Integral Methods in Science and Engineering: Progress in Numerical and Analytic Techniques

Preface
Contents
Contributors
1 Multiphase Flow Splitting in Looped Pipelines
	1.1 Introduction
	1.2 Experimental Program
	1.3 Experimental Results
	1.4 Model Development
	1.5 Results and Discussion
	1.6 Conclusion
	References
2 Green\'s Function Decomposition Method for Transport Equation
	2.1 Introduction
	2.2 Reformulation as an Integral Equation
	2.3 Methodology
		2.3.1 The Isotropic Case
		2.3.2 The Anisotropic Case
		2.3.3 The Calculation of the Coefficient of Wl,k (and Wσ)
	2.4 Numerical Results
	References
3 Integral Neutron Transport and New Computational Methods: A Review
	3.1 Introduction
	3.2 The Integral Transport Equation
	3.3 The AN Model
	3.4 The Boundary Element Approach
	3.5 The Spectral Element Approach
	3.6 Comparison of Numerical Results
	3.7 Conclusions
	References
4 Scale Invariance and Some Limits in Transport Phenomenology: Existence of a Spontaneous Scale
	4.1 Introduction
	4.2 A Geometric Invariant
	4.3 The Hyperspace Hypothesis
	4.4 SO(4,2) Symmetry Breaking
	4.5 Conclusions
	References
5 On Coherent Structures from a Diffusion-Type Model
	5.1 Introduction
	5.2 Motivation from ``Arm-Waving Arguments\'\'
	5.3 A Coherent Constituent–Mediator Model
		5.3.1 The Concept of Coherent States
		5.3.2 Modeling Coherent Fluid Constituents
		5.3.3 Modeling a Coherent Interaction Mediator
	5.4 A Simple Model with Coherence Content
	5.5 Conclusions
	References
6 Numerical Simulation of the Dynamics of Molecular Markers Involved in Cell Polarization
	6.1 Introduction
		6.1.1 One-Dimensional Case
			6.1.1.1 Simplified Model Set on the Half Line
			6.1.1.2 The Model with Dynamical Exchange of Markers at the Boundary
		6.1.2 Two-Dimensional Case: The Model with Dynamical Exchange of Markers at the Boundary
		6.1.3 Heuristics
	6.2 Numerical Analysis
		6.2.1 One-Dimensional Case
		6.2.2 Two-Dimensional Case
			6.2.2.1 Equation for μ
			6.2.2.2 Equation for c
			6.2.2.3 Equation for ρ
		6.2.3 Graphics
	6.3 Conclusion
	References
7 Analytical Study of Computational Radiative Fluxes in a Heterogeneous Medium
	7.1 Introduction
	7.2 Radiative-Conductive Transfer
	7.3 Solution by Decomposition Method
	7.4 Problem Parameter and Numerical Results
	7.5 Conclusions
	References
8 A Novel Approach to the Hankel Transform Inversion of the Neutron Diffusion Problem Using the Parseval Identity
	8.1 Introduction
	8.2 Multi-group Steady State Neutron Diffusion
	8.3 The Hankel-Transformed Problem
		8.3.1 Fast Flux Solution
		8.3.2 The Thermal Flux Solution
	8.4 Multi-regions
	8.5 Error Estimates
	8.6 Conclusions
	References
9 What Is Convergence Acceleration Anyway?
	9.1 Introduction
	9.2 Simulation of Abnormal Protein Growth
		9.2.1 Biophysical Setting
		9.2.2 Numerical Formulation
			9.2.2.1 Convergence Acceleration
			9.2.2.2 Application of Richardsons and Wynn-Epsilon Accelerations
			9.2.2.3 Dimensional Analysis
	9.3 Nuclear Reactor Kinetics
		9.3.1 Reactor Transients
			9.3.1.1 Reactor Kinetics Equations
		9.3.2 Numerical Implementation
			9.3.2.1 Application of Richardsons and Wynn-Epsilon Accelerations
			9.3.2.2 Demonstration
	9.4 Conclusion
	References
10 On the Fractal Pattern Phenomenology of Geological Fracture Signatures from a Scaling Law
	10.1 Introduction
	10.2 Geological Setting of the Studied Areas
	10.3 The Fractal Dimension and Self-similarity Analysis
