Integral Methods in Science and Engineering: Analytic Treatment and Numerical Approximations

Preface
The International Steering Committee of IMSE
Contents
Contributors
1 Singularity Subtraction for Nonlinear Weakly Singular Integral Equations of the Second Kind
	1.1 Introduction
	1.2 Singularity Subtraction
	1.3 Convergence
	1.4 Numerics
	1.5 Conclusions
	References
2 On the Flow of a Viscoplastic Fluid in a Thin Periodic Domain
	2.1 Introduction
	2.2 Statement of the Problem
	2.3 Main Convergence Result
	2.4 Conclusions and Perspectives
	References
3 q-Calculus Formalism for Non-extensive Particle Filter
	3.1 Introduction
	3.2 The Non-extensive Particle Filter
	3.3 Definition of Stable Probability Density Function
	3.4 q-Calculus
		3.4.1 q-Fourier Transform, q-Gaussian Function, and q-Stability
	3.5 Final Remarks
	Appendix: Non-extensive Tsallis' Thermostatistics
	Reference
4 Two-Operator Boundary-Domain Integral Equations for Variable-Coefficient Dirichlet Problem with General Data
	4.1 Preliminaries
	4.2 Parametrix-Based Potential Operators
	4.3 Third Green Identities and Integral Relations
	4.4 The Dirichlet Problem and Two-Operator BDIEs
	4.5 Equivalence and Invertibility of BDIE Systems
	4.6 Conclusion
	Reference
5 Two-Operator Boundary-Domain Integral Equations for Variable Coefficient Dirichlet Problem in 2D
	5.1 Preliminaries
	5.2 Parametrix and Potential Type Operators
	5.3 Invertibility of the Single Layer Potential Operator in 2D
	5.4 Dirichlet Problem and Two-Operator Third Green Identity
	5.5 Two-Operator BDIEs for Dirichlet BVP
	5.6 Equivalence and Invertibility Theorems
	5.7 Conclusion
	Reference
6 Solution of a Homogeneous Version of Love Type Integral Equation in Different Asymptotic Regimes
	6.1 Introduction
	6.2 General Properties
	6.3 Small Interval (β1)
	6.4 Large Interval (β1)
	6.5 Numerical Illustrations
	6.6 Conclusion
	References
7 A Semi-analytical Solution for One-Dimensional OilDisplacement by Miscible Gas in a Homogeneous PorousMedium
	7.1 Introduction
	7.2 Physical and Mathematical Model
	7.3 Example
	7.4 Summary and Conclusions
	References
8 Bending of Elastic Plates: Generalized Fourier Series Method for the Robin Problem
	8.1 Introduction
	8.2 The Boundary Value Problem
	8.3 The Computational Algorithm
	8.4 Numerical Example
	References
9 The Adjoint Spectral Green's Function Method Applied to Direct and Inverse Neutral Particle Source–Detector Problems
	9.1 Introduction
	9.2 The Adjoint SN Transport Problem
	9.3 The Adjoint Spectral Green's Function (SGF) Method
	9.4 Spatial Reconstruction Scheme for the SGF Solution
	9.5 Source–Detector Inverse Problems
	9.6 Numerical Examples
	9.7 Conclusions and Perspectives
	References
10 Relaxation of Periodic and Nonstandard Growth Integralsby Means of Two-Scale Convergence
	10.1 Introduction
	10.2 Preliminaries
		10.2.1 Orlicz-Sobolev Spaces
		10.2.2 Homogenization
	10.3 Proof of Theorem 1
	References
11 A Stiff Problem: Stationary Waves and Approximations
	11.1 Introduction and Statement of the Problem
	11.2 Some Explicit Computations for Standing Waves
		11.2.1 Results for the Dimension N=2
		11.2.2 Results for the Dimension N>2
	11.3 On Approaches to Solutions of the Evolution Problem
	References
12 Modelling Creep in Concrete Under a Variable External Load
	12.1 Introduction
		12.1.1 Objectives
	12.2 Problem Formulation
	12.3 Viscoelastic Model Applied to Creeping Concrete
		12.3.1 Solutions
	12.4 A Polynomial Function for the External Load
		12.4.1 Solutions
	12.5 Conclusions and Ideas for Future Work
	References
13 A Combined Boundary Element and Finite Element Modelof Cell Motion due to Chemotaxis
	13.1 Introduction
	13.2 Mathematical Model
		13.2.1 Finite Element Method for the Chemical Concentrations
		13.2.2 Boundary Integral Method for the Fluid Flow
		13.2.3 Time Integration Method
	13.3 Numerical Results
