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

✍ Scribed by Christian Constanda (editor), Paul Harris (editor)

• Publisher: Birkhäuser
• Year: 2019
• Tongue: English
• Leaves: 476
• Edition: 1st ed. 2019
• Category: Library

✦ Synopsis

This contributed volume contains a collection of articles on state-of-the-art developments on the construction of theoretical integral techniques and their application to specific problems in science and engineering. The chapters in this book are based on talks given at the Fifteenth International Conference on Integral Methods in Science and Engineering, held July 16-20, 2018 at the University of Brighton, UK, and are written by internationally recognized researchers. The topics addressed are wide ranging, and include:

• Asymptotic analysis
  
• Boundary-domain integral equations
  
• Viscoplastic fluid flow
  
• Stationary waves
  
• Interior Neumann shape optimization
  
• Self-configuring neural networks
  

This collection will be of interest to researchers in applied mathematics, physics, and mechanical and electrical engineering, as well as graduate students in these disciplines and other professionals for whom integration is an essential tool.

✦ Table of Contents

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 aSmall'' 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

✦ Subjects

Mathematics;Calculus; Differential equations