	10.4 Structural Fracture Analysis
	10.5 Fracture Lineament Map Simulation
	10.6 Conclusion
	References
11 Spectral Boundary Homogenization Problems in Perforated Domains with Robin Boundary Conditions and Large Parameters
	11.1 Introduction and Formulation of the Problem
	11.2 Preliminary Results
	11.3 Convergence Results for α=2 and κ>2
	11.4 Convergence Results for α[1, 2) and κ=2(α-1)
	11.5 Bounds for Other Values of α and κ
	References
12 A Finite Element Formulation of the Total Variation Method for Denoising a Set of Data
	12.1 Introduction
	12.2 Formulation of the Nonlinear Differential Equation
	12.3 Finite Element Method
	12.4 Numerical Results
	12.5 Conclusions
	References
13 On the Convergence of the Multi-group Isotropic Neutron LTSN Nodal Solution in Cartesian Geometry
	13.1 Introduction
	13.2 The Two-Group Discrete Ordinate (SN) Approximation to the Transport Equation in  X,Y Geometry
	13.3 The Multigroup Nodal LTSN Formulation in a Rectangle
	13.4 Error Bounds for the Discrete Ordinates Nodal Method and Two Energy Groups
	13.5 Conclusions
	References
14 Numerical Integration with Singularity by Taylor Series
	14.1 Introduction
	14.2 Taylor Series
		14.2.1 The Arithmetic of Taylor Series
		14.2.2 Basic Functions of Taylor Series
		14.2.3 Numerical Example
	14.3 Integration of Singular Functions
		14.3.1 Integrals with Algebraic and Logarithmic Singularity
		14.3.2 Cauchy Principal Value Integral
		14.3.3 Hadamard Finite–Part Integral
	14.4 Numerical Examples
		14.4.1 Integration with Algebraic and LogarithmicSingularity
		14.4.2 Cauchy Principal Value Integral
		14.4.3 Hadamard Finite Part Integral
	14.5 Conclusion
	References
15 Numerical Solutions of the 1D Convection–Diffusion–Reaction and the Burgers Equation Using Implicit Multi-stage and Finite Element Methods
	15.1 Introduction
	15.2 Statement of the Problems
		15.2.1 1D Convection–Diffusion–Reaction Equation
		15.2.2 Burgers Equation
	15.3 Numerical Methods
		15.3.1 Time Discretization
		15.3.2 Spatial Discretization
		15.3.3 Finite Element Method via Least Squares
		15.3.4 Finite Element Method via Galerkin Procedure
		15.3.5 Finite Element Method via Streamline-Upwind Petrov–Galerkin Procedure
		15.3.6 Linearization of the Convective Term
	15.4 Numerical Results
		15.4.1 1D Convection–Diffusion–Reaction Equation
		15.4.2 The Burgers Equation
	15.5 Conclusions
	References
16 Analytical Reconstruction of Monoenergetic Neutron Angular Flux in Non-multiplying Slabs Using Diffusion Synthetic Approximation
	16.1 Introduction
	16.2 The Spatial and the Angular Reconstruction Schemes of the SND Coarse-Mesh Numerical Solution
		16.2.1 The Spatial Reconstruction Scheme
		16.2.2 The Angular Reconstruction Scheme
	16.3 Numerical Results
	16.4 Conclusions
	References
17 On the Fractional Neutron Point Kinetics Equations
	17.1 Introduction
	17.2 Derivation of the Fractional Neutron Point Kinetics Equations
	17.3 The Solution of the FNPK Equations
	17.4 Numerical Results
		17.4.1 Case A
		17.4.2 Case B
		17.4.3 Case C
	17.5 Concluding Remarks
	References
18 On a Closed Form Solution of the Point Kinetics Equations with a Modified Temperature Feedback
	18.1 Introduction
	18.2 The Kinetic Model with Modified Temperature Feedback
		18.2.1 Expansions of Pj
		18.2.2 Expansion of Aj and Bj in Terms of Adomian Polynomials
		18.2.3 Solution Algorithm
