Lecture Notes in Mathematics

Green's Kernels and Meso-Scale Approximations in Perforated Domains

Authors: Maz'ya, Vladimir, Movchan, Alexander, Nieves, Michael

  • Systematic step-by-step approach to asymptotic algorithms that enables the reader to develop an insight to compound asymptotic approximations Presents a novel, well-explained method of meso-scale approximations for bodies with non-periodic multiple perforations Contains illustrations and numerical examples for a range of physically realisable configurations

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About this book

There are a wide range of applications in physics and structural mechanics involving domains with singular perturbations of the boundary. Examples include perforated domains and bodies with defects of different types. The accurate direct numerical treatment of such problems remains a challenge. Asymptotic approximations offer an alternative, efficient solution.
Green’s function is considered here as the main object of study rather than a tool for generating solutions of specific boundary value problems. The uniformity of the asymptotic approximations is the principal point of attention. We also show substantial links between Green’s functions and solutions of boundary value problems for meso-scale structures. Such systems involve a large number of small inclusions, so that a small parameter, the relative size of an inclusion, may compete with a large parameter, represented as an overall number of inclusions.
The main focus of the present text is on two topics: (a) asymptotics of Green’s kernels in domains with singularly perturbed boundaries and (b) meso-scale asymptotic approximations of physical fields in non-periodic domains with many inclusions. The novel feature of these asymptotic approximations is their uniformity with respect to the independent variables.
This book addresses the needs of mathematicians, physicists and engineers, as well as research students interested in asymptotic analysis and numerical computations for solutions to partial differential equations.

Table of contents (10 chapters)

  • Uniform Asymptotic Formulae for Green’s Functions for the Laplacian in Domains with Small Perforations

    Maz’ya, Vladimir (et al.)

    Pages 3-19

  • Mixed and Neumann Boundary Conditions for Domains with Small Holes and Inclusions: Uniform Asymptotics of Green’s Kernels

    Maz’ya, Vladimir (et al.)

    Pages 21-57

  • Green’s Function for the Dirichlet Boundary Value Problem in a Domain with Several Inclusions

    Maz’ya, Vladimir (et al.)

    Pages 59-73

  • Numerical Simulations Based on the Asymptotic Approximations

    Maz’ya, Vladimir (et al.)

    Pages 75-81

  • Other Examples of Asymptotic Approximations of Green’s Functions in Singularly Perturbed Domains

    Maz’ya, Vladimir (et al.)

    Pages 83-94

Buy this book

eBook $44.99
price for USA (gross)
  • ISBN 978-3-319-00357-3
  • Digitally watermarked, DRM-free
  • Included format: EPUB, PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $59.99
price for USA
  • ISBN 978-3-319-00356-6
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Rent the ebook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
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Bibliographic Information

Bibliographic Information
Book Title
Green's Kernels and Meso-Scale Approximations in Perforated Domains
Authors
Series Title
Lecture Notes in Mathematics
Series Volume
2077
Copyright
2013
Publisher
Springer International Publishing
Copyright Holder
Springer International Publishing Switzerland
eBook ISBN
978-3-319-00357-3
DOI
10.1007/978-3-319-00357-3
Softcover ISBN
978-3-319-00356-6
Series ISSN
0075-8434
Edition Number
1
Number of Pages
XVII, 258
Number of Illustrations and Tables
7 b/w illustrations, 10 illustrations in colour
Topics