Integral Methods in Science and Engineering, Volume 1 : Theoretical Techniques

This contributed volume contains a collection of articles on the most recent advances in integral methods. The first of two volumes, this work focuses on the construction of theoretical integral methods. Written by internationally recognized researchers, the chapters in this book are based on talks given at the Fourteenth International Conference on Integral Methods in Science and Engineering, held July 25-29, 2016, in Padova, Italy. A broad range of topics is addressed, such as: • Integral equations • Homogenization • Duality methods • Optimal design • Conformal techniques 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 to other professionals who use integration as an essential tool in their work. Preface 5 Digital Art by Walid Ben Medjedel 8 Contents 10 List of Contributors 18 1 An L1-Product-Integration Method in Astrophysics 23 1.1 Introduction 23 1.2 A Product-Integration Method in C 24 1.3 Iterative Refinement 26 1.4 Numerical Evidence 27 References 29 2 Differential Operators and Approximation Processes Generated by Markov Operators 30 2.1 Introduction 30 2.2 Canonical Elliptic Second-Order Differential Operators and Bernstein-Schnabl Operators 31 2.3 Other Classes of Differential Operators and Approximation Processes 37 2.4 Final Remarks 38 References 39 3 Analysis of Boundary-Domain Integral Equations for Variable-Coefficient Neumann BVP in 2D 41 3.1 Preliminaries 41 3.2 Parametrix-Based Potential Operators 43 3.3 BDIEs for Neumann BVP 46 3.4 Equivalence and Invertibility Theorems 47 3.5 Perturbed BDIE Systems for the Neumann Problem 50 3.6 Conclusion 52 References 53 4 A Measure of the Torsional Performances of Partially Hinged Rectangular Plates 54 4.1 Introduction 54 4.2 Variational Setting and Gap Function Definition 55 4.3 Proof of Theorem 1 59 4.4 Proof of Theorem 2 60 4.5 Proofs of Theorems 3 and 4 62 References 64 5 On a Class of Integral Equations Involving Kernels of Cosine and Sine Type 66 5.1 Introduction 66 5.2 Integral Equations Generated by an Integral Operator with Cosine and Sine Kernels 67 5.3 Operator Properties 72 5.3.1 Invertibility and Spectrum 72 5.3.2 Parseval-Type Identity and Unitary Properties 73 5.3.3 Involution 74 5.3.4 New Convolution 74 References 75 6 The Simple-Layer Potential Approach to the Dirichlet Problem: An Extension to Higher Dimensions of Muskhelishvili Method and Applications 77 6.1 Introduction 77 6.2 Muskhelishvili's Method and Its Extension to Rn 79 6.2.1 Muskhelishvili's Method 79 6.2.2 Conjugate Differential Forms 79 6.2.3 The Extension to Higher Dimensions of Muskhelishvili Method 81 6.3 The Multiple-Layer Approach 83 References 86 7 Bending of Elastic Plates: Generalized Fourier Series Method 88 7.1 Introduction 88 7.2 The Boundary Value Problem 89 7.3 Numerical Example 90 7.4 Graphical Illustrations 91 7.5 The Classical Gram–Schmidt Procedure (CGS) 92 7.6 The Modified Gram–Schmidt Procedure (MGS) 94 7.7 The Householder Reflection Procedure (HR) 95 References 98 8 Existence and Uniqueness Results for a Class of Singular Elliptic Problems in Two-Component Domains 99 8.1 Introduction 99 8.2 Setting of the Problem 100 8.3 A Priori Estimates 103 8.4 Main Results 106 References 109 9 Fredholmness of Nonlocal Singular Integral Operators with Slowly Oscillating Data 110 