	18.3 Results
	18.4 Conclusions
	References
19 Eulerian Modeling of Radionuclides in Surficial Waters: The Case of Ilha Grande Bay (RJ, Brazil)
	19.1 Introduction
	19.2 Methodology and Modeling Approach
		19.2.1 Hydrodynamical Modeling Approach
		19.2.2 Transport Modeling Approach
	19.3 Input Data and Boundary Conditions for Simulations
		19.3.1 Bathymetry
		19.3.2 Astronomical Tide
		19.3.3 Wind Speed and Direction
		19.3.4 River Discharge
		19.3.5 Hydrodynamic Model Remarks
			19.3.5.1 Future Scenario: Angra 1, 2, and 3 Operating with Discharges in Itaorna and Piraquara
			19.3.5.2 Present Scenario: Angra 1 and 2 Operating with Discharge in Piraquara Cove
	19.4 Transport Model Remarks
	19.5 Conclusions
	References
20 Fractional Calculus: Application in Modeling and Control
	20.1 Introduction
	20.2 Main Mathematical Aspects of the Theoryof Fractional Calculus
	20.3 Approximations to Fractional-Order Derivatives
	20.4 Fractional Modeling
	20.5 Fractional Control
	20.6 Conclusions
	References
21 Modified Integral Equation Method for Stationary PlateOscillations
	21.1 Introduction
	21.2 A Modified Matrix of Fundamental Solutions
	21.3 Uniquely Solvable Integral Equations
	21.4 Modification with a Finite Series
	References
22 Nonstandard Integral Equations for the Harmonic Oscillations of Thin Plates
	22.1 Prerequisites
	22.2 Fundamental Solutions
	22.3 Modified Fundamental Solutions
	22.4 Modified Integral Equations
	22.5 Numerical Example
	References
23 A Genuine Analytical Solution for the SN Multi-group Neutron Equation in Planar Geometry
	23.1 Introduction
	23.2 Time-Dependent Multi-group Transport Equation for Heterogeneous Domain
	23.3 Numerical Results
	23.4 Conclusion
	References
24 Single-Phase Flow Instabilities: Effect of Pressure Waves in a Pump–Pipe–Plenum–Choke System
	24.1 Introduction
	24.2 Single-Phase Flow Instabilities Criteria
		24.2.1 Static Instability
		24.2.2 Dynamic Instability
	24.3 Single-Phase Flow Models
		24.3.1 Incompressible Model
		24.3.2 Compressible Model
	24.4 Application and Discussion
		24.4.1 Example 1: Phase Portrait, Incompressible Model
		24.4.2 Example 2: Phase Portrait, Incompressible Model with Check-Valve
		24.4.3 Example 3: Incompressible Versus Compressible Model
		24.4.4 Example 4: Incompressible Versus Compressible Model
	24.5 Conclusions
	24.6 Nomenclature
	References
25 Two-Phase Flow Instabilities in Oil Wells: ESP Oscillatory Behavior and Casing-Heading
	25.1 Introduction
	25.2 Two-Phase Flow Modeling Overview
	25.3 Application and Discussion
		25.3.1 Example 1. ESP: Tubing and Annular Space Included in the Solution Domain. Stability Example
		25.3.2 Example 2. ESP: Neither Casing nor Annular Space Included in the Solution Domain. Instability Example
		25.3.3 Example 3. ESP: Tubing and Annular Space Included in the Solution Domain. Instability Example
		25.3.4 Example 4: Natural Flowing Well. Casing Heading
	25.4 Conclusions
	25.5 Nomenclature
	References
26 Validating a Closed Form Advection–Diffusion Solution by Experiments: Tritium Dispersion after Emission from the Brazilian Angra Dos Reis Nuclear Power Plant
	26.1 Introduction
	26.2 The Advection–Diffusion Approach
	26.3 A Closed Form Solution
		26.3.1 General Procedure
		26.3.2 A Specific Case for Application
	26.4 Experimental Data and Turbulent Parametrization
	26.5 Numerical Results
	26.6 Conclusions
	References
Index