Integral methods in science and engineering. Volume 1, Theoretical techniques

Preface
	Digital Art by Walid Ben Medjedel
Contents
List of Contributors
1 An L1-Product-Integration Method in Astrophysics
	1.1 Introduction
	1.2 A Product-Integration Method in C
	1.3 Iterative Refinement
	1.4 Numerical Evidence
	References
2 Differential Operators and Approximation Processes Generated by Markov Operators
	2.1 Introduction
	2.2 Canonical Elliptic Second-Order Differential Operators and Bernstein-Schnabl Operators
	2.3 Other Classes of Differential Operators and Approximation Processes
	2.4 Final Remarks
	References
3 Analysis of Boundary-Domain Integral Equations for Variable-Coefficient Neumann BVP in 2D
	3.1 Preliminaries
	3.2 Parametrix-Based Potential Operators
	3.3 BDIEs for Neumann BVP
	3.4 Equivalence and Invertibility Theorems
	3.5 Perturbed BDIE Systems for the Neumann Problem
	3.6 Conclusion
	References
4 A Measure of the Torsional Performances of Partially Hinged Rectangular Plates
	4.1 Introduction
	4.2 Variational Setting and Gap Function Definition
	4.3 Proof of Theorem 1
	4.4 Proof of Theorem 2
	4.5 Proofs of Theorems 3 and 4
	References
5 On a Class of Integral Equations Involving Kernels of Cosine and Sine Type
	5.1 Introduction
	5.2 Integral Equations Generated by an Integral Operator with Cosine and Sine Kernels
	5.3 Operator Properties
		5.3.1 Invertibility and Spectrum
		5.3.2 Parseval-Type Identity and Unitary Properties
		5.3.3 Involution
		5.3.4 New Convolution
	References
6 The Simple-Layer Potential Approach to the Dirichlet Problem: An Extension to Higher Dimensions of Muskhelishvili Method and Applications
	6.1 Introduction
	6.2 Muskhelishvili\'s Method and Its Extension to Rn
		6.2.1 Muskhelishvili\'s Method
		6.2.2 Conjugate Differential Forms
		6.2.3 The Extension to Higher Dimensions of Muskhelishvili Method
	6.3 The Multiple-Layer Approach
	References
7 Bending of Elastic Plates: Generalized Fourier Series Method
	7.1 Introduction
	7.2 The Boundary Value Problem
	7.3 Numerical Example
	7.4 Graphical Illustrations
	7.5 The Classical Gram–Schmidt Procedure (CGS)
	7.6 The Modified Gram–Schmidt Procedure (MGS)
	7.7 The Householder Reflection Procedure (HR)
	References
8 Existence and Uniqueness Results for a Class of Singular Elliptic Problems in Two-Component Domains
	8.1 Introduction
	8.2 Setting of the Problem
	8.3 A Priori Estimates
	8.4 Main Results
	References
9 Fredholmness of Nonlocal Singular Integral Operators with Slowly Oscillating Data
	9.1 Introduction
	9.2 Invertibility Criteria for Wiener Type Functional Operators
	9.3 Mellin Pseudodifferential Operators
	9.4 Fredholmness of the Operator N
	References
10 Multidimensional Time Fractional Diffusion Equation
	10.1 Introduction
	10.2 Preliminaries
	10.3 Fundamental Solution of the Multidimensional Time Fractional Diffusion-Wave Equation
	10.4 Fractional Moments
	10.5 Graphical Representations of the Fundamental Solution
		10.5.1 Case n=1
		10.5.2 Case n=2
	10.6 Application to Diffusion in Thermoelasticity
	References
11 On Homogenization of Nonlinear Robin Type Boundary Conditions for the n-Laplacian in n-DimensionalPerforated Domains
	11.1 Introduction
	11.2 Setting of the Problem and Homogenized Problems
		11.2.1 Table of Homogenized Problems
	11.3 Preliminary Results
	11.4 Critical Relation for the Adsorption
	11.5 Critical Size for Perforations
	11.6 Extreme Cases
	References
12 Interior Transmission Eigenvalues for Anisotropic Media
	12.1 Introduction
	12.2 Problem Formulation
	12.3 Boundary Integral Equations
	12.4 Numerical Results
	12.5 Summary and Outlook
	References
13 Improvement of the Inside-Outside Duality Method
	13.1 Introduction and Motivation
	13.2 Problem Formulation
	13.3 The Inside-Outside Duality Method
	13.4 Improvement and Numerical Results
		Product Gaussian Quadrature
		13.4.1 Lebedev Quadrature
		13.4.2 Spherical t-Design
		13.4.3 Numerical Results
	13.5 Summary and Outlook
	References
14 A Note on Optimal Design for Thin Structures in the Orlicz–Sobolev Setting
	14.1 Introduction
	14.2 Preliminaries