9.1 Introduction 110 9.2 Invertibility Criteria for Wiener Type Functional Operators 113 9.3 Mellin Pseudodifferential Operators 115 9.4 Fredholmness of the Operator N 116 References 119 10 Multidimensional Time Fractional Diffusion Equation 121 10.1 Introduction 121 10.2 Preliminaries 122 10.3 Fundamental Solution of the Multidimensional Time Fractional Diffusion-Wave Equation 124 10.4 Fractional Moments 125 10.5 Graphical Representations of the Fundamental Solution 126 10.5.1 Case n=1 126 10.5.2 Case n=2 127 10.6 Application to Diffusion in Thermoelasticity 128 References 130 11 On Homogenization of Nonlinear Robin Type Boundary Conditions for the n-Laplacian in n-DimensionalPerforated Domains 132 11.1 Introduction 132 11.2 Setting of the Problem and Homogenized Problems 135 11.2.1 Table of Homogenized Problems 138 11.3 Preliminary Results 139 11.4 Critical Relation for the Adsorption 141 11.5 Critical Size for Perforations 144 11.6 Extreme Cases 147 References 150 12 Interior Transmission Eigenvalues for Anisotropic Media 152 12.1 Introduction 152 12.2 Problem Formulation 153 12.3 Boundary Integral Equations 154 12.4 Numerical Results 156 12.5 Summary and Outlook 159 References 159 13 Improvement of the Inside-Outside Duality Method 161 13.1 Introduction and Motivation 161 13.2 Problem Formulation 162 13.3 The Inside-Outside Duality Method 163 13.4 Improvement and Numerical Results 164 Product Gaussian Quadrature 164 13.4.1 Lebedev Quadrature 165 13.4.2 Spherical t-Design 166 13.4.3 Numerical Results 167 13.5 Summary and Outlook 169 References 170 14 A Note on Optimal Design for Thin Structures in the Orlicz–Sobolev Setting 172 14.1 Introduction 172 14.2 Preliminaries 175 14.3 Proof of Theorem 1 178 References 181 15 On the Radiative Conductive Transfer Equation: A Heuristic Convergence Criterion by Stability Analysis 183 15.1 Introduction 183 15.2 The Radiative Conductive Transfer Equation in Cylinder Geometry 184 15.3 Solution by the Decomposition Method 185 15.4 A Heuristic Convergence Criterion by Stability Analysis 189 15.5 Numerical Results 190 15.6 Conclusions 191 References 192 16 An Indirect Boundary Integral Equation Method for Boundary Value Problems in Elastostatics 193 16.1 Preliminaries 193 16.2 Indirect Method 195 References 200 17 An Instability Result for Suspension Bridges 202 17.1 Introduction 202 17.2 The Isoenergetic Poincaré Map and the Asymptotic Behavior of Its Linearization Lk 205 17.3 Some Related Problems and Questions 207 17.4 Numerical Examples 208 References 211 18 A New Diffeomorph Conformal Methodology to Solve Flow Problems with Complex Boundaries by an Equivalent Plane Parallel Problem 213 18.1 Introduction 213 18.2 Coordinate Transformations 213 18.3 Transformation by Invariance 215 18.4 Application to a Navier-Stokes Problem 216 18.5 Application to an Advection-Diffusion Problem 217 18.6 A Numerical Example 217 18.7 Conclusion 221 References 222 19 A New Family of Boundary-Domain Integral Equations for the Mixed Exterior Stationary Heat Transfer Problem with Variable Coefficient 223 19.1 Introduction 223 19.2 Preliminaries 224 19.3 Boundary Value Problem 226 19.4 Parametrices and Remainders 226 19.5 Surface and Volume Potentials 227 19.6 Third Green Identities and Integral Relations 229 19.7 Boundary-Domain Integral Equation System 230 19.8 Invertibility 232 References 