research from a range of mathematical fields.
| Author | : S. Nemat-Nasser |
| Publisher | : Elsevier |
| Total Pages | : 429 |
| Release | : 2017-01-31 |
| Genre | : Technology & Engineering |
| ISBN | : 1483145832 |
Variational Methods in the Mechanics of Solids contains the proceedings of the International Union of Theoretical and Applied Mechanics Symposium on Variational Methods in the Mechanics of Solids, held at Northwestern University in Evanston, Illinois, on September 11-13, 1978. The papers focus on advances in the application of variational methods to a variety of mathematically and technically significant problems in solid mechanics. The discussions are organized around three themes: thermomechanical behavior of composites, elastic and inelastic boundary value problems, and elastic and inelastic dynamic problems. This book is comprised of 58 chapters and opens by addressing some questions of asymptotic expansions connected with composite and with perforated materials. The following chapters explore mathematical and computational methods in plasticity; variational irreversible thermodynamics of open physical-chemical continua; macroscopic behavior of elastic material with periodically spaced rigid inclusions; and application of the Lanczos method to structural vibration. Finite deformation of elastic beams and complementary theorems of solid mechanics are also considered, along with numerical contact elastostatics; periodic solutions in plasticity and viscoplasticity; and the convergence of the mixed finite element method in linear elasticity. This monograph will appeal to practitioners of mathematicians as well as theoretical and applied mechanics.
| Author | : P.A. Lagerstrom |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 263 |
| Release | : 2013-03-09 |
| Genre | : Mathematics |
| ISBN | : 1475719906 |
Content and Aims of this Book Earlier drafts of the manuscript of this book (James A. Boa was then coau thor) contained discussions of many methods and examples of singular perturba tion problems. The ambitious plans of covering a large number of topics were later abandoned in favor of the present goal: a thorough discussion of selected ideas and techniques used in the method of matched asymptotic expansions. Thus many problems and methods are not covered here: the method of av eraging and the related method of multiple scales are mentioned mainly to give reasons why they are not discussed further. Examples which required too sophis ticated and involved calculations, or advanced knowledge of a special field, are not treated; for instance, to the author's regret some very interesting applications to fluid mechanics had to be omitted for this reason. Artificial mathematical examples introduced to show some exotic or unexpected behavior are omitted, except when they are analytically simple and are needed to illustrate mathematical phenomena important for realistic problems. Problems of numerical analysis are not discussed.
| Author | : Mark Kachanov |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 393 |
| Release | : 2013-01-21 |
| Genre | : Science |
| ISBN | : 9400757158 |
The book contains state-of the-art reviews in the area of effective properties of heterogeneous materials - the classical field at interface of materials science and solid mechanics. The primary focus is on thermo-mechanical properties, materials science applications, as well as computational aspects and new opportunities provided by rapidly increasing computer powers. The reviews are at the level that is appropriate for a substantial community of researchers working in this field, both at universities and in the industry, and to graduate students. The book can be used as supplementary reading to graduate level courses.
| Author | : Galina Filipuk |
| Publisher | : Springer |
| Total Pages | : 273 |
| Release | : 2018-09-24 |
| Genre | : Mathematics |
| ISBN | : 3319991485 |
These proceedings provide methods, techniques, different mathematical tools and recent results in the study of formal and analytic solutions to Diff. (differential, partial differential, difference, q-difference, q-difference-differential.... ) Equations. They consist of selected contributions from the conference "Formal and Analytic Solutions of Diff. Equations", held at Alcalá de Henares, Spain during September 4-8, 2017. Their topics include summability and asymptotic study of both ordinary and partial differential equations. The volume is divided into four parts. The first paper is a survey of the elements of nonlinear analysis. It describes the algorithms to obtain asymptotic expansion of solutions of nonlinear algebraic, ordinary differential, partial differential equations, and of systems of such equations. Five works on formal and analytic solutions of PDEs are followed by five papers on the study of solutions of ODEs. The proceedings conclude with five works on related topics, generalizations and applications. All contributions have been peer reviewed by anonymous referees chosen among the experts on the subject. The volume will be of interest to graduate students and researchers in theoretical and applied mathematics, physics and engineering seeking an overview of the recent trends in the theory of formal and analytic solutions of functional (differential, partial differential, difference, q-difference, q-difference-differential) equations in the complex domain.
| Author | : Amer Society Composi |
| Publisher | : CRC Press |
| Total Pages | : 1338 |
| Release | : 1994-09-23 |
| Genre | : Technology & Engineering |
| ISBN | : 9781566762205 |
| Author | : Habib Ammari |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 278 |
| Release | : 2006 |
| Genre | : Mathematics |
| ISBN | : 0821839683 |
Recent developments in inverse problems, multi-scale analysis and effective medium theory reveal that these fields share several fundamental concepts. This book is the proceedings of the research conference, ``Workshop in Seoul: Inverse Problems, Multi-Scale Analysis and Homogenization,'' held at Seoul National University, June 22-24, 2005. It highlights the benefits of sharing ideas among these areas, of merging the expertise of scientists working there, and of directing interest towards challenging issues such as imaging nanoscience and biological imaging. Contributions are written by prominent experts and are of interest to researchers and graduate students interested in partial differential equations and applications.