	14.3 Proof of Theorem 1
	References
15 On the Radiative Conductive Transfer Equation: A Heuristic Convergence Criterion by Stability Analysis
	15.1 Introduction
	15.2 The Radiative Conductive Transfer Equation in Cylinder Geometry
	15.3 Solution by the Decomposition Method
	15.4 A Heuristic Convergence Criterion by Stability Analysis
	15.5 Numerical Results
	15.6 Conclusions
	References
16 An Indirect Boundary Integral Equation Method for Boundary Value Problems in Elastostatics
	16.1 Preliminaries
	16.2 Indirect Method
	References
17 An Instability Result for Suspension Bridges
	17.1 Introduction
	17.2 The Isoenergetic Poincaré Map and the Asymptotic Behavior of Its Linearization  Lk
	17.3 Some Related Problems and Questions
	17.4 Numerical Examples
	References
18 A New Diffeomorph Conformal Methodology to Solve Flow Problems with Complex Boundaries by an Equivalent Plane Parallel Problem
	18.1 Introduction
	18.2 Coordinate Transformations
	18.3 Transformation by Invariance
	18.4 Application to a Navier-Stokes Problem
	18.5 Application to an Advection-Diffusion Problem
	18.6 A Numerical Example
	18.7 Conclusion
	References
19 A New Family of Boundary-Domain Integral Equations for the Mixed Exterior Stationary Heat Transfer Problem with Variable Coefficient
	19.1 Introduction
	19.2 Preliminaries
	19.3 Boundary Value Problem
	19.4 Parametrices and Remainders
	19.5 Surface and Volume Potentials
	19.6 Third Green Identities and Integral Relations
	19.7 Boundary-Domain Integral Equation System
	19.8 Invertibility
	References
20 Radiation Conditions and Integral Representations for Clifford Algebra-Valued Null-Solutions of the Iterated Helmholtz Operator
	20.1 Introduction
	20.2 Fundamental Solutions
	20.3 Integral Representations
	References
21 A Wiener-Hopf System of Equations in the Steady-State Propagation of a Rectilinear Crack in an Infinite Elastic Plate
	21.1 Introduction
	21.2 Boundary Conditions
	21.3 Wiener-Hopf Factorization
	References
22 Mono-Energetic Neutron Space-Kinetics in Full Cylinder Symmetry: Simulating Power Decrease
	22.1 Introduction
	22.2 The Model
	22.3 Variable Separation
	22.4 A Closed Form Solution in Cylinder Geometry
	22.5 Numerical Results
	22.6 Conclusion
	References
23 Asymptotic Solutions of Maxwell\'s Equations in a Layered Periodic Structure
	23.1 Introduction
	23.2 Maxwell\'s Equations and the Floquet–Bloch Solutions
	23.3 Two-Scale Solutions of the Maxwell Equations
	23.4 Boundary Value Problem in a Half-Space
	23.5 Conclusions
	References
24 Some Properties of the Fractional Circle Zernike Polynomials
	24.1 Introduction
	24.2 Preliminaries
	24.3 Fractional Circle Zernike Polynomials
	24.4 Graphical Representation of Fractional Circle Zernike Polynomials
	References
25 Double Laplace Transform and Explicit Fractional Analogue of 2D Laplacian
	25.1 Introduction
	25.2 Double Laplace Transforms. Definition and Main Properties
	25.3 Application of the Double Laplace Transform to Explicit Fractional Laplacian
		25.3.1 Power Type Fractional Laplacian α,β
		25.3.2 Skewed Fractional Laplacian α,β; Q-+in the Quarter-Plane
	25.4 Discussion and Outlook
	References
26 Stability of the Laplace Single Layer Boundary Integral Operator in Sobolev Spaces
	26.1 Introduction
	26.2 Strong Domain Variational Formulation
	26.3 Ultra-Weak Domain Variational Formulation
	26.4 Single Layer Potential
	References
27 Spectral Lanczos\' Tau Method for Systems of Nonlinear Integro-Differential Equations
	27.1 Introduction
	27.2 Preliminaries
	27.3 The Tau Method for Integro-Differential Problems
	27.4 Nonlinear Approach for Integro-Differential Problems
	27.5 Contributions to Stability
	27.6 Numerical Results
	27.7 Conclusions
	References
28 Discreteness, Periodicity, Holomorphy, and Factorization
	28.1 Introduction
	28.2 Discreteness
	28.3 Periodicity
	28.4 Holomorphy
	28.5 Factorization
		28.5.1 Conical Case
	Conclusion
	References
29 Modes Coupling Seismic Waves and VibratingBuildings: Existence
	29.1 Introduction
	29.2 Low Frequency Analysis
	29.3 High Frequency Asymptotic Analysis
	References
Index