233 20 Radiation Conditions and Integral Representations for Clifford Algebra-Valued Null-Solutions of the Iterated Helmholtz Operator 235 20.1 Introduction 235 20.2 Fundamental Solutions 237 20.3 Integral Representations 241 References 243 21 A Wiener-Hopf System of Equations in the Steady-State Propagation of a Rectilinear Crack in an Infinite Elastic Plate 244 21.1 Introduction 244 21.2 Boundary Conditions 247 21.3 Wiener-Hopf Factorization 250 References 254 22 Mono-Energetic Neutron Space-Kinetics in Full Cylinder Symmetry: Simulating Power Decrease 255 22.1 Introduction 255 22.2 The Model 256 22.3 Variable Separation 257 22.4 A Closed Form Solution in Cylinder Geometry 257 22.5 Numerical Results 260 22.6 Conclusion 263 References 263 23 Asymptotic Solutions of Maxwell's Equations in a Layered Periodic Structure 265 23.1 Introduction 265 23.2 Maxwell's Equations and the Floquet–Bloch Solutions 266 23.3 Two-Scale Solutions of the Maxwell Equations 267 23.4 Boundary Value Problem in a Half-Space 268 23.5 Conclusions 270 References 270 24 Some Properties of the Fractional Circle Zernike Polynomials 271 24.1 Introduction 271 24.2 Preliminaries 272 24.3 Fractional Circle Zernike Polynomials 274 24.4 Graphical Representation of Fractional Circle Zernike Polynomials 278 References 281 25 Double Laplace Transform and Explicit Fractional Analogue of 2D Laplacian 283 25.1 Introduction 283 25.2 Double Laplace Transforms. Definition and Main Properties 285 25.3 Application of the Double Laplace Transform to Explicit Fractional Laplacian 289 25.3.1 Power Type Fractional Laplacian α,β 289 25.3.2 Skewed Fractional Laplacian α,β; Q-+in the Quarter-Plane 290 25.4 Discussion and Outlook 297 References 298 26 Stability of the Laplace Single Layer Boundary Integral Operator in Sobolev Spaces 299 26.1 Introduction 299 26.2 Strong Domain Variational Formulation 301 26.3 Ultra-Weak Domain Variational Formulation 304 26.4 Single Layer Potential 305 References 309 27 Spectral Lanczos' Tau Method for Systems of Nonlinear Integro-Differential Equations 310 27.1 Introduction 310 27.2 Preliminaries 311 27.3 The Tau Method for Integro-Differential Problems 313 27.4 Nonlinear Approach for Integro-Differential Problems 314 27.5 Contributions to Stability 315 27.6 Numerical Results 316 27.7 Conclusions 319 References 319 28 Discreteness, Periodicity, Holomorphy, and Factorization 320 28.1 Introduction 320 28.2 Discreteness 321 28.3 Periodicity 322 28.4 Holomorphy 323 28.5 Factorization 323 28.5.1 Conical Case 325 Conclusion 328 References 329 29 Modes Coupling Seismic Waves and VibratingBuildings: Existence 330 29.1 Introduction 330 29.2 Low Frequency Analysis 332 29.3 High Frequency Asymptotic Analysis 335 References 338 Index 340 Front Matter ....Pages i-xxiii An L1-Product-Integration Method in Astrophysics (M. Ahues Blanchait, H. Kaboul)....Pages 1-7 Differential Operators and Approximation Processes Generated by Markov Operators (F. Altomare, M. Cappelletti Montano, V. Leonessa, I. Raşa)....Pages 9-19 Analysis of Boundary-Domain Integral Equations for Variable-Coefficient Neumann BVP in 2D (T. G. Ayele, T. T. Dufera, S. E. Mikhailov)....Pages 21-33 A Measure of the Torsional Performances of Partially Hinged Rectangular Plates (E. Berchio, D. Buoso, F. Gazzola)....Pages 35-46 On a Class of Integral Equations Involving Kernels