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

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.

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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.

---

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.
Content:
Front Matter....Pages i-xvii
Front Matter....Pages 1-1
Uniform Asymptotic Formulae for Green’s Functions for the Laplacian in Domains with Small Perforations....Pages 3-19
Mixed and Neumann Boundary Conditions for Domains with Small Holes and Inclusions: Uniform Asymptotics of Green’s Kernels....Pages 21-57
Green’s Function for the Dirichlet Boundary Value Problem in a Domain with Several Inclusions....Pages 59-73
Numerical Simulations Based on the Asymptotic Approximations....Pages 75-81
Other Examples of Asymptotic Approximations of Green’s Functions in Singularly Perturbed Domains....Pages 83-94
Front Matter....Pages 95-95
Green’s Tensor for the Dirichlet Boundary Value Problem in a Domain with a Single Inclusion....Pages 97-137
Green’s Tensor in Bodies with Multiple Rigid Inclusions....Pages 139-167
Green’s Tensor for the Mixed Boundary Value Problem in a Domain with a Small Hole....Pages 169-188
Front Matter....Pages 189-189
Meso-scale Approximations for Solutions of Dirichlet Problems....Pages 191-219
Mixed Boundary Value Problems in Multiply-Perforated Domains....Pages 221-247
Back Matter....Pages 249-260

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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.
Content:
Front Matter....Pages i-xvii
Front Matter....Pages 1-1
Uniform Asymptotic Formulae for Green’s Functions for the Laplacian in Domains with Small Perforations....Pages 3-19
Mixed and Neumann Boundary Conditions for Domains with Small Holes and Inclusions: Uniform Asymptotics of Green’s Kernels....Pages 21-57
Green’s Function for the Dirichlet Boundary Value Problem in a Domain with Several Inclusions....Pages 59-73
Numerical Simulations Based on the Asymptotic Approximations....Pages 75-81
Other Examples of Asymptotic Approximations of Green’s Functions in Singularly Perturbed Domains....Pages 83-94
Front Matter....Pages 95-95
Green’s Tensor for the Dirichlet Boundary Value Problem in a Domain with a Single Inclusion....Pages 97-137
Green’s Tensor in Bodies with Multiple Rigid Inclusions....Pages 139-167
Green’s Tensor for the Mixed Boundary Value Problem in a Domain with a Small Hole....Pages 169-188
Front Matter....Pages 189-189
Meso-scale Approximations for Solutions of Dirichlet Problems....Pages 191-219
Mixed Boundary Value Problems in Multiply-Perforated Domains....Pages 221-247
Back Matter....Pages 249-260
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