	13.4 Conclusions and Future Work
	References
14 Numerical Calculation by Quadruple Precision Higher OrderTaylor Series Method of the Pythagorean Problemof Three Bodies
	14.1 Introduction
	14.2 Taylor Series of Ordinary Differential Equations
		14.2.1 Solution of Simple Differential Equations by Taylor Series
		14.2.2 Calculation of Square Root of Taylor Series
	14.3 Quad Precision Calculation
	14.4 Calculation of Three-Body Problem of Pythagoras
	14.5 Conclusion
	References
15 Shape Optimization for Interior Neumann and Transmission Eigenvalues
	15.1 Introduction
		15.1.1 Contribution of the Paper
		15.1.2 Outline of the Paper
	15.2 Shape Optimization for Interior Neumann Eigenvalues
	15.3 Shape Optimization for Interior Transmission Eigenvalues
	15.4 Summary and Outlook
	References
16 On the Integro-Differential Radiative Conductive Transfer Equation: A Modified Decomposition Method
	16.1 Introduction
	16.2 The Integro-Differential Radiative Conductive Transfer Equation
	16.3 Solution by the Modified Decomposition Method
	16.4 Numerical Results and Discussion
		16.4.1 Consistency
		16.4.2 A Convergence Criterion by Stability Analysis
	16.5 Conclusions
	References
17 Periodic Transmission Problems for the Heat Equation
	17.1 Introduction
	17.2 Preliminaries and Notation
	17.3 A Periodic Non-ideal Transmission Problem
	17.4 A Periodic Ideal Transmission Problem
	References
18 On United Boundary-Domain Integro-Differential Equations for Variable Coefficient Dirichlet Problem with General Right-Hand Side
	18.1 Introduction
	18.2 Co-normal Derivatives and the Boundary Value Problem
	18.3 Parametrix and Potential Type Operators
	18.4 The Third Green Identity and Integral Relations
	18.5 United Boundary-Domain Integro-Differential Equations
		18.5.1 United Boundary-Domain Integro-Differential Problem
		18.5.2 United Boundary-Domain Integro-Differential Equation
	18.6 Conclusion
	References
19 Rescaling and Trace Operators in Fractional Sobolev Spaces on Bounded Lipschitz Domains with Periodic Structure
	19.1 Introduction
	19.2 Function Spaces
	19.3 Rescaling Norms on Oscillating Lipschitz Manifold
	19.4 Unfolding in Sobolev–Slobodetskii Spaces in Perforated Domains
	19.5 Rescaling of the Trace Theorem in W2s
	References
20 Design and Performance of a Multiphase Flow Manifold
	20.1 Nomenclature
	20.2 Introduction
	20.3 Experimental Setup
		20.3.1 Flow Loop
		20.3.2 BFM Test Section
		20.3.3 Data Acquisition System
		20.3.4 Test Matrix
		20.3.5 Experimental Results
	20.4 Modeling
		20.4.1 Main Manifold Diameter
		20.4.2 Main Manifold Length
	20.5 Comparison Study
	References
21 On the Polarization Matrix for a Perforated Strip
	21.1 Introduction
	21.2 Some General Properties of the Polarization Matrix
		21.2.1 The Case of a Symmetric Hole
	21.3 The Case of a ``Big'' Rectangular Hole
	21.4 The Case of a ``Small'' Symmetric Hole
	References
22 Operator Perturbation Approach for Fourth Order Elliptic Equations with Variable Coefficients
	22.1 Periodic Boundary Value Problem
		22.1.1 Problem in the Weak and Operator Form
		Potential Polarization Field
		22.1.2 Orthogonal Decomposition of the 4th-Order Differential Operator on Ker and Im
		22.1.3 Periodic Fundamental Solution of the Biharmonic Equation
	22.2 Neumann Series and Its Convergence Estimate by Spectral Properties
	22.3 Bounds on C0(x)
		22.3.1 Voigt-Reuss Bounds for the Effective Coefficients
	References
23 Extension of the Fully Lagrangian Approach for the Integration of the Droplet Number Density on Caustic Formations
	23.1 Introduction
	23.2 The Number Density in a Finite Volume
	23.3 The Calculation of the Hessian in the Second Order FLA for Multiple Dimensions
	23.4 Calculation of the Hessian Magnitude H Across the Caustic Formation
	23.5 Droplets in a Periodic Two-Dimensional Array of Taylor Vortices
	23.6 Conclusion
	References