of Cosine and Sine Type (L. P. Castro, R. C. Guerra, N. M. Tuan)....Pages 47-57 The Simple-Layer Potential Approach to the Dirichlet Problem: An Extension to Higher Dimensions of Muskhelishvili Method and Applications (A. Cialdea)....Pages 59-69 Bending of Elastic Plates: Generalized Fourier Series Method (C. Constanda, D. Doty)....Pages 71-81 Existence and Uniqueness Results for a Class of Singular Elliptic Problems in Two-Component Domains (P. Donato, F. Raimondi)....Pages 83-93 Fredholmness of Nonlocal Singular Integral Operators with Slowly Oscillating Data (G. Fernández-Torres, Yu. I. Karlovich)....Pages 95-105 Multidimensional Time Fractional Diffusion Equation (M. Ferreira, N. Vieira)....Pages 107-117 On Homogenization of Nonlinear Robin Type Boundary Conditions for the n-Laplacian in n-Dimensional Perforated Domains (D. Gómez, E. Pérez, A. V. Podol’skii, T. A. Shaposhnikova)....Pages 119-138 Interior Transmission Eigenvalues for Anisotropic Media (A. Kleefeld, D. Colton)....Pages 139-147 Improvement of the Inside-Outside Duality Method (A. Kleefeld, E. Reichwein)....Pages 149-159 A Note on Optimal Design for Thin Structures in the Orlicz–Sobolev Setting (P. A. Kozarzewski, E. Zappale)....Pages 161-171 On the Radiative Conductive Transfer Equation: A Heuristic Convergence Criterion by Stability Analysis (C. A. Ladeia, J. C. L. Fernandes, B. E. J. Bodmann, M. T. Vilhena)....Pages 173-182 An Indirect Boundary Integral Equation Method for Boundary Value Problems in Elastostatics (A. Malaspina)....Pages 183-191 An Instability Result for Suspension Bridges (C. Marchionna, S. Panizzi)....Pages 193-203 A New Diffeomorph Conformal Methodology to Solve Flow Problems with Complex Boundaries by an Equivalent Plane Parallel Problem (A. Meneghetti, B. E. J. Bodmann, M. T. Vilhena)....Pages 205-214 A New Family of Boundary-Domain Integral Equations for the Mixed Exterior Stationary Heat Transfer Problem with Variable Coefficient (S. E. Mikhailov, C. F. Portillo)....Pages 215-226 Radiation Conditions and Integral Representations for Clifford Algebra-Valued Null-Solutions of the Iterated Helmholtz Operator (D. Mitrea, N. Okamoto)....Pages 227-235 A Wiener-Hopf System of Equations in the Steady-State Propagation of a Rectilinear Crack in an Infinite Elastic Plate (A. Nobili, E. Radi, L. Lanzoni)....Pages 237-247 Mono-Energetic Neutron Space-Kinetics in Full Cylinder Symmetry: Simulating Power Decrease (F. R. Oliveira, B. E. J. Bodmann, M. T. Vilhena, F. Carvalho da Silva)....Pages 249-258 Asymptotic Solutions of Maxwell’s Equations in a Layered Periodic Structure (M. V. Perel, M. S. Sidorenko)....Pages 259-264 Some Properties of the Fractional Circle Zernike Polynomials (M. M. Rodrigues, N. Vieira)....Pages 265-276 Double Laplace Transform and Explicit Fractional Analogue of 2D Laplacian (S. Rogosin, M. Dubatovskaya)....Pages 277-292 Stability of the Laplace Single Layer Boundary Integral Operator in Sobolev Spaces (O. Steinbach)....Pages 293-303 Spectral Lanczos’ Tau Method for Systems of Nonlinear Integro-Differential Equations (P. B. Vasconcelos, J. Matos, M. S. Trindade)....Pages 305-314 Discreteness, Periodicity, Holomorphy, and Factorization (V. B. Vasilyev)....Pages 315-324 Modes Coupling Seismic Waves and Vibrating Buildings: Existence (D. Volkov, S. Zheltukhin)....Pages 325-334 Back Matter ....Pages 335-337