24 The Nodal LTSN Solution in a Rectangular Domain: A New Method to Determine the Outgoing Angular Flux at theBoundary
	24.1 Introduction
	24.2 The LTSN Transport Equations in 2D
	24.3 Numerical Results for Case 1
	24.4 An Alternative to Determine the Unknown Angular Fluxes at the Boundaries
	24.5 Numerical Results for Case 2
	24.6 Conclusion
	Reference
25 Image Processing for UAV Autonomous Navigation Applying Self-configuring Neural Network
	25.1 Introduction
	25.2 Applied Model
		25.2.1 Platform Used
		25.2.2 Kalman Filter Applied to Autonomous Navigation
	25.3 Neural Network Applied to Autonomous Navigation
		25.3.1 MPCA Metaheuristic for ANN Optimal Architecture
	25.4 Experiment Results
	25.5 Final Remarks
	Reference
26 Towards the Super-Massive Black Hole Seeds
	26.1 Introduction
	26.2 The Forward Problem: Conservation Law to the Ancient Black Holes
	26.3 Mathematical Framework for the Inverse Solution
		26.3.1 Regularization
		26.3.2 Optimization
	26.4 Identifying Black Hole Initial Distribution
	26.5 Final Remarks
	Reference
27 Decomposition of Solutions of the Wave Equationinto Poincaré Wavelets
	27.1 Introduction
	27.2 Statement of the Problem
	27.3 Affine Poincaré CWA
	27.4 Wavelet Analysis for Solutions in Homogeneous Medium
	27.5 Decomposition of Solutions for an Inhomogeneous Medium
	27.6 Conclusions
	References
28 The Method of Fundamental Solutions for Computing Interior Transmission Eigenvalues of Inhomogeneous Media
	28.1 Introduction
	28.2 The ITEP and the Modified MFS
	28.3 Approximation Analysis
	28.4 Numerical Examples
	28.5 Conclusion
	References
29 Tensor Product Approach to Quantum Control
	29.1 Introduction
	29.2 Optimal Quantum Control
		29.2.1 Dynamic Optimisation Problem
		29.2.2 First-Order Optimisation Framework
		29.2.3 GRAPE Algorithm
	29.3 Tensor Train Format and the tAMEn Algorithm
	29.4 Numerical Experiments
	29.5 Conclusions and Future Work
	References
30 Epidemic Genetic Algorithm for Solving Inverse Problems: Parallel Algorithms
	30.1 Introduction
	30.2 Inverse Problem
	30.3 Parallel Genetic Algorithm with Epidemic Operator
		30.3.1 Parallel Strategies for Epidemic-GA
	30.4 Numerical Results
	30.5 Conclusion
	Reference
31 A Chemical Kinetics Extension to the Advection-Diffusion Equation by NOx and SO2
	31.1 Introduction
	31.2 Tropospheric Chemistry
	31.3 The Extended Advection-Diffusion Equation
	31.4 Model Validation and Effects Due to Chemical Reactions
	31.5 Conclusion
	References
32 On the Development of an Alternative Proposition of Cross Wavelet Analysis for Transient Discrimination Problems
	32.1 Introduction
	32.2 Developments
		32.2.1 Classic Definitions
		32.2.2 Alternative Definitions for Cross Wavelet Spectrum and Wavelet Coherence
	32.3 Signal Composition and Transient Analysis
	32.4 Discussion and Conclusions
	References
33 A Simple Non-linear Transfer Function for a Wiener-Hammerstein Model to Simulate Guitar Distortion and Overdrive Effects
	33.1 Introduction
	33.2 The Development of the NLTF
	33.3 Model Validation
	33.4 Results
	33.5 Discussion
	33.6 Conclusions
	References
34 Existence of Nonlinear Problems: An Applicative and Computational Approach
	34.1 Introduction
	34.2 Preliminaries
	34.3 Fixed Point Problem Under Constraint Inequality for (F,ψ)-Rational Type Contraction
	34.4 Some Consequences
		34.4.1 Common Fixed Point Problem Under One Constraint Equality for (F,ψ)-Rational Type Contraction
	34.5 Application to Integral Equation
	References
35 Solving Existence Problems via F-Reich Contraction
	35.1 Introduction and Basic Facts
	35.2 F-Reich Contraction
	35.3 Applications
		35.3.1 Application to Concentration of a Diffusing Substance
		35.3.2 Application to Integral Equation
	References
36 On the Convergence of Dynamic Iterations in Terms ofModel Parameters
	36.1 Introduction
	36.2 Convergence Analysis
	36.3 Numerical Examples
	36.4 Concluding Remarks and Future Work
	Reference
Index